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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
SyntaxcgcdOLD 26201 Extend class notation to include the gdc function. (New usage is discouraged.)
 class  gcd OLD ( A ,  B )
 
Definitiondf-gcdOLD 26202*  gcd OLD ( A ,  B ) is the largest natural number that evenly divides both  A and  B. (Contributed by Jeff Hoffman, 17-Jun-2008.) (New usage is discouraged.)
 |-  gcd OLD ( A ,  B )  =  sup ( { x  e.  NN  |  ( ( A  /  x )  e.  NN  /\  ( B  /  x )  e. 
 NN ) } ,  NN ,  <  )
 
Theoremee7.2aOLD 26203 Lemma for Euclid's Elements, Book 7, proposition 2. The original mentions the smaller measure being 'continually subtracted' from the larger. Many authors interpret this phrase as  A mod  B. Here, just one subtraction step is proved to preserve the  gcd OLD. The  rec function will be used in other proofs for iterated subtraction. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( A  e.  NN  /\  B  e.  NN )  ->  ( A  <  B  ->  gcd OLD ( A ,  B )  = 
 gcd OLD ( A ,  ( B  -  A ) ) ) )
 
19.11  Mathbox for Wolf Lammen

Most of the theorems in the section "Logical implication" are about handling chains of implications:  ph  ->  ( ps  ->  ( ch  ->  .... With respect to chains, a rich set of rules clarify

- how to swap antecedents (com12, ...);

- how to drop antecedents (ax-mp, pm2.43, ...);

- how to add antecedents (a1i, ...)

- how to replace an antecedent (syl, ...);

- how to replace a consequent (ax-mp, syl, ...);

- what is, when an antecedent equals the consequent (ax-1, id, ...).

In all these cases, the operands of the chain have no inner structure, or it is of no importance. These chains are called "simple" here.

There is less support, when the operands are structured themselves. Some kinds of inner structure involving the  -. operator are best handled by the symmetric operators  /\ and  \/. But a nested, simple chain has no such convenient replacement. I can focus on antecedents here, since a consequent representing a chain is, in conjunction with its antecedents, just an extended simple chain again.

The following theorems show, how operations on nested chains appear somehow mirrored: The minor premises of the syllogisms look reverted, in comparison to their normal counterparts, and while adding an antecedent to a chain via a1i 11 is easy, in nested chains they can be easily dropped.

 
Theoremwl-jarri 26204 Dropping a nested antecedent. This theorem is one of two reversions of ja 155. Since ja 155 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

ax46 2238 is an instance of this idea.

 |-  (
 ( ph  ->  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theoremwl-jarli 26205 Dropping a nested consequent. This theorem is one of two reversions of ja 155. Since ja 155 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

ax46 2238 is an instance of this idea.

 |-  (
 ( ph  ->  ps )  ->  ch )   =>    |-  ( -.  ph  ->  ch )
 
Theoremwl-mps 26206 Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ( ph  ->  ch )  ->  th )   =>    |-  (
 ( ph  ->  ps )  ->  th )
 
Theoremwl-syls1 26207 Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ps  ->  ch )   &    |-  ( ( ph  ->  ch )  ->  th )   =>    |-  (
 ( ph  ->  ps )  ->  th )
 
Theoremwl-syls2 26208 Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ( ph  ->  ch )  ->  th )   =>    |-  (
 ( ps  ->  ch )  ->  th )
 
Theoremwl-adnestant 26209 A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid (see wl-adnestantALT 26210) (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   =>    |-  ( ( ph  ->  ps )  ->  ch )
 
Theoremwl-adnestantALT 26210 Proof of wl-adnestant 26209 not based on ax-3 7. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   =>    |-  ( ( ph  ->  ps )  ->  ch )
 
Theoremwl-adnestantd 26211 Deduction version of wl-adnestant 26209. Generalization of a2i 13, imim12i 55, imim1i 56 and imim2i 14, which can be proved by specializing its hypotheses, and some trivial rearrangements. This theorem clarifies in a more general way, under what conditions a wff may be introduced as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid (see wl-adnestantALT 26210). (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  th ) )
 
Theoremwl-bitr1 26212 Closed form of bitri 241. Place before bitri 241. [ +33] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ps  <->  ch )  ->  ( ph 
 <->  ch ) ) )
 
Theoremwl-bitri 26213 An inference from transitive law for logical equivalence. [ -5] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  ps )   &    |-  ( ps  <->  ch )   =>    |-  ( ph  <->  ch )
 
Theoremwl-bitrd 26214 Deduction form of bitri 241. [ -7] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   &    |-  ( ph  ->  ( ch  <->  th ) )   =>    |-  ( ph  ->  ( ps  <->  th ) )
 
Theoremwl-bibi1 26215 Theorem *4.86 of [WhiteheadRussell] p. 122. Place this (and the following theorems) after bitr1. [ +22] (Contributed by NM, 3-Jan-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ph  <->  ch )  <->  ( ps  <->  ch ) ) )
 
Theoremwl-bibi1i 26216 Inference adding a biconditional to the right in an equivalence. Move after bibi1. [ -8] (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph 
 <->  ps )   =>    |-  ( ( ph  <->  ch )  <->  ( ps  <->  ch ) )
 
Theoremwl-bibi1d 26217 Deduction adding a biconditional to the right in an equivalence. Move after bibi1i. [ -9] (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( ( ps  <->  th )  <->  ( ch  <->  th ) ) )
 
