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Theorem grpinvpropd 14559
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 14415 . . . . . . . 8  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
54adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( 0g `  K
)  =  ( 0g
`  L ) )
61, 5eqeq12d 2310 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x ( +g  `  K ) y )  =  ( 0g `  K )  <-> 
( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
76anass1rs 782 . . . . 5  |-  ( ( ( ph  /\  y  e.  B )  /\  x  e.  B )  ->  (
( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  ( x
( +g  `  L ) y )  =  ( 0g `  L ) ) )
87riotabidva 6337 . . . 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 4122 . . 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 6321 . . . 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 4114 . . 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 6321 . . . 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 4114 . . 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 2336 . 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 2296 . . 3  |-  ( Base `  K )  =  (
Base `  K )
16 eqid 2296 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
17 eqid 2296 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
18 eqid 2296 . . 3  |-  ( inv g `  K )  =  ( inv g `  K )
1915, 16, 17, 18grpinvfval 14536 . 2  |-  ( inv g `  K )  =  ( y  e.  ( Base `  K
)  |->  ( iota_ x  e.  ( Base `  K
) ( x ( +g  `  K ) y )  =  ( 0g `  K ) ) )
20 eqid 2296 . . 3  |-  ( Base `  L )  =  (
Base `  L )
21 eqid 2296 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
22 eqid 2296 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
23 eqid 2296 . . 3  |-  ( inv g `  L )  =  ( inv g `  L )
2420, 21, 22, 23grpinvfval 14536 . 2  |-  ( inv g `  L )  =  ( y  e.  ( Base `  L
)  |->  ( iota_ x  e.  ( Base `  L
) ( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
2514, 19, 243eqtr4g 2353 1  |-  ( ph  ->  ( inv g `  K )  =  ( inv g `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   +g cplusg 13224   0gc0g 13416   inv gcminusg 14379
This theorem is referenced by:  grpsubpropd  14582  grpsubpropd2  14583  mulgpropd  14616  invrpropd  15496  rlmvneg  15975  matinvg  27576
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-minusg 14506
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