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Theorem List for Metamath Proof Explorer - 25401-25500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnalf 25401 Not all sets hold  F. as true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  A. x  F.
 
Theoremextt 25402 There exists a set that holds  T. as true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  E. x  T.
 
Theoremnextnt 25403 There does not exist a set, such that  T. is not true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  E. x  -.  T.
 
Theoremnextf 25404 There does not exist a set, such that  F. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  E. x  F.
 
Theoremunnf 25405 There does not exist exactly one set, such that  F. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  E! x  F.
 
Theoremunnt 25406 There does not exist exactly one set, such that  T. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  E! x  T.
 
Theoremmont 25407 There does not exist at most one set, such that  T. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  -.  E* x  T.
 
Theoremmof 25408 There exist at most one set, such that  F. is true. (Contributed by Anthony Hart, 13-Sep-2011.)
 |-  E* x  F.
 
18.8.3  Misc. Single Axiom Systems
 
Theoremmeran1 25409 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
 |-  ( -.  ( -.  ( -.  ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  \/  ( -.  ( -.  th  \/  ph )  \/  ( ch 
 \/  ( ta  \/  ph ) ) ) )
 
Theoremmeran2 25410 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
 |-  ( -.  ( -.  ( -.  ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  \/  ( -.  ( -.  ta  \/  th )  \/  ( ch 
 \/  ( ph  \/  th ) ) ) )
 
Theoremmeran3 25411 A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
 |-  ( -.  ( -.  ( -.  ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  \/  ( -.  ( -.  ch  \/  ph )  \/  ( ta 
 \/  ( th  \/  ph ) ) ) )
 
Theoremwaj-ax 25412 A single axiom for propositional calculus offered by Wajsberg. (Contributed by Anthony Hart, 13-Aug-2011.)
 |-  (
 ( ph  -/\  ( ps  -/\  ch ) )  -/\  ( ( ( th  -/\ 
 ch )  -/\  (
 ( ph  -/\  th )  -/\  ( ph  -/\  th )
 ) )  -/\  ( ph  -/\  ( ph  -/\  ps )
 ) ) )
 
Theoremlukshef-ax2 25413 A single axiom for propositional calculus offered by Lukasiewicz. (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  (
 ( ph  -/\  ( ps  -/\  ch ) )  -/\  ( ( ph  -/\  ( ch  -/\  ph ) )  -/\  ( ( th  -/\  ps )  -/\  ( ( ph  -/\  th )  -/\  ( ph  -/\  th )
 ) ) ) )
 
Theoremarg-ax 25414 ? (Contributed by Anthony Hart, 14-Aug-2011.)
 |-  (
 ( ph  -/\  ( ps  -/\  ch ) )  -/\  ( ( ph  -/\  ( ps  -/\  ch ) ) 
 -/\  ( ( th  -/\ 
 ch )  -/\  (
 ( ch  -/\  th )  -/\  ( ph  -/\  th )
 ) ) ) )
 
18.8.4  Connective Symmetry
 
Theoremnegsym1 25415 In the paper "On Variable Functors of Propositional Arguments", Lukasiewicz introduced a system that can handle variable connectives. This was done by introducing a variable, marked with a lowercase delta, which takes a wff as input. In the system, "delta  ph " means that "something is true of 
ph." "delta  ph " can be substituted with  -.  ph,  ps  /\ 
ph,  A. x ph, etc.

Later on, Meredith discovered a single axiom, in the form of  ( delta delta  F.  -> delta  ph  ). This represents the shortest theorem in the extended propositional calculus that cannot be derived as an instance of a theorem in propositional calculus.

A symmetry with  -.. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  ( -.  -.  F.  ->  -.  ph )
 
Theoremimsym1 25416 A symmetry with  ->.

See negsym1 25415 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  (
 ( ps  ->  ( ps  ->  F.  ) )  ->  ( ps  ->  ph )
 )
 
Theorembisym1 25417 A symmetry with 
<->.

See negsym1 25415 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  (
 ( ps  <->  ( ps  <->  F.  ) )  ->  ( ps  <->  ph ) )
 
Theoremconsym1 25418 A symmetry with  /\.

See negsym1 25415 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  (
 ( ps  /\  ( ps  /\  F.  ) ) 
 ->  ( ps  /\  ph )
 )
 
Theoremdissym1 25419 A symmetry with  \/.

See negsym1 25415 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  (
 ( ps  \/  ( ps  \/  F.  ) ) 
 ->  ( ps  \/  ph ) )
 
Theoremnandsym1 25420 A symmetry with  -/\.

