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Theorem grpinvpropd 15601
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  ->  ( invg `  K )  =  ( invg `  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 15447 . . . . . . . 8  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
54adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( 0g `  K
)  =  ( 0g
`  L ) )
61, 5eqeq12d 2457 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( ( x ( +g  `  K ) y )  =  ( 0g `  K )  <-> 
( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
76anass1rs 805 . . . . 5  |-  ( ( ( ph  /\  y  e.  B )  /\  x  e.  B )  ->  (
( x ( +g  `  K ) y )  =  ( 0g `  K )  <->  ( x
( +g  `  L ) y )  =  ( 0g `  L ) ) )
87riotabidva 6069 . . . 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 4378 . . 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 6053 . . . 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 4370 . . 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 6053 . . . 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 4370 . . 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 2483 . 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 2443 . . 3  |-  ( Base `  K )  =  (
Base `  K )
16 eqid 2443 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
17 eqid 2443 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
18 eqid 2443 . . 3  |-  ( invg `  K )  =  ( invg `  K )
1915, 16, 17, 18grpinvfval 15576 . 2  |-  ( invg `  K )  =  ( y  e.  ( Base `  K
)  |->  ( iota_ x  e.  ( Base `  K
) ( x ( +g  `  K ) y )  =  ( 0g `  K ) ) )
20 eqid 2443 . . 3  |-  ( Base `  L )  =  (
Base `  L )
21 eqid 2443 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
22 eqid 2443 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
23 eqid 2443 . . 3  |-  ( invg `  L )  =  ( invg `  L )
2420, 21, 22, 23grpinvfval 15576 . 2  |-  ( invg `  L )  =  ( y  e.  ( Base `  L
)  |->  ( iota_ x  e.  ( Base `  L
) ( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
2514, 19, 243eqtr4g 2500 1  |-  ( ph  ->  ( invg `  K )  =  ( invg `  L
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    e. cmpt 4350   ` cfv 5418   iota_crio 6051  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   0gc0g 14378   invgcminusg 15411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-0g 14380  df-minusg 15546
This theorem is referenced by:  grpsubpropd  15626  grpsubpropd2  15627  mulgpropd  15660  invrpropd  16790  rlmvneg  17288  matinvg  18319
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