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Theorem efgmnvl 16211
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 4858 . 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 16209 . . . . . . 7  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
43fveq2d 5695 . . . . . 6  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  ( M `
 <. a ,  ( 1o  \  b )
>. ) )
5 df-ov 6094 . . . . . 6  |-  ( a M ( 1o  \ 
b ) )  =  ( M `  <. a ,  ( 1o  \ 
b ) >. )
64, 5syl6eqr 2493 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  ( a M ( 1o  \ 
b ) ) )
7 2oconcl 6943 . . . . . 6  |-  ( b  e.  2o  ->  ( 1o  \  b )  e.  2o )
82efgmval 16209 . . . . . 6  |-  ( ( a  e.  I  /\  ( 1o  \  b
)  e.  2o )  ->  ( a M ( 1o  \  b
) )  =  <. a ,  ( 1o  \ 
( 1o  \  b
) ) >. )
97, 8sylan2 474 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M ( 1o  \  b ) )  =  <. a ,  ( 1o  \ 
( 1o  \  b
) ) >. )
10 1on 6927 . . . . . . . . . . 11  |-  1o  e.  On
1110onordi 4823 . . . . . . . . . 10  |-  Ord  1o
12 ordtr 4733 . . . . . . . . . 10  |-  ( Ord 
1o  ->  Tr  1o )
13 trsucss 4804 . . . . . . . . . 10  |-  ( Tr  1o  ->  ( b  e.  suc  1o  ->  b  C_  1o ) )
1411, 12, 13mp2b 10 . . . . . . . . 9  |-  ( b  e.  suc  1o  ->  b 
C_  1o )
15 df-2o 6921 . . . . . . . . 9  |-  2o  =  suc  1o
1614, 15eleq2s 2535 . . . . . . . 8  |-  ( b  e.  2o  ->  b  C_  1o )
1716adantl 466 . . . . . . 7  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  b  C_  1o )
18 dfss4 3584 . . . . . . 7  |-  ( b 
C_  1o  <->  ( 1o  \ 
( 1o  \  b
) )  =  b )
1917, 18sylib 196 . . . . . 6  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( 1o  \  ( 1o  \  b ) )  =  b )
2019opeq2d 4066 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  ( 1o 
\  ( 1o  \ 
b ) ) >.  =  <. a ,  b
>. )
216, 9, 203eqtrd 2479 . . . 4  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  <. a ,  b >. )
22 fveq2 5691 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( M `  <. a ,  b >. )
)
23 df-ov 6094 . . . . . . 7  |-  ( a M b )  =  ( M `  <. a ,  b >. )
2422, 23syl6eqr 2493 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( a M b ) )
2524fveq2d 5695 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  ( M `  ( M `  A ) )  =  ( M `
 ( a M b ) ) )
26 id 22 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  A  =  <. a ,  b >. )
2725, 26eqeq12d 2457 . . . 4  |-  ( A  =  <. a ,  b
>.  ->  ( ( M `
 ( M `  A ) )  =  A  <->  ( M `  ( a M b ) )  =  <. a ,  b >. )
)
2821, 27syl5ibrcom 222 . . 3  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( A  =  <. a ,  b >.  ->  ( M `  ( M `  A ) )  =  A ) )
2928rexlimivv 2846 . 2  |-  ( E. a  e.  I  E. b  e.  2o  A  =  <. a ,  b
>.  ->  ( M `  ( M `  A ) )  =  A )
301, 29sylbi 195 1  |-  ( A  e.  ( I  X.  2o )  ->  ( M `
 ( M `  A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716    \ cdif 3325    C_ wss 3328   <.cop 3883   Tr wtr 4385   Ord word 4718   suc csuc 4721    X. cxp 4838   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1oc1o 6913   2oc2o 6914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1o 6920  df-2o 6921
This theorem is referenced by:  efginvrel1  16225  efgredlemc  16242
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