See negsym1 25415 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  (
 ( ps  -/\  ( ps  -/\  F.  ) ) 
 ->  ( ps  -/\  ph )
 )
 
Theoremunisym1 25421 A symmetry with  A..

See negsym1 25415 for more information. (Contributed by Anthony Hart, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

 |-  ( A. x A. x  F.  ->  A. x ph )
 
Theoremexisym1 25422 A symmetry with  E..

See negsym1 25415 for more information. (Contributed by Anthony Hart, 4-Sep-2011.)

 |-  ( E. x E. x  F.  ->  E. x ph )
 
Theoremunqsym1 25423 A symmetry with  E!.

See negsym1 25415 for more information. (Contributed by Anthony Hart, 6-Sep-2011.)

 |-  ( E! x E! x  F.  ->  E! x ph )
 
Theoremamosym1 25424 A symmetry with  E*.

See negsym1 25415 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)

 |-  ( E* x E* x  F.  ->  E* x ph )
 
Theoremsubsym1 25425 A symmetry with  [ x  / 
y ].

See negsym1 25415 for more information. (Contributed by Anthony Hart, 11-Sep-2011.)

 |-  ( [ x  /  y ] [ x  /  y ]  F.  ->  [ x  /  y ] ph )
 
18.9  Mathbox for Chen-Pang He
 
18.9.1  Ordinal topology
 
Theoremontopbas 25426 An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.)
 |-  ( B  e.  On  ->  B  e.  TopBases )
 
Theoremonsstopbas 25427 The class of ordinal numbers is a subclass of the class of topological bases. (Contributed by Chen-Pang He, 8-Oct-2015.)
 |-  On  C_  TopBases
 
Theoremonpsstopbas 25428 The class of ordinal numbers is a proper subclass of the class of topological bases. (Contributed by Chen-Pang He, 9-Oct-2015.)
 |-  On  C.  TopBases
 
Theoremontgval 25429 The topology generated from an ordinal number  B is 
suc  U. B. (Contributed by Chen-Pang He, 10-Oct-2015.)
 |-  ( B  e.  On  ->  (
 topGen `  B )  = 
 suc  U. B )
 
Theoremontgsucval 25430 The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.)
 |-  ( A  e.  On  ->  (
 topGen `  suc  A )  =  suc  A )
 
Theoremonsuctop 25431 A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.)
 |-  ( A  e.  On  ->  suc 
 A  e.  Top )
 
Theoremonsuctopon 25432 One of the topologies on an ordinal number is its successor. (Contributed by Chen-Pang He, 7-Nov-2015.)
 |-  ( A  e.  On  ->  suc 
 A  e.  (TopOn `  A ) )
 
Theoremordtoplem 25433 Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.)
 |-  ( U. A  e.  On  ->  suc  U. A  e.  S )   =>    |-  ( Ord  A  ->  ( A  =/=  U. A  ->  A  e.  S ) )
 
Theoremordtop 25434 An ordinal is a topology iff it is not its supremum (union), proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 1-Nov-2015.)
 |-  ( Ord  J  ->  ( J  e.  Top  <->  J  =/=  U. J ) )
 
Theoremonsucconi 25435 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
 |-  A  e.  On   =>    |- 
 suc  A  e.  Con
 
Theoremonsuccon 25436 A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
 |-  ( A  e.  On  ->  suc 
 A  e.  Con )
 
Theoremordtopcon 25437 An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.)
 |-  ( Ord  J  ->  ( J  e.  Top  <->  J  e.  Con ) )
 
Theoremonintopsscon 25438 An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
 |-  ( On  i^i  Top )  C_  Con
 
Theoremonsuct0 25439 A successor ordinal number is a T0 space. (Contributed by Chen-Pang He, 8-Nov-2015.)
 |-  ( A  e.  On  ->  suc 
 A  e.  Kol2 )
 
Theoremordtopt0 25440 An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.)
 |-  ( Ord  J  ->  ( J  e.  Top  <->  J  e.  Kol2 )
 )
 
Theoremonsucsuccmpi 25441 The successor of a successor ordinal number is a compact topology, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 18-Oct-2015.)
 |-  A  e.  On   =>    |- 
 suc  suc  A  e.  Comp
 
Theoremonsucsuccmp 25442 The successor of a successor ordinal number is a compact topology. (Contributed by Chen-Pang He, 18-Oct-2015.)
 |-  ( A  e.  On  ->  suc 
 suc  A  e.  Comp )
 
