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Theorem grpinvpropd 14858
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 14714 . . . . . . . 8  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
54adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( 0g `  K
)  =  ( 0g
`  L ) )
61, 5eqeq12d 2449 . . . . . 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 6558 . . . 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 4287 . . 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 6542 . . . 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 4279 . . 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 6542 . . . 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 4279 . . 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 2475 . 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 2435 . . 3  |-  ( Base `  K )  =  (
Base `  K )
16 eqid 2435 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
17 eqid 2435 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
18 eqid 2435 . . 3  |-  ( inv g `  K )  =  ( inv g `  K )
1915, 16, 17, 18grpinvfval 14835 . 2  |-  ( inv g `  K )  =  ( y  e.  ( Base `  K
)  |->  ( iota_ x  e.  ( Base `  K
) ( x ( +g  `  K ) y )  =  ( 0g `  K ) ) )
20 eqid 2435 . . 3  |-  ( Base `  L )  =  (
Base `  L )
21 eqid 2435 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
22 eqid 2435 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
23 eqid 2435 . . 3  |-  ( inv g `  L )  =  ( inv g `  L )
2420, 21, 22, 23grpinvfval 14835 . 2  |-  ( inv g `  L )  =  ( y  e.  ( Base `  L
)  |->  ( iota_ x  e.  ( Base `  L
) ( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
2514, 19, 243eqtr4g 2492 1  |-  ( ph  ->  ( inv g `  K )  =  ( inv g `  L
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   iota_crio 6534   Basecbs 13461   +g cplusg 13521   0gc0g 13715   inv gcminusg 14678
This theorem is referenced by:  grpsubpropd  14881  grpsubpropd2  14882  mulgpropd  14915  invrpropd  15795  rlmvneg  16270  matinvg  27431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-riota 6541  df-0g 13719  df-minusg 14805
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