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Theorem efgmnvl 15023
Description: The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypothesis
Ref Expression
efgmval.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
Assertion
Ref Expression
efgmnvl  |-  ( A  e.  ( I  X.  2o )  ->  ( M `
 ( M `  A ) )  =  A )
Distinct variable group:    y, z, I
Allowed substitution hints:    A( y, z)    M( y, z)

Proof of Theorem efgmnvl
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp2 4707 . 2  |-  ( A  e.  ( I  X.  2o )  <->  E. a  e.  I  E. b  e.  2o  A  =  <. a ,  b >. )
2 efgmval.m . . . . . . . 8  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
32efgmval 15021 . . . . . . 7  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
43fveq2d 5529 . . . . . 6  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  ( M `
 <. a ,  ( 1o  \  b )
>. ) )
5 df-ov 5861 . . . . . 6  |-  ( a M ( 1o  \ 
b ) )  =  ( M `  <. a ,  ( 1o  \ 
b ) >. )
64, 5syl6eqr 2333 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  ( a M ( 1o  \ 
b ) ) )
7 2oconcl 6502 . . . . . 6  |-  ( b  e.  2o  ->  ( 1o  \  b )  e.  2o )
82efgmval 15021 . . . . . 6  |-  ( ( a  e.  I  /\  ( 1o  \  b
)  e.  2o )  ->  ( a M ( 1o  \  b
) )  =  <. a ,  ( 1o  \ 
( 1o  \  b
) ) >. )
97, 8sylan2 460 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M ( 1o  \  b ) )  =  <. a ,  ( 1o  \ 
( 1o  \  b
) ) >. )
10 1on 6486 . . . . . . . . . . 11  |-  1o  e.  On
1110onordi 4497 . . . . . . . . . 10  |-  Ord  1o
12 ordtr 4406 . . . . . . . . . 10  |-  ( Ord 
1o  ->  Tr  1o )
13 trsucss 4478 . . . . . . . . . 10  |-  ( Tr  1o  ->  ( b  e.  suc  1o  ->  b  C_  1o ) )
1411, 12, 13mp2b 9 . . . . . . . . 9  |-  ( b  e.  suc  1o  ->  b 
C_  1o )
15 df-2o 6480 . . . . . . . . 9  |-  2o  =  suc  1o
1614, 15eleq2s 2375 . . . . . . . 8  |-  ( b  e.  2o  ->  b  C_  1o )
1716adantl 452 . . . . . . 7  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  b  C_  1o )
18 dfss4 3403 . . . . . . 7  |-  ( b 
C_  1o  <->  ( 1o  \ 
( 1o  \  b
) )  =  b )
1917, 18sylib 188 . . . . . 6  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( 1o  \  ( 1o  \  b ) )  =  b )
2019opeq2d 3803 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  ( 1o 
\  ( 1o  \ 
b ) ) >.  =  <. a ,  b
>. )
216, 9, 203eqtrd 2319 . . . 4  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  <. a ,  b >. )
22 fveq2 5525 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( M `  <. a ,  b >. )
)
23 df-ov 5861 . . . . . . 7  |-  ( a M b )  =  ( M `  <. a ,  b >. )
2422, 23syl6eqr 2333 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( a M b ) )
2524fveq2d 5529 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  ( M `  ( M `  A ) )  =  ( M `
 ( a M b ) ) )
26 id 19 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  A  =  <. a ,  b >. )
2725, 26eqeq12d 2297 . . . 4  |-  ( A  =  <. a ,  b
>.  ->  ( ( M `
 ( M `  A ) )  =  A  <->  ( M `  ( a M b ) )  =  <. a ,  b >. )
)
2821, 27syl5ibrcom 213 . . 3  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( A  =  <. a ,  b >.  ->  ( M `  ( M `  A ) )  =  A ) )
2928rexlimivv 2672 . 2  |-  ( E. a  e.  I  E. b  e.  2o  A  =  <. a ,  b
>.  ->  ( M `  ( M `  A ) )  =  A )
301, 29sylbi 187 1  |-  ( A  e.  ( I  X.  2o )  ->  ( M `
 ( M `  A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    \ cdif 3149    C_ wss 3152   <.cop 3643   Tr wtr 4113   Ord word 4391   suc csuc 4394    X. cxp 4687   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473
This theorem is referenced by:  efginvrel1  15037  efgredlemc  15054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1o 6479  df-2o 6480
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