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Theorem List for Metamath Proof Explorer - 27601-27700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremax4567 27601 Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1533 as the inference rule. This proof extends the idea of ax467 2108 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)
 |-  (
 ( A. x A. y  -.  A. x A. y
 ( A. y ph  ->  ps )  ->  ( ph  ->  A. y ( A. y ph  ->  ps )
 ) )  ->  ( A. y ph  ->  A. y ps ) )
 
Theoremax4567to4 27602 Re-derivation of sp 1716 from ax4567 27601. Note that ax9 1889 is used for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x ph  ->  ph )
 
Theoremax4567to5 27603 Re-derivation of ax5o 1717 from ax4567 27601. Note that only propositional calculus is required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x ( A. x ph 
 ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremax4567to6 27604 Re-derivation of ax6o 1723 from ax4567 27601. Note that neither ax6o 1723 nor ax-7 1708 are required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax4567to7 27605 Re-derivation of ax-7 1708 from ax4567 27601. Note that ax-7 1708 is not required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax10ext 27606* This theorem shows that, given axext4 2267, we can derive a version of ax10 1884. However, it is weaker than ax10 1884 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  z  ->  A. z  z  =  x )
 
18.19.4  Principia Mathematica * 13 and * 14
 
Theorempm13.13a 27607 One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  A )  ->  [. A  /  x ]. ph )
 
Theorempm13.13b 27608 Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( [. A  /  x ].
 ph  /\  x  =  A )  ->  ph )
 
Theorempm13.14 27609 Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( [. A  /  x ].
 ph  /\  -.  ph )  ->  x  =/=  A )
 
Theorempm13.192 27610* Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( A. x ( x  =  A  <->  x  =  y )  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.193 27611 Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  y )  <->  ( [ y  /  x ] ph  /\  x  =  y ) )
 
Theorempm13.194 27612 Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  y )  <->  ( [ y  /  x ] ph  /\  ph  /\  x  =  y ) )
 
Theorempm13.195 27613* Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3015. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( y  =  A  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.196a 27614* Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( -.  ph  <->  A. y ( [
 y  /  x ] ph  ->  y  =/=  x ) )
 
Theorem2sbc6g 27615* Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
 
Theorem2sbc5g 27616* Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
 
Theoremiotain 27617 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( E! x ph  ->  |^| { x  |  ph }  =  (
 iota x ph ) )
 
Theoremiotaexeu 27618 The iota class exists. This theorem does not require ax-nul 4149 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  e. 
 _V )
 
Theoremiotasbc 27619* Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define  iota in terms of a function of  ( iota x ph ). Their definition differs in that a function of  ( iota x ph ) evaluates to "false" when there isn't a single  x that satisfies  ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( A. x ( ph  <->  x  =  y
 )  /\  ps )
 ) )
 
Theoremiotasbc2 27620* Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  (
 ( E! x ph  /\ 
 E! x ps )  ->  ( [. ( iota
 x ph )  /  y ]. [. ( iota x ps )  /  z ]. ch  <->  E. y E. z
 ( A. x ( ph  <->  x  =  y )  /\  A. x ( ps  <->  x  =  z
 )  /\  ch )
 ) )
 
Theorempm14.12 27621* Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  A. x A. y ( ( ph  /\  [. y  /  x ].
 ph )  ->  x  =  y ) )
 
Theorempm14.122a 27622* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  [. A  /  x ]. ph )
 ) )
 
Theorempm14.122b 27623* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  ( ( A. x (
 ph  ->  x  =  A )  /\  [. A  /  x ].
 ph )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
 
Theorempm14.122c 27624* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
 
Theorempm14.123a 27625* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  [. A  /  z ]. [. B  /  w ]. ph ) ) )
 
Theorempm14.123b 27626* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  [. A  /  z ]. [. B  /  w ]. ph )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  E. z E. w ph ) ) )
 
Theorempm14.123c 27627* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  E. z E. w ph ) ) )
 
Theorempm14.18 27628 Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ps  ->  [. ( iota x ph )  /  x ]. ps ) )
 
Theoremiotaequ 27629* Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( iota x x  =  y )  =  y
 
Theoremiotavalb 27630* Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5230. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y
 ) 
 <->  ( iota x ph )  =  y )
 )
 