Theoremwl-bibi2d 26218 Deduction adding a biconditional to the left in an equivalence. Move after bibi1d. [ -25] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( ( th  <->  ps )  <->  ( th  <->  ch ) ) )
 
Theoremwl-pm5.74lem 26219 Moving a common antecedent on one side of an equivalence. Place before pm5.74 236. [ +25] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ch ) )   &    |-  ( -.  ph  ->  ch )   =>    |-  (
 ( ph  ->  ps )  <->  ch )
 
Theoremwl-pm5.74 26220 Distribution of implication over biconditional. Theorem *5.74 of [ WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.) Replace and move biimt 326.. albi 1573 before it. [ -22] (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps  <->  ch ) )  <->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
 
Theoremwl-pm5.32 26221 Distribution of implication over biconditional. Theorem *5.32 of [ WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Oct-2013.) Replace. [ -43] (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps  <->  ch ) )  <->  ( ( ph  /\ 
 ps )  <->  ( ph  /\  ch ) ) )
 
Theoremwl-bitr 26222 Theorem *4.22 of [WhiteheadRussell] p. 117. Replace. [ -4] (Contributed by NM, 3-Jan-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ( ph  <->  ps )  /\  ( ps 
 <->  ch ) )  ->  ( ph  <->  ch ) )
 
Theoremwl-pm2.86i 26223 Inference based on pm2.86 96. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( ph  ->  ch )
 )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theoremwl-dedlem0a 26224 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  <->  ( ( ch 
 ->  ph )  ->  ( ps  /\  ph ) ) ) )
 
Theoremwl-nfnbi 26225 Being free does not depend on an outside negation in an expression. This theorem is slightly more general than nfn 1811 or nfnd 1809. (Contributed by Wolf Lammen, 5-May-2018.)
 |-  ( F/ x ph  <->  F/ x  -.  ph )
 
Theoremwl-exeq 26226 The semantics of  E. x y  =  z. (Contributed by Wolf Lammen, 27-Apr-2018.)
 |-  ( E. x  y  =  z 
 <->  ( y  =  z  \/  A. x  x  =  y  \/  A. x  x  =  z
 ) )
 
Theoremwl-aleq 26227 The semantics of  A. x y  =  z. (Contributed by Wolf Lammen, 27-Apr-2018.)
 |-  ( A. x  y  =  z 
 <->  ( y  =  z 
 /\  ( A. x  x  =  y  <->  A. x  x  =  z ) ) )
 
19.12  Mathbox for Brendan Leahy
 
Theoremsupaddc 26228* The supremum function distributes over addition in a sense similar to that in supmul1 9965. (Contributed by Brendan Leahy, 25-Sep-2017.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   &    |-  ( ph  ->  B  e.  RR )   &    |-  C  =  { z  |  E. v  e.  A  z  =  ( v  +  B ) }   =>    |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  B )  =  sup ( C ,  RR ,  <  ) )
 
Theoremsupadd 26229* The supremum function distributes over addition in a sense similar to that in supmul 9968. (Contributed by Brendan Leahy, 26-Sep-2017.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )   &    |-  ( ph  ->  B 
 C_  RR )   &    |-  ( ph  ->  B  =/=  (/) )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  B  y  <_  x )   &    |-  C  =  {
 z  |  E. v  e.  A  E. b  e.  B  z  =  ( v  +  b ) }   =>    |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  +  sup ( B ,  RR ,  <  ) )  =  sup ( C ,  RR ,  <  ) )
 
Theoremrabiun2 26230* Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017.)
 |-  { x  e.  U_ y  e.  A  B  |  ph }  =  U_ y  e.  A  { x  e.  B  |  ph
 }
 
Theoremltflcei 26231 Theorem to move the floor function across a strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( |_ `  A )  <  B  <->  A  <  -u ( |_ `  -u B ) ) )
 
Theoremleceifl 26232 Theorem to move the floor function across a non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( -u ( |_ `  -u A )  <_  B  <->  A  <_  ( |_ `  B ) ) )
 
Theoremlxflflp1 26233 Theorem to move floor function between strict and non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( ( |_ `  A )  <_  B  <->  A  <  ( ( |_ `  B )  +  1 ) ) )
 
Theoremmblfinlem 26234* Lemma for ismblfin 26237, effectively one direction of the same fact for open sets, made necessary by Viaclovsky's slightly different defintion of outer measure. Note that unlike the main theorem, this holds for sets of infinite measure. (Contributed by Brendan Leahy, 21-Feb-2018.)
 |-  (
 ( A  e.  ( topGen `
  ran  (,) )  /\  M  e.  RR  /\  M  <  ( vol * `  A ) )  ->  E. s  e.  ( Clsd `  ( topGen `  ran  (,) ) ) ( s 
 C_  A  /\  M  <  ( vol * `  s ) ) )
 
