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Theorem grpinvpropd 14793
Description: If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.)
Hypotheses
Ref Expression
grpinvpropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
grpinvpropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
grpinvpropd.3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
Assertion
Ref Expression
grpinvpropd  |-  ( ph  ->  ( inv g `  K )  =  ( inv g `  L
) )
Distinct variable groups:    x, y, B    x, K, y    x, L, y    ph, x, y

Proof of Theorem grpinvpropd
StepHypRef Expression
1 grpinvpropd.3 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
2 grpinvpropd.1 . . . . . . . . 9  |-  ( ph  ->  B  =  ( Base `  K ) )
3 grpinvpropd.2 . . . . . . . . 9  |-  ( ph  ->  B  =  ( Base `  L ) )
42, 3, 1grpidpropd 14649 . . . . . . . 8  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
54adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( 0g `  K
)  =  ( 0g
`  L ) )
61, 5eqeq12d 2401 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x ( +g  `  K ) y )  =  ( 0g `  K )  <-> 
( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
76anass1rs 783 . . . . 5  |-  ( ( ( ph  /\  y  e.  B )  /\  x  e.  B )  ->  (
( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  ( x
( +g  `  L ) y )  =  ( 0g `  L ) ) )
87riotabidva 6502 . . . 4  |-  ( (
ph  /\  y  e.  B )  ->  ( iota_ x  e.  B ( x ( +g  `  K
) y )  =  ( 0g `  K
) )  =  (
iota_ x  e.  B
( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
98mpteq2dva 4236 . . 3  |-  ( ph  ->  ( y  e.  B  |->  ( iota_ x  e.  B
( x ( +g  `  K ) y )  =  ( 0g `  K ) ) )  =  ( y  e.  B  |->  ( iota_ x  e.  B ( x ( +g  `  L ) y )  =  ( 0g `  L ) ) ) )
102riotaeqdv 6486 . . . 4  |-  ( ph  ->  ( iota_ x  e.  B
( x ( +g  `  K ) y )  =  ( 0g `  K ) )  =  ( iota_ x  e.  (
Base `  K )
( x ( +g  `  K ) y )  =  ( 0g `  K ) ) )
112, 10mpteq12dv 4228 . . 3  |-  ( ph  ->  ( y  e.  B  |->  ( iota_ x  e.  B
( x ( +g  `  K ) y )  =  ( 0g `  K ) ) )  =  ( y  e.  ( Base `  K
)  |->  ( iota_ x  e.  ( Base `  K
) ( x ( +g  `  K ) y )  =  ( 0g `  K ) ) ) )
123riotaeqdv 6486 . . . 4  |-  ( ph  ->  ( iota_ x  e.  B
( x ( +g  `  L ) y )  =  ( 0g `  L ) )  =  ( iota_ x  e.  (
Base `  L )
( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
133, 12mpteq12dv 4228 . . 3  |-  ( ph  ->  ( y  e.  B  |->  ( iota_ x  e.  B
( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )  =  ( y  e.  ( Base `  L
)  |->  ( iota_ x  e.  ( Base `  L
) ( x ( +g  `  L ) y )  =  ( 0g `  L ) ) ) )
149, 11, 133eqtr3d 2427 . 2  |-  ( ph  ->  ( y  e.  (
Base `  K )  |->  ( iota_ x  e.  (
Base `  K )
( x ( +g  `  K ) y )  =  ( 0g `  K ) ) )  =  ( y  e.  ( Base `  L
)  |->  ( iota_ x  e.  ( Base `  L
) ( x ( +g  `  L ) y )  =  ( 0g `  L ) ) ) )
15 eqid 2387 . . 3  |-  ( Base `  K )  =  (
Base `  K )
16 eqid 2387 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
17 eqid 2387 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
18 eqid 2387 . . 3  |-  ( inv g `  K )  =  ( inv g `  K )
1915, 16, 17, 18grpinvfval 14770 . 2  |-  ( inv g `  K )  =  ( y  e.  ( Base `  K
)  |->  ( iota_ x  e.  ( Base `  K
) ( x ( +g  `  K ) y )  =  ( 0g `  K ) ) )
20 eqid 2387 . . 3  |-  ( Base `  L )  =  (
Base `  L )
21 eqid 2387 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
22 eqid 2387 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
23 eqid 2387 . . 3  |-  ( inv g `  L )  =  ( inv g `  L )
2420, 21, 22, 23grpinvfval 14770 . 2  |-  ( inv g `  L )  =  ( y  e.  ( Base `  L
)  |->  ( iota_ x  e.  ( Base `  L
) ( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
2514, 19, 243eqtr4g 2444 1  |-  ( ph  ->  ( inv g `  K )  =  ( inv g `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   iota_crio 6478   Basecbs 13396   +g cplusg 13456   0gc0g 13650   inv gcminusg 14613
This theorem is referenced by:  grpsubpropd  14816  grpsubpropd2  14817  mulgpropd  14850  invrpropd  15730  rlmvneg  16205  matinvg  27142
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-riota 6485  df-0g 13654  df-minusg 14740
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