Theoremiotasbc5 27631* Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( y  =  ( iota x ph )  /\  ps ) ) )
 
Theorempm14.24 27632* Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  A. y
 ( [. y  /  x ].
 ph 
 <->  y  =  ( iota
 x ph ) ) )
 
Theoremiotavalsb 27633* Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x ( ph  <->  x  =  y
 )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps )
 )
 
Theoremsbiota1 27634 Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  <->  [. ( iota x ph )  /  x ]. ps ) )
 
Theoremsbaniota 27635 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( E. x ( ph  /\  ps ) 
 <-> 
 [. ( iota x ph )  /  x ]. ps ) )
 
Theoremeubi 27636 Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x ( ph  <->  ps )  ->  ( E! x ph  <->  E! x ps )
 )
 
Theoremiotasbcq 27637 Theorem *14.272 in [WhiteheadRussell] p. 193. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x ( ph  <->  ps )  ->  ( [. ( iota x ph )  /  y ]. ch  <->  [. ( iota x ps )  /  y ]. ch )
 )
 
18.19.5  Set Theory
 
Theoremelnev 27638* Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.)
 |-  ( A  e.  _V  <->  { x  |  -.  x  =  A }  =/=  _V )
 
TheoremrusbcALT 27639 A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (Proof modification is discouraged.)
 |-  { x  |  x  e/  x }  e/  _V
 
Theoremcompel 27640 Equivalence between two ways of saying "is a member of the complement of  A." (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
 
Theoremcompeq 27641* Equality between two ways of saying "the complement of  A." (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( _V  \  A )  =  { x  |  -.  x  e.  A }
 
Theoremcompne 27642 The complement of  A is not equal to  A. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( _V  \  A )  =/= 
 A
 
TheoremcompneOLD 27643 Obsolete proof of compne 27642 as of 28-Jun-2015. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( _V  \  A )  =/= 
 A
 
Theoremcompab 27644 Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  ( _V  \  { z  | 
 ph } )  =  { z  |  -.  ph
 }
 
Theoremconss34 27645 Contrpositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( A  C_  B  <->  ( _V  \  B )  C_  ( _V  \  A ) )
 
Theoremconss2 27646 Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( A  C_  ( _V  \  B ) 
 <->  B  C_  ( _V  \  A ) )
 
Theoremconss1 27647 Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  (
 ( _V  \  A )  C_  B  <->  ( _V  \  B )  C_  A )
 
Theoremralbidar 27648 More general form of ralbida 2557. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( ph  ->  A. x  e.  A  ph )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  A  ch ) )
 
Theoremrexbidar 27649 More general form of rexbida 2558. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( ph  ->  A. x  e.  A  ph )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  A  ch ) )
 
Theoremdropab1 27650 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( A. x  x  =  y  ->  { <. x ,  z >.  |  ph }  =  { <. y ,  z >.  |  ph } )
 
Theoremdropab2 27651 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( A. x  x  =  y  ->  { <. z ,  x >.  |  ph }  =  { <. z ,  y >.  |  ph } )
 
Theoremipo0 27652 If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  (  _I  Po  A  <->  A  =  (/) )
 
Theoremifr0 27653 A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  (  _I  Fr  A  <->  A  =  (/) )
 
Theoremordpss 27654 ordelpss 4420 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( Ord  B  ->  ( A  e.  B  ->  A  C.  B ) )
 
Theoremfvsb 27655* Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( E! y  A F y  ->  ( [. ( F `  A )  /  x ]. ph  <->  E. x ( A. y ( A F y 
 <->  y  =  x ) 
 /\  ph ) ) )
 
Theoremfveqsb 27656* Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( x  =  ( F `  A )  ->  ( ph 
 <->  ps ) )   &    |-  F/ x ps   =>    |-  ( E! y  A F y  ->  ( ps 
 <-> 
 E. x ( A. y ( A F y 
 <->  y  =  x ) 
 /\  ph ) ) )
 
TheoremxrltneNEW 27657 'Less than' implies not equal for extended reals. (Contributed by Andrew Salmon, 11-Nov-2011.)
 |-  (
 ( A  e.  RR*  /\  A  <  B ) 
 ->  A  =/=  B )
 
Theoremxpexb 27658 A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
 |-  (
 ( A  X.  B )  e.  _V  <->  ( B  X.  A )  e.  _V )
 