Theoremmblfinlem2 26235* The difference between two sets measurable by the criterion in ismblfin 26237 is itself measurable by the same. Proposition 0.3 of [Viaclovsky7] p. 3. (Contributed by Brendan Leahy, 25-Mar-2018.)
 |-  (
 ( ( A  C_  RR  /\  ( vol * `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol * `  B )  e.  RR )  /\  ( ( vol
 * `  A )  =  sup ( { y  |  E. b  e.  ( Clsd `  ( topGen `  ran  (,) ) ) ( b 
 C_  A  /\  y  =  ( vol `  b
 ) ) } ,  RR ,  <  )  /\  ( vol * `  B )  =  sup ( {
 y  |  E. b  e.  ( Clsd `  ( topGen `  ran  (,) ) ) ( b 
 C_  B  /\  y  =  ( vol `  b
 ) ) } ,  RR ,  <  ) ) )  ->  sup ( {
 y  |  E. b  e.  ( Clsd `  ( topGen `  ran  (,) ) ) ( b 
 C_  ( A  \  B )  /\  y  =  ( vol `  b
 ) ) } ,  RR ,  <  )  =  ( vol * `  ( A  \  B ) ) )
 
Theoremmblfinlem3 26236* Backward direction of ismblfin 26237. (Contributed by Brendan Leahy, 28-Mar-2018.)
 |-  (
 ( ( A  C_  RR  /\  ( vol * `  A )  e.  RR )  /\  A  e.  dom  vol )  ->  ( vol * `
  A )  = 
 sup ( { y  |  E. b  e.  ( Clsd `  ( topGen `  ran  (,) ) ) ( b 
 C_  A  /\  y  =  ( vol `  b
 ) ) } ,  RR ,  <  ) )
 
Theoremismblfin 26237* Measurability in terms of inner and outer measure. Proposition 7 of [Viaclovsky8] p. 3. (Contributed by Brendan Leahy, 4-Mar-2018.) (Revised by Brendan Leahy, 28-Mar-2018.)
 |-  (
 ( A  C_  RR  /\  ( vol * `  A )  e.  RR )  ->  ( A  e.  dom 
 vol 
 <->  ( vol * `  A )  =  sup ( { y  |  E. b  e.  ( Clsd `  ( topGen `  ran  (,) )
 ) ( b  C_  A  /\  y  =  ( vol `  b )
 ) } ,  RR ,  <  ) ) )
 
Theoremovoliunnfl 26238* ovoliun 19393 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 21-Nov-2017.)
 |-  (
 ( f  Fn  NN  /\ 
 A. n  e.  NN  ( ( f `  n )  C_  RR  /\  ( vol * `  (
 f `  n )
 )  e.  RR )
 )  ->  ( vol * `
  U_ m  e.  NN  ( f `  m ) )  <_  sup ( ran  seq  1 (  +  ,  ( m  e.  NN  |->  ( vol * `  (
 f `  m )
 ) ) ) , 
 RR* ,  <  ) )   =>    |-  ( ( A  ~<_  NN  /\  A. x  e.  A  x  ~<_  NN )  ->  U. A  =/=  RR )
 
Theoremex-ovoliunnfl 26239* Demonstration of ovoliunnfl 26238. (Contributed by Brendan Leahy, 21-Nov-2017.)
 |-  (
 ( A  ~<_  NN  /\  A. x  e.  A  x  ~<_  NN )  ->  U. A  =/=  RR )
 
Theoremvoliunnfl 26240* voliun 19440 is incompatible with the Feferman-Levy model; in that model, therefore, the Lebesgue measure as we've defined it isn't actually a measure. (Contributed by Brendan Leahy, 16-Dec-2017.)
 |-  S  =  seq  1 (  +  ,  G )   &    |-  G  =  ( n  e.  NN  |->  ( vol `  ( f `  n ) ) )   &    |-  ( ( A. n  e.  NN  ( ( f `
  n )  e. 
 dom  vol  /\  ( vol `  ( f `  n ) )  e.  RR )  /\ Disj  n  e.  NN (
 f `  n )
 )  ->  ( vol ` 
 U_ n  e.  NN  ( f `  n ) )  =  sup ( ran  S ,  RR* ,  <  ) )   =>    |-  ( ( A  ~<_  NN  /\  A. x  e.  A  x  ~<_  NN )  ->  U. A  =/=  RR )
 
Theoremvolsupnfl 26241* volsup 19442 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 2-Jan-2018.)
 |-  (
 ( f : NN --> dom  vol  /\  A. n  e. 
 NN  ( f `  n )  C_  ( f `
  ( n  +  1 ) ) ) 
 ->  ( vol `  U. ran  f )  =  sup ( ( vol " ran  f ) ,  RR* ,  <  ) )   =>    |-  ( ( A  ~<_  NN  /\  A. x  e.  A  x  ~<_  NN )  ->  U. A  =/=  RR )
 
Theorem0mbf 26242 The empty function is measurable. (Contributed by Brendan Leahy, 28-Mar-2018.)
 |-  (/)  e. MblFn
 
Theoremmbfresfi 26243* Measurability of a piecewise function across arbitrarily many subsets. (Contributed by Brendan Leahy, 31-Mar-2018.)
 |-  ( ph  ->  F : A --> CC )   &    |-  ( ph  ->  S  e.  Fin )   &    |-  ( ph  ->  A. s  e.  S  ( F  |`  s )  e. MblFn )   &    |-  ( ph  ->  U. S  =  A )   =>    |-  ( ph  ->  F  e. MblFn )
 
Theoremmbfposadd 26244* If the sum of two measurable functions is measurable, the sum of their nonnegative parts is measurable. (Contributed by Brendan Leahy, 2-Apr-2018.)
 |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn
 )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e. MblFn )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( if ( 0  <_  B ,  B ,  0 )  +  if ( 0 
 <_  C ,  C , 
 0 ) ) )  e. MblFn )
 