Theoremlimsucncmpi 25443 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
 |-  Lim  A   =>    |-  -. 
 suc  A  e.  Comp
 
Theoremlimsucncmp 25444 The successor of a limit ordinal is not compact. (Contributed by Chen-Pang He, 20-Oct-2015.)
 |-  ( Lim  A  ->  -.  suc  A  e.  Comp )
 
Theoremordcmp 25445 An ordinal topology is compact iff the underlying set is its supremum (union) only when the ordinal is  1o. (Contributed by Chen-Pang He, 1-Nov-2015.)
 |-  ( Ord  A  ->  ( A  e.  Comp 
 <->  ( U. A  =  U.
 U. A  ->  A  =  1o ) ) )
 
Theoremssoninhaus 25446 The ordinal topologies  1o and  2o are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
 |-  { 1o ,  2o }  C_  ( On  i^i  Haus )
 
Theoremonint1 25447 The ordinal T1 spaces are  1o and  2o, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.)
 |-  ( On  i^i  Fre )  =  { 1o ,  2o }
 
Theoremoninhaus 25448 The ordinal Hausdorff spaces are 
1o and  2o. (Contributed by Chen-Pang He, 10-Nov-2015.)
 |-  ( On  i^i  Haus )  =  { 1o ,  2o }
 
18.10  Mathbox for Jeff Hoffman
 
18.10.1  Inferences for finite induction on generic function values
 
Theoremfveleq 25449 Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
 |-  ( A  =  B  ->  ( ( ph  ->  ( F `  A )  e.  P )  <->  ( ph  ->  ( F `  B )  e.  P ) ) )
 
Theoremfindfvcl 25450* Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)
 |-  ( ph  ->  ( F `  (/) )  e.  P )   &    |-  ( y  e.  om  ->  (
 ph  ->  ( ( F `
  y )  e.  P  ->  ( F ` 
 suc  y )  e.  P ) ) )   =>    |-  ( A  e.  om  ->  (
 ph  ->  ( F `  A )  e.  P ) )
 
Theoremfindreccl 25451* Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.)
 |-  (
 z  e.  P  ->  ( G `  z )  e.  P )   =>    |-  ( C  e.  om 
 ->  ( A  e.  P  ->  ( rec ( G ,  A ) `  C )  e.  P ) )
 
Theoremfindabrcl 25452* Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  (
 z  e.  P  ->  ( G `  z )  e.  P )   =>    |-  ( ( C  e.  om  /\  A  e.  P )  ->  (
 ( x  e.  _V  |->  ( rec ( G ,  A ) `  x ) ) `  C )  e.  P )
 
18.10.2  gdc.mm
 
Theoremnnssi2 25453 Convert a theorem for real/complex numbers into one for natural numbers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  NN  C_  D   &    |-  ( B  e.  NN  ->  ph )   &    |-  ( ( A  e.  D  /\  B  e.  D  /\  ph )  ->  ps )   =>    |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ps )
 
Theoremnnssi3 25454 Convert a theorem for real/complex numbers into one for natural numbers. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  NN  C_  D   &    |-  ( C  e.  NN  ->  ph )   &    |-  ( ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  /\  ph )  ->  ps )   =>    |-  (
 ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ps )
 
Theoremnndivsub 25455 Please add description here. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  (
 ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A  /  C )  e.  NN  /\  A  <  B ) )  ->  ( ( B  /  C )  e. 
 NN 
 <->  ( ( B  -  A )  /  C )  e.  NN ) )
 
Theoremnndivlub 25456 A factor of a natural number cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  (
 ( A  e.  NN  /\  B  e.  NN )  ->  ( ( A  /  B )  e.  NN  ->  B  <_  A )
 )
 
SyntaxcgcdOLD 25457 Extend class notation to include the gdc function.
 class  gcd OLD ( A ,  B )
 
Definitiondf-gcdOLD 25458*  gcd OLD ( A ,  B ) is the largest natural number that evenly divides both  A and  B. (Contributed by Jeff Hoffman, 17-Jun-2008.)
 |-  gcd OLD ( A ,  B )  =  sup ( { x  e.  NN  |  ( ( A  /  x )  e.  NN  /\  ( B  /  x )  e. 
 NN ) } ,  NN ,  <  )
 
Theoremee7.2aOLD 25459 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 ) ) ) )
 
18.11  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 10 is easy, in nested chains they can be easily dropped.