Theoremxpexcnv 27659 A condition where the converse of xpex 4801 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
 |-  (
 ( B  =/=  (/)  /\  ( A  X.  B )  e. 
 _V )  ->  A  e.  _V )
 
Theoremtrelpss 27660 An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 4377, ax-reg 7306 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)
 |-  (
 ( Tr  A  /\  B  e.  A )  ->  B  C.  A )
 
18.19.6  Arithmetic
 
Theoremaddcomgi 27661 Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( A  +  B )  =  ( B  +  A )
 
18.19.7  Geometry
 
Syntaxcplusr 27662 Introduce the operation of vector addition.
 class  + r
 
Syntaxcminusr 27663 Introduce the operation of vector subtraction.
 class  - r
 
Syntaxctimesr 27664 Introduce the operation of scalar multiplication.
 class  . v
 
Syntaxcptdfc 27665  PtDf is a predicate that is crucial for the definition of lines as well as proving a number of important theorems.
 class  PtDf ( A ,  B )
 
Syntaxcrr3c 27666  RR 3 is a class.
 class  RR 3
 
Syntaxcline3 27667  line 3 is a class.
 class  line 3
 
Definitiondf-addr 27668* Define the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  + r  =  ( x  e.  _V ,  y  e. 
 _V  |->  ( v  e. 
 RR  |->  ( ( x `
  v )  +  ( y `  v
 ) ) ) )
 
Definitiondf-subr 27669* Define the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  - r  =  ( x  e.  _V ,  y  e. 
 _V  |->  ( v  e. 
 RR  |->  ( ( x `
  v )  -  ( y `  v
 ) ) ) )
 
Definitiondf-mulv 27670* Define the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  . v  =  ( x  e.  _V ,  y  e. 
 _V  |->  ( v  e. 
 RR  |->  ( x  x.  ( y `  v
 ) ) ) )
 
Theoremaddrval 27671* Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A + r B )  =  (
 v  e.  RR  |->  ( ( A `  v
 )  +  ( B `
  v ) ) ) )
 
Theoremsubrval 27672* Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A - r B )  =  (
 v  e.  RR  |->  ( ( A `  v
 )  -  ( B `
  v ) ) ) )
 
Theoremmulvval 27673* Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A . v B )  =  (
 v  e.  RR  |->  ( A  x.  ( B `
  v ) ) ) )
 
Theoremaddrfv 27674 Vector addition at a value. The operation takes each vector  A and  B and forms a new vector whose values are the sum of each of the values of  A and  B. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A + r B ) `  C )  =  ( ( A `  C )  +  ( B `  C ) ) )
 
Theoremsubrfv 27675 Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A - r B ) `  C )  =  ( ( A `  C )  -  ( B `  C ) ) )
 
Theoremmulvfv 27676 Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A . v B ) `  C )  =  ( A  x.  ( B `  C ) ) )
 
Theoremaddrfn 27677 Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A + r B )  Fn  RR )
 
Theoremsubrfn 27678 Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A - r B )  Fn  RR )
 
Theoremmulvfn 27679 Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A . v B )  Fn  RR )
 
Theoremaddrcom 27680 Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A + r B )  =  ( B + r A ) )
 
Definitiondf-ptdf 27681* Define the predicate  PtDf, which is a utility definition used to shorten definitions and simplify proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)
 |-  PtDf ( A ,  B )  =  ( x  e. 
 RR  |->  ( ( ( x . v ( B - r A ) ) +v A ) " { 1 ,  2 ,  3 } ) )
 
Definitiondf-rr3 27682 Define the set of all points  RR 3. We define each point  A as a function to allow the use of vector addition and subtraction as well as scalar multiplication in our proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)
 |-  RR 3  =  ( RR  ^m 
 { 1 ,  2 ,  3 } )
 
Definitiondf-line3 27683* Define the set of all lines. A line is an infinite subset of  RR 3 that satisfies a  PtDf property. (Contributed by Andrew Salmon, 15-Jul-2012.)
 |-  line 3  =  { x  e. 
 ~P RR 3  |  ( 2o  ~<_  x  /\  A. y  e.  x  A. z  e.  x  (
 z  =/=  y  ->  ran  PtDf ( y ,  z
 )  =  x ) ) }
 