Theoremcnambfre 26245 A real-valued, a.e. continuous function is measurable. (Contributed by Brendan Leahy, 4-Apr-2018.)
 |-  (
 ( F : A --> RR  /\  A  e.  dom  vol  /\  ( vol * `  ( A  \  ( ( `' ( ( ( topGen `  ran  (,) )t  A )  CnP  ( topGen `
  ran  (,) ) )  o.  _E  ) " { F } ) ) )  =  0 ) 
 ->  F  e. MblFn )
 
Theoremitg2addnclem 26246* An alternate expression for the 
S.2 integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017.) (Revised by Brendan Leahy, 10-Mar-2018.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( E. y  e.  RR+  ( z  e.  RR  |->  if (
 ( g `  z
 )  =  0 ,  0 ,  ( ( g `  z )  +  y ) ) )  o R  <_  F 
 /\  x  =  (
 S.1 `  g )
 ) }   =>    |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  =  sup ( L ,  RR* ,  <  )
 )
 
Theoremitg2addnclem2 26247* Lemma for itg2addnc 26249. The function described is a simple function. (Contributed by Brendan Leahy, 29-Oct-2017.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   =>    |-  ( ( ( ph  /\  h  e.  dom  S.1 )  /\  v  e.  RR+ )  ->  ( x  e. 
 RR  |->  if ( ( ( ( ( |_ `  (
 ( F `  x )  /  ( v  / 
 3 ) ) )  -  1 )  x.  ( v  /  3
 ) )  <_  ( h `  x )  /\  ( h `  x )  =/=  0 ) ,  ( ( ( |_ `  ( ( F `  x )  /  (
 v  /  3 )
 ) )  -  1
 )  x.  ( v 
 /  3 ) ) ,  ( h `  x ) ) )  e.  dom  S.1 )
 
Theoremitg2addnclem3 26248* Lemma incomprehensible in isolation split off to shorten proof of itg2addnc 26249. (Contributed by Brendan Leahy, 11-Mar-2018.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   &    |-  ( ph  ->  G : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  ( S.2 `  G )  e.  RR )   =>    |-  ( ph  ->  ( E. h  e.  dom  S.1 ( E. y  e.  RR+  ( z  e. 
 RR  |->  if ( ( h `
  z )  =  0 ,  0 ,  ( ( h `  z )  +  y
 ) ) )  o R  <_  ( F  o F  +  G )  /\  s  =  (
 S.1 `  h )
 )  ->  E. t E. u ( E. f  e.  dom  S.1 E. g  e. 
 dom  S.1 ( ( E. c  e.  RR+  ( z  e.  RR  |->  if (
 ( f `  z
 )  =  0 ,  0 ,  ( ( f `  z )  +  c ) ) )  o R  <_  F 
 /\  t  =  (
 S.1 `  f )
 )  /\  ( E. d  e.  RR+  ( z  e.  RR  |->  if (
 ( g `  z
 )  =  0 ,  0 ,  ( ( g `  z )  +  d ) ) )  o R  <_  G 
 /\  u  =  (
 S.1 `  g )
 ) )  /\  s  =  ( t  +  u ) ) ) )
 
Theoremitg2addnc 26249 Alternate proof of itg2add 19643 using the "buffer zone" definition from the first lemma, in which every simple function in the set is divided into to by dividing its buffer by a third and finding the largest allowable function locked to a grid laid out in increments of the new, smaller buffer up to the original simple function. The measurability of this function follows from that of the augend, and subtracting it from the original simple function yields another simple function by i1fsub 19592, which is allowable by the fact that the grid must have a mark between one third and two thirds the original buffer. This has two advantages over the current approach: first, eliminating ax-cc 8307, and second, weakening the measurability hypothesis to only the augend. (Contributed by Brendan Leahy, 31-Oct-2017.) (Revised by Brendan Leahy, 13-Mar-2018.)
 |-  ( ph  ->  F  e. MblFn )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  (
 S.2 `  F )  e.  RR )   &    |-  ( ph  ->  G : RR --> ( 0 [,)  +oo ) )   &    |-  ( ph  ->  ( S.2 `  G )  e.  RR )   =>    |-  ( ph  ->  ( S.2 `  ( F  o F  +  G ) )  =  (
 ( S.2 `  F )  +  ( S.2 `  G ) ) )
 
Theoremitg2gt0cn 26250* itg2gt0 19644 holds on functions continuous on an open interval in the absence of ax-cc 8307. The fourth hypothesis is made unnecessary by the continuity hypothesis. (Contributed by Brendan Leahy, 16-Nov-2017.)
 |-  ( ph  ->  X  <  Y )   &    |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )   &    |-  ( ( ph  /\  x  e.  ( X (,) Y ) ) 
 ->  0  <  ( F `
  x ) )   &    |-  ( ph  ->  ( F  |`  ( X (,) Y ) )  e.  (
 ( X (,) Y ) -cn-> CC ) )   =>    |-  ( ph  ->  0  <  ( S.2 `  F ) )
 
