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Theorem efgmnvl 16605
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 5023 . 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 16603 . . . . . . 7  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
43fveq2d 5876 . . . . . 6  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  ( M `
 <. a ,  ( 1o  \  b )
>. ) )
5 df-ov 6298 . . . . . 6  |-  ( a M ( 1o  \ 
b ) )  =  ( M `  <. a ,  ( 1o  \ 
b ) >. )
64, 5syl6eqr 2526 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  ( a M ( 1o  \ 
b ) ) )
7 2oconcl 7165 . . . . . 6  |-  ( b  e.  2o  ->  ( 1o  \  b )  e.  2o )
82efgmval 16603 . . . . . 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 7149 . . . . . . . . . . 11  |-  1o  e.  On
1110onordi 4988 . . . . . . . . . 10  |-  Ord  1o
12 ordtr 4898 . . . . . . . . . 10  |-  ( Ord 
1o  ->  Tr  1o )
13 trsucss 4969 . . . . . . . . . 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 7143 . . . . . . . . 9  |-  2o  =  suc  1o
1614, 15eleq2s 2575 . . . . . . . 8  |-  ( b  e.  2o  ->  b  C_  1o )
1716adantl 466 . . . . . . 7  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  b  C_  1o )
18 dfss4 3737 . . . . . . 7  |-  ( b 
C_  1o  <->  ( 1o  \ 
( 1o  \  b
) )  =  b )
1917, 18sylib 196 . . . . . 6  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( 1o  \  ( 1o  \  b ) )  =  b )
2019opeq2d 4226 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  ( 1o 
\  ( 1o  \ 
b ) ) >.  =  <. a ,  b
>. )
216, 9, 203eqtrd 2512 . . . 4  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  <. a ,  b >. )
22 fveq2 5872 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( M `  <. a ,  b >. )
)
23 df-ov 6298 . . . . . . 7  |-  ( a M b )  =  ( M `  <. a ,  b >. )
2422, 23syl6eqr 2526 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( a M b ) )
2524fveq2d 5876 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  ( M `  ( M `  A ) )  =  ( M `
 ( a M b ) ) )
26 id 22 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  A  =  <. a ,  b >. )
2725, 26eqeq12d 2489 . . . 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 2964 . 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 1379    e. wcel 1767   E.wrex 2818    \ cdif 3478    C_ wss 3481   <.cop 4039   Tr wtr 4546   Ord word 4883   suc csuc 4886    X. cxp 5003   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1oc1o 7135   2oc2o 7136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1o 7142  df-2o 7143
This theorem is referenced by:  efginvrel1  16619  efgredlemc  16636
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