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Theorem pmtrprfv3 16059
Description: In a transposition of two given points, all other points are mapped to themselves. (Contributed by AV, 17-Mar-2019.)
Hypothesis
Ref Expression
pmtrfval.t  |-  T  =  (pmTrsp `  D )
Assertion
Ref Expression
pmtrprfv3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( T `  { X ,  Y }
) `  Z )  =  Z )

Proof of Theorem pmtrprfv3
StepHypRef Expression
1 simp1 988 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  D  e.  V )
2 simp1 988 . . . . 5  |-  ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  ->  X  e.  D )
323ad2ant2 1010 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  X  e.  D )
4 simp22 1022 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Y  e.  D )
5 prssi 4124 . . . 4  |-  ( ( X  e.  D  /\  Y  e.  D )  ->  { X ,  Y }  C_  D )
63, 4, 5syl2anc 661 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  { X ,  Y }  C_  D
)
7 pr2nelem 8269 . . . . . . . . 9  |-  ( ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y )  ->  { X ,  Y }  ~~  2o )
873expia 1190 . . . . . . . 8  |-  ( ( X  e.  D  /\  Y  e.  D )  ->  ( X  =/=  Y  ->  { X ,  Y }  ~~  2o ) )
983adant3 1008 . . . . . . 7  |-  ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  ->  ( X  =/=  Y  ->  { X ,  Y }  ~~  2o ) )
109com12 31 . . . . . 6  |-  ( X  =/=  Y  ->  (
( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  ->  { X ,  Y }  ~~  2o ) )
11103ad2ant1 1009 . . . . 5  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  (
( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  ->  { X ,  Y }  ~~  2o ) )
1211impcom 430 . . . 4  |-  ( ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  { X ,  Y }  ~~  2o )
13123adant1 1006 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  { X ,  Y }  ~~  2o )
14 simp23 1023 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Z  e.  D )
15 pmtrfval.t . . . 4  |-  T  =  (pmTrsp `  D )
1615pmtrfv 16057 . . 3  |-  ( ( ( D  e.  V  /\  { X ,  Y }  C_  D  /\  { X ,  Y }  ~~  2o )  /\  Z  e.  D )  ->  (
( T `  { X ,  Y }
) `  Z )  =  if ( Z  e. 
{ X ,  Y } ,  U. ( { X ,  Y }  \  { Z } ) ,  Z ) )
171, 6, 13, 14, 16syl31anc 1222 . 2  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( T `  { X ,  Y }
) `  Z )  =  if ( Z  e. 
{ X ,  Y } ,  U. ( { X ,  Y }  \  { Z } ) ,  Z ) )
18 necom 2715 . . . . . . 7  |-  ( X  =/=  Z  <->  Z  =/=  X )
1918biimpi 194 . . . . . 6  |-  ( X  =/=  Z  ->  Z  =/=  X )
20193ad2ant2 1010 . . . . 5  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  Z  =/=  X )
21203ad2ant3 1011 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Z  =/=  X )
22 necom 2715 . . . . . . 7  |-  ( Y  =/=  Z  <->  Z  =/=  Y )
2322biimpi 194 . . . . . 6  |-  ( Y  =/=  Z  ->  Z  =/=  Y )
24233ad2ant3 1011 . . . . 5  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  Z  =/=  Y )
25243ad2ant3 1011 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Z  =/=  Y )
2621, 25nelprd 3994 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  -.  Z  e.  { X ,  Y } )
27 iffalse 3894 . . 3  |-  ( -.  Z  e.  { X ,  Y }  ->  if ( Z  e.  { X ,  Y } ,  U. ( { X ,  Y }  \  { Z }
) ,  Z )  =  Z )
2826, 27syl 16 . 2  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  if ( Z  e.  { X ,  Y } ,  U. ( { X ,  Y }  \  { Z }
) ,  Z )  =  Z )
2917, 28eqtrd 2491 1  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( T `  { X ,  Y }
) `  Z )  =  Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2642    \ cdif 3420    C_ wss 3423   ifcif 3886   {csn 3972   {cpr 3974   U.cuni 4186   class class class wbr 4387   ` cfv 5513   2oc2o 7011    ~~ cen 7404  pmTrspcpmtr 16046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-om 6574  df-1o 7017  df-2o 7018  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-pmtr 16047
This theorem is referenced by:  pmtr3ncomlem1  16078
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