Theoremibladdnclem 26251* Lemma for ibladdnc 26252; cf ibladdlem 19703, whose fifth hypothesis is rendered unnecessary by the weakened hypotheses of itg2addnc 26249. (Contributed by Brendan Leahy, 31-Oct-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  D  =  ( B  +  C ) )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e. MblFn )   &    |-  ( ph  ->  (
 S.2 `  ( x  e.  RR  |->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 ) ) )  e.  RR )   &    |-  ( ph  ->  ( S.2 `  ( x  e.  RR  |->  if (
 ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) ) )  e. 
 RR )   =>    |-  ( ph  ->  ( S.2 `  ( x  e. 
 RR  |->  if ( ( x  e.  A  /\  0  <_  D ) ,  D ,  0 ) ) )  e.  RR )
 
Theoremibladdnc 26252* Choice-free analogue of itgadd 19708. A measurability hypothesis is necessitated by the loss of mbfadd 19545; for large classes of functions, such as continuous functions, it should be relatively easy to show. (Contributed by Brendan Leahy, 1-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e.  L ^1 )
 
Theoremitgaddnclem1 26253* Lemma for itgaddnc 26255; cf. itgaddlem1 19706. (Contributed by Brendan Leahy, 7-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  0  <_  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  C )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgaddnclem2 26254* Lemma for itgaddnc 26255; cf. itgaddlem2 19707. (Contributed by Brendan Leahy, 10-Nov-2017.) (Revised by Brendan Leahy, 3-Apr-2018.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  RR )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgaddnc 26255* Choice-free analogue of itgadd 19708. (Contributed by Brendan Leahy, 11-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  +  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremiblsubnc 26256* Choice-free analogue of iblsub 19705. (Contributed by Brendan Leahy, 11-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e.  L ^1 )
 
Theoremitgsubnc 26257* Choice-free analogue of itgsub 19709. (Contributed by Brendan Leahy, 11-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  C )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( B  -  C ) )  e. MblFn
 )   =>    |-  ( ph  ->  S. A ( B  -  C )  _d x  =  ( S. A B  _d x  -  S. A C  _d x ) )
 
Theoremiblabsnclem 26258* Lemma for iblabsnc 26259; cf. iblabslem 19711. (Contributed by Brendan Leahy, 7-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( abs `  ( F `  B ) ) ,  0 ) )   &    |-  ( ph  ->  ( x  e.  A  |->  ( F `  B ) )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  A )  ->  ( F `  B )  e. 
 RR )   =>    |-  ( ph  ->  ( G  e. MblFn  /\  ( S.2 `  G )  e.  RR ) )
 
Theoremiblabsnc 26259* Choice-free analogue of iblabs 19712. As with ibladdnc 26252, a measurability hypothesis is needed. (Contributed by Brendan Leahy, 7-Nov-2017.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B )
 )  e. MblFn )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B ) )  e.  L ^1 )
 
Theoremiblmulc2nc 26260* Choice-free analogue of iblmulc2 19714. (Contributed by Brendan Leahy, 17-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   =>    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e.  L ^1 )
 
Theoremitgmulc2nclem1 26261* Lemma for itgmulc2nc 26263; cf. itgmulc2lem1 19715. (Contributed by Brendan Leahy, 17-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   &    |-  ( ph  ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  0 
 <_  C )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  0  <_  B )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgmulc2nclem2 26262* Lemma for itgmulc2nc 26263; cf. itgmulc2lem2 19716. (Contributed by Brendan Leahy, 19-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   &    |-  ( ph  ->  C  e.  RR )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgmulc2nc 26263* Choice-free analogue of itgmulc2 19717. (Contributed by Brendan Leahy, 19-Nov-2017.)
 |-  ( ph  ->  C  e.  CC )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( C  x.  B ) )  e. MblFn )   =>    |-  ( ph  ->  ( C  x.  S. A B  _d x )  =  S. A ( C  x.  B )  _d x )
 
Theoremitgabsnc 26264* Choice-free analogue of itgabs 19718. (Contributed by Brendan Leahy, 19-Nov-2017.) (Revised by Brendan Leahy, 19-Jun-2018.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  L ^1 )   &    |-  ( ph  ->  ( x  e.  A  |->  ( abs `  B )
 )  e. MblFn )   &    |-  ( ph  ->  ( y  e.  A  |->  ( ( * `  S. A B  _d x )  x.  [_ y  /  x ]_ B ) )  e. MblFn
 )   =>    |-  ( ph  ->  ( abs `  S. A B  _d x )  <_  S. A ( abs `  B )  _d x )
 
Theorembddiblnc 26265* Choice-free proof of bddibl 19723. (Contributed by Brendan Leahy, 2-Nov-2017.) (Revised by Brendan Leahy, 6-Nov-2017.)
 |-  (
 ( F  e. MblFn  /\  ( vol `  dom  F )  e.  RR  /\  E. x  e.  RR  A. y  e.  dom  F ( abs `  ( F `  y
 ) )  <_  x )  ->  F  e.  L ^1 )
 
Theoremcnicciblnc 26266 Choice-free proof of cniccibl 19724. (Contributed by Brendan Leahy, 2-Nov-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  F  e.  ( ( A [,] B ) -cn-> CC ) )  ->  F  e.  L ^1 )
 
