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Theorem pmtr3ncomlem1 16081
Description: Lemma 1 for pmtr3ncom 16083. (Contributed by AV, 17-Mar-2018.)
Hypotheses
Ref Expression
pmtr3ncom.t  |-  T  =  (pmTrsp `  D )
pmtr3ncom.f  |-  F  =  ( T `  { X ,  Y }
)
pmtr3ncom.g  |-  G  =  ( T `  { Y ,  Z }
)
Assertion
Ref Expression
pmtr3ncomlem1  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( G  o.  F
) `  X )  =/=  ( ( F  o.  G ) `  X
) )

Proof of Theorem pmtr3ncomlem1
StepHypRef Expression
1 necom 2717 . . . . 5  |-  ( Y  =/=  Z  <->  Z  =/=  Y )
21biimpi 194 . . . 4  |-  ( Y  =/=  Z  ->  Z  =/=  Y )
323ad2ant3 1011 . . 3  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  Z  =/=  Y )
433ad2ant3 1011 . 2  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Z  =/=  Y )
5 simp1 988 . . . . . . . 8  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  D  e.  V )
6 simp1 988 . . . . . . . . . 10  |-  ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  ->  X  e.  D )
763ad2ant2 1010 . . . . . . . . 9  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  X  e.  D )
8 simp2 989 . . . . . . . . . 10  |-  ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  ->  Y  e.  D )
983ad2ant2 1010 . . . . . . . . 9  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Y  e.  D )
10 prssi 4127 . . . . . . . . 9  |-  ( ( X  e.  D  /\  Y  e.  D )  ->  { X ,  Y }  C_  D )
117, 9, 10syl2anc 661 . . . . . . . 8  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  { X ,  Y }  C_  D
)
12 simp1 988 . . . . . . . . . . 11  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  X  =/=  Y )
13123ad2ant3 1011 . . . . . . . . . 10  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  X  =/=  Y )
147, 9, 133jca 1168 . . . . . . . . 9  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y ) )
15 pr2nelem 8272 . . . . . . . . 9  |-  ( ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y )  ->  { X ,  Y }  ~~  2o )
1614, 15syl 16 . . . . . . . 8  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  { X ,  Y }  ~~  2o )
175, 11, 163jca 1168 . . . . . . 7  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  ( D  e.  V  /\  { X ,  Y }  C_  D  /\  { X ,  Y }  ~~  2o ) )
18 pmtr3ncom.t . . . . . . . 8  |-  T  =  (pmTrsp `  D )
1918pmtrf 16063 . . . . . . 7  |-  ( ( D  e.  V  /\  { X ,  Y }  C_  D  /\  { X ,  Y }  ~~  2o )  ->  ( T `  { X ,  Y }
) : D --> D )
2017, 19syl 16 . . . . . 6  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  ( T `  { X ,  Y } ) : D --> D )
21 pmtr3ncom.f . . . . . . 7  |-  F  =  ( T `  { X ,  Y }
)
2221feq1i 5649 . . . . . 6  |-  ( F : D --> D  <->  ( T `  { X ,  Y } ) : D --> D )
2320, 22sylibr 212 . . . . 5  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  F : D --> D )
24 ffn 5657 . . . . 5  |-  ( F : D --> D  ->  F  Fn  D )
2523, 24syl 16 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  F  Fn  D )
26 fvco2 5865 . . . 4  |-  ( ( F  Fn  D  /\  X  e.  D )  ->  ( ( G  o.  F ) `  X
)  =  ( G `
 ( F `  X ) ) )
2725, 7, 26syl2anc 661 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( G  o.  F
) `  X )  =  ( G `  ( F `  X ) ) )
2821fveq1i 5790 . . . . 5  |-  ( F `
 X )  =  ( ( T `  { X ,  Y }
) `  X )
2918pmtrprfv 16061 . . . . . 6  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y
) )  ->  (
( T `  { X ,  Y }
) `  X )  =  Y )
305, 14, 29syl2anc 661 . . . . 5  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( T `  { X ,  Y }
) `  X )  =  Y )
3128, 30syl5eq 2504 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  ( F `  X )  =  Y )
3231fveq2d 5793 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  ( G `  ( F `  X ) )  =  ( G `  Y
) )
33 pmtr3ncom.g . . . . 5  |-  G  =  ( T `  { Y ,  Z }
)
3433fveq1i 5790 . . . 4  |-  ( G `
 Y )  =  ( ( T `  { Y ,  Z }
) `  Y )
35 simp3 990 . . . . . . 7  |-  ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  ->  Z  e.  D )
36353ad2ant2 1010 . . . . . 6  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Z  e.  D )
37 simp3 990 . . . . . . 7  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  Y  =/=  Z )
38373ad2ant3 1011 . . . . . 6  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Y  =/=  Z )
399, 36, 383jca 1168 . . . . 5  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  ( Y  e.  D  /\  Z  e.  D  /\  Y  =/=  Z ) )
