MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgmnvl Unicode version

Theorem efgmnvl 15039
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 4723 . 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 15037 . . . . . . 7  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( a M b )  =  <. a ,  ( 1o  \ 
b ) >. )
43fveq2d 5545 . . . . . 6  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  ( M `
 <. a ,  ( 1o  \  b )
>. ) )
5 df-ov 5877 . . . . . 6  |-  ( a M ( 1o  \ 
b ) )  =  ( M `  <. a ,  ( 1o  \ 
b ) >. )
64, 5syl6eqr 2346 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  ( a M ( 1o  \ 
b ) ) )
7 2oconcl 6518 . . . . . 6  |-  ( b  e.  2o  ->  ( 1o  \  b )  e.  2o )
82efgmval 15037 . . . . . 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 6502 . . . . . . . . . . 11  |-  1o  e.  On
1110onordi 4513 . . . . . . . . . 10  |-  Ord  1o
12 ordtr 4422 . . . . . . . . . 10  |-  ( Ord 
1o  ->  Tr  1o )
13 trsucss 4494 . . . . . . . . . 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 6496 . . . . . . . . 9  |-  2o  =  suc  1o
1614, 15eleq2s 2388 . . . . . . . 8  |-  ( b  e.  2o  ->  b  C_  1o )
1716adantl 452 . . . . . . 7  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  b  C_  1o )
18 dfss4 3416 . . . . . . 7  |-  ( b 
C_  1o  <->  ( 1o  \ 
( 1o  \  b
) )  =  b )
1917, 18sylib 188 . . . . . 6  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( 1o  \  ( 1o  \  b ) )  =  b )
2019opeq2d 3819 . . . . 5  |-  ( ( a  e.  I  /\  b  e.  2o )  -> 
<. a ,  ( 1o 
\  ( 1o  \ 
b ) ) >.  =  <. a ,  b
>. )
216, 9, 203eqtrd 2332 . . . 4  |-  ( ( a  e.  I  /\  b  e.  2o )  ->  ( M `  (
a M b ) )  =  <. a ,  b >. )
22 fveq2 5541 . . . . . . 7  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( M `  <. a ,  b >. )
)
23 df-ov 5877 . . . . . . 7  |-  ( a M b )  =  ( M `  <. a ,  b >. )
2422, 23syl6eqr 2346 . . . . . 6  |-  ( A  =  <. a ,  b
>.  ->  ( M `  A )  =  ( a M b ) )
2524fveq2d 5545 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  ( M `  ( M `  A ) )  =  ( M `
 ( a M b ) ) )
26 id 19 . . . . 5  |-  ( A  =  <. a ,  b
>.  ->  A  =  <. a ,  b >. )
2725, 26eqeq12d 2310 . . . 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 2685 . 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 1632    e. wcel 1696   E.wrex 2557    \ cdif 3162    C_ wss 3165   <.cop 3656   Tr wtr 4129   Ord word 4407   suc csuc 4410    X. cxp 4703   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489
This theorem is referenced by:  efginvrel1  15053  efgredlemc  15070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1o 6495  df-2o 6496
  Copyright terms: Public domain W3C validator