Theoremitggt0cn 26267* itggt0 19725 holds for continuous functions in the absence of ax-cc 8307. (Contributed by Brendan Leahy, 16-Nov-2017.)
 |-  ( ph  ->  X  <  Y )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  L ^1 )   &    |-  (
 ( ph  /\  x  e.  ( X (,) Y ) )  ->  B  e.  RR+ )   &    |-  ( ph  ->  ( x  e.  ( X (,) Y )  |->  B )  e.  ( ( X (,) Y )
 -cn-> CC ) )   =>    |-  ( ph  ->  0  <  S. ( X (,) Y ) B  _d x )
 
Theoremftc1cnnclem 26268* Lemma for ftc1cnnc 26269; cf. ftc1lem4 19915. The stronger assumptions of ftc1cn 19919 are exploited to make use of weaker theorems. (Contributed by Brendan Leahy, 19-Nov-2017.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `  t )  _d t
 )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  c  e.  ( A (,) B ) )   &    |-  H  =  ( z  e.  ( ( A [,] B ) 
 \  { c }
 )  |->  ( ( ( G `  z )  -  ( G `  c ) )  /  ( z  -  c
 ) ) )   &    |-  ( ph  ->  E  e.  RR+ )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\  y  e.  ( A (,) B ) ) 
 ->  ( ( abs `  (
 y  -  c ) )  <  R  ->  ( abs `  ( ( F `  y )  -  ( F `  c ) ) )  <  E ) )   &    |-  ( ph  ->  X  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( X  -  c ) )  <  R )   &    |-  ( ph  ->  Y  e.  ( A [,] B ) )   &    |-  ( ph  ->  ( abs `  ( Y  -  c ) )  <  R )   =>    |-  ( ( ph  /\  X  <  Y )  ->  ( abs `  ( ( ( ( G `  Y )  -  ( G `  X ) )  /  ( Y  -  X ) )  -  ( F `  c ) ) )  <  E )
 
Theoremftc1cnnc 26269* Choice-free proof of ftc1cn 19919. (Contributed by Brendan Leahy, 20-Nov-2017.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `  t )  _d t
 )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  F  e.  L ^1 )   =>    |-  ( ph  ->  ( RR  _D  G )  =  F )
 
Theoremftc1anclem1 26270 Lemma for ftc1anc 26278- the absolute value of a real-valued measurable function is measurable. Would be trivial with cncombf 19542, but this proof avoids ax-cc 8307. (Contributed by Brendan Leahy, 18-Jun-2018.)
 |-  (
 ( F : A --> RR  /\  F  e. MblFn )  ->  ( abs  o.  F )  e. MblFn )
 
Theoremftc1anclem2 26271* Lemma for ftc1anc 26278- restriction of an integrable function to the absolute value of its real or imaginary part. (Contributed by Brendan Leahy, 19-Jun-2018.)
 |-  (
 ( F : A --> CC  /\  F  e.  L ^1  /\  Z  e.  {
 0 ,  1 } )  ->  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  A ,  ( abs `  ( Re `  ( ( F `
  t )  /  ( _i ^ Z ) ) ) ) ,  0 ) ) )  e.  RR )
 
Theoremftc1anclem3 26272 Lemma for ftc1anc 26278- the absolute value of the sum of a simple function and  _i times another simple function is itself a simple function. (Contributed by Brendan Leahy, 27-May-2018.)
 |-  (
 ( F  e.  dom  S.1  /\  G  e.  dom  S.1 )  ->  ( abs  o.  ( F  o F  +  ( ( RR  X.  { _i } )  o F  x.  G ) ) )  e.  dom  S.1 )
 
Theoremftc1anclem4 26273* Lemma for ftc1anc 26278. (Contributed by Brendan Leahy, 17-Jun-2018.)
 |-  (
 ( F  e.  dom  S.1  /\  G  e.  L ^1 
 /\  G : RR --> RR )  ->  ( S.2 `  ( t  e.  RR  |->  ( abs `  ( ( G `  t )  -  ( F `  t ) ) ) ) )  e.  RR )
 
Theoremftc1anclem5 26274* Lemma for ftc1anc 26278, the existence of a simple function the integral of whose pointwise difference from the function is less than a given positive real. (Contributed by Brendan Leahy, 17-Jun-2018.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `  t )  _d t
 )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B )  C_  D )   &    |-  ( ph  ->  D 
 C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  F : D --> CC )   =>    |-  ( ( ph  /\  Y  e.  RR+ )  ->  E. f  e.  dom  S.1 ( S.2 `  (
 t  e.  RR  |->  ( abs `  ( ( Re `  if ( t  e.  D ,  ( F `  t ) ,  0 ) )  -  ( f `  t
 ) ) ) ) )  <  Y )
 
Theoremftc1anclem6 26275* Lemma for ftc1anc 26278- construction of simple functions within an arbitrary absolute distance of the given function. Similar to Lemma 565Ib of [Fremlin5] p. 218, but without Fremlin's additional step of converting the simple function into a continuous one, which is unnecessary to this lemma's use; also, two simple functions are used to allow for complex-valued  F. (Contributed by Brendan Leahy, 31-May-2018.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `  t )  _d t
 )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B )  C_  D )   &    |-  ( ph  ->  D 
 C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  F : D --> CC )   =>    |-  ( ( ph  /\  Y  e.  RR+ )  ->  E. f  e.  dom  S.1 E. g  e. 
 dom  S.1 ( S.2 `  (
 t  e.  RR  |->  ( abs `  ( if ( t  e.  D ,  ( F `  t
 ) ,  0 )  -  ( ( f `
  t )  +  ( _i  x.  (
 g `  t )
 ) ) ) ) ) )  <  Y )
 