18.20  Mathbox for Glauco Siliprandi
 
18.20.1  Miscellanea
 
Theoremssrexf 27684 restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B  ->  ( E. x  e.  A  ph  ->  E. x  e.  B  ph ) )
 
Theoremfnvinran 27685 the function value belongs to its codomain. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ( ph  /\  C  e.  A )  ->  ( F `  C )  e.  B )
 
Theoremevth2f 27686* A version of evth2 18458 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x F   &    |-  F/_ y F   &    |-  F/_ x X   &    |-  F/_ y X   &    |-  X  =  U. J   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  X  =/=  (/) )   =>    |-  ( ph  ->  E. x  e.  X  A. y  e.  X  ( F `  x )  <_  ( F `
  y ) )
 
Theoremelunif 27687* A version of eluni 3830 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  e.  U. B 
 <-> 
 E. x ( A  e.  x  /\  x  e.  B ) )
 
Theoremrzalf 27688 A version of rzal 3555 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ x  A  =  (/)   =>    |-  ( A  =  (/)  ->  A. x  e.  A  ph )
 
Theoremfvelrnbf 27689 A version of fvelrnb 5570 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/_ x F   =>    |-  ( F  Fn  A  ->  ( B  e.  ran 
 F 
 <-> 
 E. x  e.  A  ( F `  x )  =  B ) )
 
Theoremrfcnpre1 27690 If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than a given extended real B is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x B   &    |-  F/_ x F   &    |-  F/ x ph   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  X  =  U. J   &    |-  A  =  { x  e.  X  |  B  <  ( F `  x ) }   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  A  e.  J )
 
Theoremubelsupr 27691* If U belongs to A and U is an upper bound, then U is the sup of A. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( A  C_  RR  /\  U  e.  A  /\  A. x  e.  A  x  <_  U )  ->  U  =  sup ( A ,  RR ,  <  ) )
 
Theoremfsumcnf 27692* A finite sum of functions to complex numbers from a common topological space is continuous, without disjoint var constraint x ph. The class expression for B normally contains free variables k and x to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  K  =  ( TopOpen ` fld )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  (
 ( ph  /\  k  e.  A )  ->  ( x  e.  X  |->  B )  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  sum_ k  e.  A  B )  e.  ( J  Cn  K ) )
 
Theoremmulltgt0 27693 The product of a negative and a positive number is negative. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( ( A  e.  RR  /\  A  <  0
 )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  x.  B )  <  0 )
 
Theoremrspcegf 27694 A version of rspcev 2884 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/ x ps   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  B  /\  ps )  ->  E. x  e.  B  ph )
 
Theoremrabexgf 27695 A version of rabexg 4164 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x A   =>    |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
 
Theoremfcnre 27696 A function continuous with respect to the standard topology, is a real mapping. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  K  =  ( topGen `  ran  (,) )   &    |-  T  =  U. J   &    |-  C  =  ( J  Cn  K )   &    |-  ( ph  ->  F  e.  C )   =>    |-  ( ph  ->  F : T --> RR )
 
Theoremsumsnd 27697* A sum of a singleton is the term. The deduction version of sumsn 12213. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( ph  ->  F/_ k B )   &    |-  F/ k ph   &    |-  ( ( ph  /\  k  =  M ) 
 ->  A  =  B )   &    |-  ( ph  ->  M  e.  V )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  sum_
 k  e.  { M } A  =  B )
 
Theoremevthf 27698* A version of evth 18457 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x F   &    |-  F/_ y F   &    |-  F/_ x X   &    |-  F/_ y X   &    |- 
 F/ x ph   &    |-  F/ y ph   &    |-  X  =  U. J   &    |-  K  =  (
 topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  X  =/=  (/) )   =>    |-  ( ph  ->  E. x  e.  X  A. y  e.  X  ( F `  y )  <_  ( F `
  x ) )
 
Theoremcnfex 27699 The class of continuous functions between two topologies is a set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top )  ->  ( J  Cn  K )  e.  _V )
 
Theoremfnchoice 27700* For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  ( A  e.  Fin  ->  E. f
 ( f  Fn  A  /\  A. x  e.  A  ( x  =/=  (/)  ->  (
 f `  x )  e.  x ) ) )
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