 
Theoremwl-jarri 25460 Dropping a nested antecedent. This theorem is one of two reversions of ja 153. Since ja 153 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 2167 is an instance of this idea.

 |-  (
 ( ph  ->  ps )  ->  ch )   =>    |-  ( ps  ->  ch )
 
Theoremwl-jarli 25461 Dropping a nested consequent. This theorem is one of two reversions of ja 153. Since ja 153 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 2167 is an instance of this idea.

 |-  (
 ( ph  ->  ps )  ->  ch )   =>    |-  ( -.  ph  ->  ch )
 
Theoremwl-mps 25462 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 25463 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 25464 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 25465 A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid (see wl-adnestantALT 25466) (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   &    |-  ( ps  ->  ch )   =>    |-  ( ( ph  ->  ps )  ->  ch )
 
Theoremwl-adnestantALT 25466 Proof of wl-adnestant 25465 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 25467 Deduction version of wl-adnestant 25465. Generalization of a2i 12, imim12i 53, imim1i 54 and imim2i 13, 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-adnestantdALTOLD 25468). (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-adnestantdALTOLD 25468 Proof of wl-adnestantd 25467 not based on ax-3 7. (Contributed by Wolf Lammen, 4-Oct-2013.) (Moved to embantd 50 in main set.mm and may be deleted by mathbox owner, WL. --NM 14-Jan-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   =>    |-  ( ph  ->  (
 ( ps  ->  ch )  ->  th ) )
 
Theoremwl-bitr1 25469 Closed form of bitri 240. Place before bitri 240. [ +33] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (
 ( ps  <->  ch )  ->  ( ph 
 <->  ch ) ) )
 
Theoremwl-bitri 25470 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 25471 Deduction form of bitri 240. [ -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 25472 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 25473 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 25474 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 25475 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 25476 Moving a common antecedent on one side of an equivalence. Place before pm5.74 235. [ +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 25477 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 325.. albi 1564 before it. [ -22] (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ps  <->  ch ) )  <->  ( ( ph  ->  ps )  <->  ( ph  ->  ch ) ) )
 
Theoremwl-pm5.32 25478 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 25479 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 25480 Inference based on pm2.86 94. (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 25481 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 ) ) ) )
 
18.12  Mathbox for Brendan Leahy
 
Theoremsupaddc 25482* The supremum function distributes over addition in a sense similar to that in supmul1 9809. (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 25483* The supremum function distributes over addition in a sense similar to that in supmul 9812. (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 25484* 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 25485 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 25486 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 25487 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 ) ) )
 
Theoremovoliunnfl 25488* ovoliun 18968 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 25489* Demonstration of ovoliunnfl 25488. (Contributed by Brendan Leahy, 21-Nov-2017.)
 |-  (
 ( A  ~<_  NN  /\  A. x  e.  A  x  ~<_  NN )  ->  U. A  =/=  RR )
 
Theoremvoliunnfl 25490* voliun 19015 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 25491* volsup 19017 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 )
 
Theoremitg2addnclem 25492* 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.)
 |-  L  =  { x  |  E. g  e.  dom  S.1 ( E. y  e.  RR+  ( g  o F ( z  e.  RR ,  w  e.  RR  |->  if ( z  =  0 ,  0 ,  ( z  +  w ) ) ) ( RR  X.  { y } ) )  o R  <_  F  /\  x  =  ( S.1 `  g ) ) }   =>    |-  ( F : RR --> ( 0 [,]  +oo )  ->  ( S.2 `  F )  = 
 sup ( L ,  RR*
 ,  <  ) )
 
Theoremitg2addnclem2 25493* Lemma for itg2addnc 25494. 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 )
 
Theoremitg2addnc 25494 Alternate proof of itg2add 19218 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 19167, 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 8151, and second, weakening the measurability hypothesis to only the augend. (Contributed by Brendan Leahy, 31-Oct-2017.)
 |-  ( 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 25495* itg2gt0 19219 holds on functions continuous on an open interval in the absence of ax-cc 8151. 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 25496* Lemma for ibladdnc 25497; cf ibladdlem 19278, whose fifth hypothesis is rendered unnecessary by the weakened hypotheses of itg2addnc 25494. (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 25497* Choice-free analogue of itgadd 19283. A measurability hypothesis is necessitated by the loss of mbfadd 19120; 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 25498* Lemma for itgaddnc 25500; cf. itgaddlem1 19281. (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 25499* Lemma for itgaddnc 25500; cf. itgaddlem2 19282. (Contributed by Brendan Leahy, 10-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  ->  S. A ( B  +  C )  _d x  =  ( S. A B  _d x  +  S. A C  _d x ) )
 
Theoremitgaddnc 25500* Choice-free analogue of itgadd 19283. (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 ) )
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