4018pmtrprfv 16061 . . . . 5  |-  ( ( D  e.  V  /\  ( Y  e.  D  /\  Z  e.  D  /\  Y  =/=  Z
) )  ->  (
( T `  { Y ,  Z }
) `  Y )  =  Z )
415, 39, 40syl2anc 661 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( T `  { Y ,  Z }
) `  Y )  =  Z )
4234, 41syl5eq 2504 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  ( G `  Y )  =  Z )
4327, 32, 423eqtrd 2496 . 2  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( G  o.  F
) `  X )  =  Z )
44 prssi 4127 . . . . . . . . 9  |-  ( ( Y  e.  D  /\  Z  e.  D )  ->  { Y ,  Z }  C_  D )
458, 35, 44syl2anc 661 . . . . . . . 8  |-  ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  ->  { Y ,  Z }  C_  D )
46453ad2ant2 1010 . . . . . . 7  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  { Y ,  Z }  C_  D
)
47 pr2nelem 8272 . . . . . . . 8  |-  ( ( Y  e.  D  /\  Z  e.  D  /\  Y  =/=  Z )  ->  { Y ,  Z }  ~~  2o )
4839, 47syl 16 . . . . . . 7  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  { Y ,  Z }  ~~  2o )
495, 46, 483jca 1168 . . . . . 6  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  ( D  e.  V  /\  { Y ,  Z }  C_  D  /\  { Y ,  Z }  ~~  2o ) )
5018pmtrf 16063 . . . . . . 7  |-  ( ( D  e.  V  /\  { Y ,  Z }  C_  D  /\  { Y ,  Z }  ~~  2o )  ->  ( T `  { Y ,  Z }
) : D --> D )
5133feq1i 5649 . . . . . . 7  |-  ( G : D --> D  <->  ( T `  { Y ,  Z } ) : D --> D )
5250, 51sylibr 212 . . . . . 6  |-  ( ( D  e.  V  /\  { Y ,  Z }  C_  D  /\  { Y ,  Z }  ~~  2o )  ->  G : D --> D )
5349, 52syl 16 . . . . 5  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  G : D --> D )
54 ffn 5657 . . . . 5  |-  ( G : D --> D  ->  G  Fn  D )
5553, 54syl 16 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  G  Fn  D )
56 fvco2 5865 . . . 4  |-  ( ( G  Fn  D  /\  X  e.  D )  ->  ( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
5755, 7, 56syl2anc 661 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( F  o.  G
) `  X )  =  ( F `  ( G `  X ) ) )
5833fveq1i 5790 . . . . 5  |-  ( G `
 X )  =  ( ( T `  { Y ,  Z }
) `  X )
59 id 22 . . . . . 6  |-  ( D  e.  V  ->  D  e.  V )
60 3anrot 970 . . . . . . 7  |-  ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  <->  ( Y  e.  D  /\  Z  e.  D  /\  X  e.  D )
)
6160biimpi 194 . . . . . 6  |-  ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  ->  ( Y  e.  D  /\  Z  e.  D  /\  X  e.  D
) )
62 3anrot 970 . . . . . . . 8  |-  ( ( Y  =/=  Z  /\  Y  =/=  X  /\  Z  =/=  X )  <->  ( Y  =/=  X  /\  Z  =/= 
X  /\  Y  =/=  Z ) )
63 necom 2717 . . . . . . . . 9  |-  ( Y  =/=  X  <->  X  =/=  Y )
64 necom 2717 . . . . . . . . 9  |-  ( Z  =/=  X  <->  X  =/=  Z )
65 biid 236 . . . . . . . . 9  |-  ( Y  =/=  Z  <->  Y  =/=  Z )
6663, 64, 653anbi123i 1177 . . . . . . . 8  |-  ( ( Y  =/=  X  /\  Z  =/=  X  /\  Y  =/=  Z )  <->  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )
6762, 66bitr2i 250 . . . . . . 7  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  <->  ( Y  =/=  Z  /\  Y  =/= 
X  /\  Z  =/=  X ) )
6867biimpi 194 . . . . . 6  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  ( Y  =/=  Z  /\  Y  =/=  X  /\  Z  =/= 
X ) )
6918pmtrprfv3 16062 . . . . . 6  |-  ( ( D  e.  V  /\  ( Y  e.  D  /\  Z  e.  D  /\  X  e.  D
)  /\  ( Y  =/=  Z  /\  Y  =/= 
X  /\  Z  =/=  X ) )  ->  (
( T `  { Y ,  Z }
) `  X )  =  X )
7059, 61, 68, 69syl3an 1261 . . . . 5  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( T `  { Y ,  Z }
) `  X )  =  X )
7158, 70syl5eq 2504 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  ( G `  X )  =  X )
7271fveq2d 5793 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  ( F `  ( G `  X ) )  =  ( F `  X
) )
7357, 72, 313eqtrd 2496 . 2  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( F  o.  G
) `  X )  =  Y )
744, 43, 733netr4d 2753 1  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( G  o.  F
) `  X )  =/=  ( ( F  o.  G ) `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644    C_ wss 3426   {cpr 3977   class class class wbr 4390    o. ccom 4942    Fn wfn 5511   -->wf 5512   ` cfv 5516   2oc2o 7014    ~~ cen 7407  pmTrspcpmtr 16049
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 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-om 6577  df-1o 7020  df-2o 7021  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-pmtr 16050
This theorem is referenced by:  pmtr3ncomlem2  16082
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