Theoremftc1anclem7 26276* Lemma for ftc1anc 26278. (Contributed by Brendan Leahy, 13-May-2018.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `  t )  _d t
 )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B )  C_  D )   &    |-  ( ph  ->  D 
 C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  F : D --> CC )   =>    |-  ( ( ( ( ( ( ( ph  /\  ( f  e.  dom  S.1  /\  g  e.  dom  S.1 ) )  /\  ( S.2 `  ( t  e. 
 RR  |->  ( abs `  ( if ( t  e.  D ,  ( F `  t
 ) ,  0 )  -  ( ( f `
  t )  +  ( _i  x.  (
 g `  t )
 ) ) ) ) ) )  <  (
 y  /  2 )
 )  /\  E. r  e.  ( ran  f  u. 
 ran  g ) r  =/=  0 )  /\  y  e.  RR+ )  /\  ( u  e.  ( A [,] B )  /\  w  e.  ( A [,] B )  /\  u  <_  w ) )  /\  ( abs `  ( w  -  u ) )  < 
 ( ( y  / 
 2 )  /  (
 2  x.  sup (
 ( abs " ( ran  f  u.  ran  g
 ) ) ,  RR ,  <  ) ) ) )  ->  ( ( S.2 `  ( t  e. 
 RR  |->  if ( t  e.  ( u (,) w ) ,  ( abs `  ( ( f `  t )  +  ( _i  x.  ( g `  t ) ) ) ) ,  0 ) ) )  +  ( S.2 `  ( t  e. 
 RR  |->  if ( t  e.  ( u (,) w ) ,  ( abs `  ( ( F `  t )  -  (
 ( f `  t
 )  +  ( _i 
 x.  ( g `  t ) ) ) ) ) ,  0 ) ) ) )  <  ( ( y 
 /  2 )  +  ( y  /  2
 ) ) )
 
Theoremftc1anclem8 26277* Lemma for ftc1anc 26278. (Contributed by Brendan Leahy, 29-May-2018.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `  t )  _d t
 )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B )  C_  D )   &    |-  ( ph  ->  D 
 C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  F : D --> CC )   =>    |-  ( ( ( ( ( ( ( ph  /\  ( f  e.  dom  S.1  /\  g  e.  dom  S.1 ) )  /\  ( S.2 `  ( t  e. 
 RR  |->  ( abs `  ( if ( t  e.  D ,  ( F `  t
 ) ,  0 )  -  ( ( f `
  t )  +  ( _i  x.  (
 g `  t )
 ) ) ) ) ) )  <  (
 y  /  2 )
 )  /\  E. r  e.  ( ran  f  u. 
 ran  g ) r  =/=  0 )  /\  y  e.  RR+ )  /\  ( u  e.  ( A [,] B )  /\  w  e.  ( A [,] B )  /\  u  <_  w ) )  /\  ( abs `  ( w  -  u ) )  < 
 ( ( y  / 
 2 )  /  (
 2  x.  sup (
 ( abs " ( ran  f  u.  ran  g
 ) ) ,  RR ,  <  ) ) ) )  ->  ( S.2 `  ( t  e.  RR  |->  if ( t  e.  ( u (,) w ) ,  ( ( abs `  (
 ( F `  t
 )  -  ( ( f `  t )  +  ( _i  x.  ( g `  t
 ) ) ) ) )  +  ( abs `  ( ( f `  t )  +  ( _i  x.  ( g `  t ) ) ) ) ) ,  0 ) ) )  < 
 y )
 
Theoremftc1anc 26278* ftc1a 19913 holds for functions that obey the triangle inequality in the absence of ax-cc 8307. Theorem 565Ma of [Fremlin5] p. 220. (Contributed by Brendan Leahy, 11-May-2018.)
 |-  G  =  ( x  e.  ( A [,] B )  |->  S. ( A (,) x ) ( F `  t )  _d t
 )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  ( A (,) B )  C_  D )   &    |-  ( ph  ->  D 
 C_  RR )   &    |-  ( ph  ->  F  e.  L ^1 )   &    |-  ( ph  ->  F : D --> CC )   &    |-  ( ph  ->  A. s  e.  ( (,) " ( ( A [,] B )  X.  ( A [,] B ) ) ) ( abs `  S. s ( F `
  t )  _d t )  <_  ( S.2 `  ( t  e. 
 RR  |->  if ( t  e.  s ,  ( abs `  ( F `  t
 ) ) ,  0 ) ) ) )   =>    |-  ( ph  ->  G  e.  ( ( A [,] B ) -cn-> CC ) )
 
Theoremftc2nc 26279* Choice-free proof of ftc2 19920. (Contributed by Brendan Leahy, 19-Jun-2018.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   &    |-  ( ph  ->  ( RR  _D  F )  e.  ( ( A (,) B ) -cn-> CC ) )   &    |-  ( ph  ->  ( RR  _D  F )  e.  L ^1 )   &    |-  ( ph  ->  F  e.  (
 ( A [,] B ) -cn-> CC ) )   =>    |-  ( ph  ->  S. ( A (,) B ) ( ( RR 
 _D  F ) `  t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
 
