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Theorem efgmnvl 17412
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 4870 . 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 17410 . . . . . . 7  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
43fveq2d 5891 . . . . . 6  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  ( M `
 <. a ,  ( 1o  \  b )
>. ) )
5 df-ov 6317 . . . . . 6  |-  ( a M ( 1o  \ 
b ) )  =  ( M `  <. a ,  ( 1o  \ 
b ) >. )
64, 5syl6eqr 2513 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  ( a M ( 1o  \ 
b ) ) )
7 2oconcl 7230 . . . . . 6  |-  ( b  e.  2o  ->  ( 1o  \  b )  e.  2o )
82efgmval 17410 . . . . . 6  |-  ( ( a  e.  I  /\  ( 1o  \  b
)  e.  2o )  ->  ( a M ( 1o  \  b
) )  =  <. a ,  ( 1o  \ 
( 1o  \  b
) ) >. )
97, 8sylan2 481 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M ( 1o  \  b ) )  =  <. a ,  ( 1o  \ 
( 1o  \  b
) ) >. )
10 1on 7214 . . . . . . . . . . 11  |-  1o  e.  On
1110onordi 5545 . . . . . . . . . 10  |-  Ord  1o
12 ordtr 5455 . . . . . . . . . 10  |-  ( Ord 
1o  ->  Tr  1o )
13 trsucss 5526 . . . . . . . . . 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 7208 . . . . . . . . 9  |-  2o  =  suc  1o
1614, 15eleq2s 2557 . . . . . . . 8  |-  ( b  e.  2o  ->  b  C_  1o )
1716adantl 472 . . . . . . 7  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  b  C_  1o )
18 dfss4 3688 . . . . . . 7  |-  ( b 
C_  1o  <->  ( 1o  \ 
( 1o  \  b
) )  =  b )
1917, 18sylib 201 . . . . . 6  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( 1o  \  ( 1o  \  b ) )  =  b )
2019opeq2d 4186 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  ( 1o 
\  ( 1o  \ 
b ) ) >.  =  <. a ,  b
>. )
216, 9, 203eqtrd 2499 . . . 4  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  <. a ,  b >. )
22 fveq2 5887 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( M `  <. a ,  b >. )
)
23 df-ov 6317 . . . . . . 7  |-  ( a M b )  =  ( M `  <. a ,  b >. )
2422, 23syl6eqr 2513 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( a M b ) )
2524fveq2d 5891 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  ( M `  ( M `  A ) )  =  ( M `
 ( a M b ) ) )
26 id 22 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  A  =  <. a ,  b >. )
2725, 26eqeq12d 2476 . . . 4  |-  ( A  =  <. a ,  b
>.  ->  ( ( M `
 ( M `  A ) )  =  A  <->  ( M `  ( a M b ) )  =  <. a ,  b >. )
)
2821, 27syl5ibrcom 230 . . 3  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( A  =  <. a ,  b >.  ->  ( M `  ( M `  A ) )  =  A ) )
2928rexlimivv 2895 . 2  |-  ( E. a  e.  I  E. b  e.  2o  A  =  <. a ,  b
>.  ->  ( M `  ( M `  A ) )  =  A )
301, 29sylbi 200 1  |-  ( A  e.  ( I  X.  2o )  ->  ( M `
 ( M `  A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897   E.wrex 2749    \ cdif 3412    C_ wss 3415   <.cop 3985   Tr wtr 4510    X. cxp 4850   Ord word 5440   suc csuc 5443   ` cfv 5600  (class class class)co 6314    |-> cmpt2 6316   1oc1o 7200   2oc2o 7201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-ord 5444  df-on 5445  df-suc 5447  df-iota 5564  df-fun 5602  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-1o 7207  df-2o 7208
This theorem is referenced by:  efginvrel1  17426  efgredlemc  17443
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