Theoremdvreasin 26280 Real derivative of arcsine. (Contributed by Brendan Leahy, 3-Aug-2017.)
 |-  ( RR  _D  (arcsin  |`  ( -u 1 (,) 1 ) ) )  =  ( x  e.  ( -u 1 (,) 1 )  |->  ( 1 
 /  ( sqr `  (
 1  -  ( x ^ 2 ) ) ) ) )
 
Theoremdvreacos 26281 Real derivative of arccosine. (Contributed by Brendan Leahy, 3-Aug-2017.)
 |-  ( RR  _D  (arccos  |`  ( -u 1 (,) 1 ) ) )  =  ( x  e.  ( -u 1 (,) 1 )  |->  ( -u 1  /  ( sqr `  (
 1  -  ( x ^ 2 ) ) ) ) )
 
Theoremareacirclem2 26282* Antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.)
 |-  ( R  e.  RR+  ->  ( RR  _D  ( t  e.  ( -u R (,) R )  |->  ( ( R ^ 2 )  x.  ( (arcsin `  (
 t  /  R )
 )  +  ( ( t  /  R )  x.  ( sqr `  (
 1  -  ( ( t  /  R ) ^ 2 ) ) ) ) ) ) ) )  =  ( t  e.  ( -u R (,) R )  |->  ( 2  x.  ( sqr `  ( ( R ^
 2 )  -  (
 t ^ 2 ) ) ) ) ) )
 
Theoremareacirclem3 26283* Continuity of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.)
 |-  ( R  e.  RR+  ->  (
 t  e.  ( -u R (,) R )  |->  ( 2  x.  ( sqr `  ( ( R ^
 2 )  -  (
 t ^ 2 ) ) ) ) )  e.  ( ( -u R (,) R ) -cn-> CC ) )
 
Theoremareacirclem4 26284* Endpoint-inclusive continuity of Cartesian ordinate of circle. (Contributed by Brendan Leahy, 29-Aug-2017.)
 |-  (
 ( R  e.  RR  /\  0  <_  R )  ->  ( t  e.  ( -u R [,] R ) 
 |->  ( sqr `  (
 ( R ^ 2
 )  -  ( t ^ 2 ) ) ) )  e.  (
 ( -u R [,] R ) -cn-> CC ) )
 
Theoremareacirclem1 26285* Integrability of cross-section of circle. (Contributed by Brendan Leahy, 26-Aug-2017.)
 |-  (
 ( R  e.  RR  /\  0  <_  R )  ->  ( t  e.  ( -u R [,] R ) 
 |->  ( 2  x.  ( sqr `  ( ( R ^ 2 )  -  ( t ^ 2
 ) ) ) ) )  e.  L ^1 )
 
Theoremareacirclem5 26286* Endpoint-inclusive continuity of antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 31-Aug-2017.)
 |-  ( R  e.  RR+  ->  (
 t  e.  ( -u R [,] R )  |->  ( ( R ^ 2
 )  x.  ( (arcsin `  ( t  /  R ) )  +  (
 ( t  /  R )  x.  ( sqr `  (
 1  -  ( ( t  /  R ) ^ 2 ) ) ) ) ) ) )  e.  ( (
 -u R [,] R ) -cn-> CC ) )
 
Theoremareacirclem6 26287* Finding the cross-section of a circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.)
 |-  S  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  (
 ( x ^ 2
 )  +  ( y ^ 2 ) ) 
 <_  ( R ^ 2
 ) ) }   =>    |-  ( ( R  e.  RR  /\  0  <_  R  /\  t  e. 
 RR )  ->  ( S " { t }
 )  =  if (
 ( abs `  t )  <_  R ,  ( -u ( sqr `  ( ( R ^ 2 )  -  ( t ^ 2
 ) ) ) [,] ( sqr `  (
 ( R ^ 2
 )  -  ( t ^ 2 ) ) ) ) ,  (/) ) )
 
Theoremareacirc 26288* The area of a circle of radius  R is  pi  x.  R ^ 2. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.)
 |-  S  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  (
 ( x ^ 2
 )  +  ( y ^ 2 ) ) 
 <_  ( R ^ 2
 ) ) }   =>    |-  ( ( R  e.  RR  /\  0  <_  R )  ->  (area `  S )  =  ( pi  x.  ( R ^ 2 ) ) )
 
19.13  Mathbox for Jeff Hankins
 
19.13.1  Miscellany
 
Theorema1i13 26289 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ps  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theorema1i4 26290 Add an antecedent to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ta )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i14 26291 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
 |-  ( ps  ->  ( ch  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i24 26292 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)
 |-  ( ph  ->  ( ch  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theorema1i34 26293 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)
 |-  ( ph  ->  ( ps  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp5d 26294 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  ->  ( ( th  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5g 26295 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ph  /\  ps )  ->  ( ( ( ch 
 /\  th )  /\  ta )  ->  et ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5j 26296 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ( ps  /\  ch )  /\  th )  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5k 26297 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ps  /\  ( ch 
 /\  th ) )  /\  ta )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp5l 26298 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ( ( ps  /\  ch )  /\  ( th  /\  ta ) )  ->  et )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp56 26299 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  ( th  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
 
Theoremexp58 26300 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  (
 ( ( ph  /\  ps )  /\  ( ( ch 
 /\  th )  /\  ta ) )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) ) ) )
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