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

Theorem grpinvpropd 15962
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 15799 . . . . . . . 8  |-  ( ph  ->  ( 0g `  K
)  =  ( 0g
`  L ) )
54adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( 0g `  K
)  =  ( 0g
`  L ) )
61, 5eqeq12d 2489 . . . . . 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 6272 . . . 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 4538 . . 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 6256 . . . 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 4530 . . 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 6256 . . . 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 4530 . . 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 2516 . 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 2467 . . 3  |-  ( Base `  K )  =  (
Base `  K )
16 eqid 2467 . . 3  |-  ( +g  `  K )  =  ( +g  `  K )
17 eqid 2467 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
18 eqid 2467 . . 3  |-  ( invg `  K )  =  ( invg `  K )
1915, 16, 17, 18grpinvfval 15937 . 2  |-  ( invg `  K )  =  ( y  e.  ( Base `  K
)  |->  ( iota_ x  e.  ( Base `  K
) ( x ( +g  `  K ) y )  =  ( 0g `  K ) ) )
20 eqid 2467 . . 3  |-  ( Base `  L )  =  (
Base `  L )
21 eqid 2467 . . 3  |-  ( +g  `  L )  =  ( +g  `  L )
22 eqid 2467 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
23 eqid 2467 . . 3  |-  ( invg `  L )  =  ( invg `  L )
2420, 21, 22, 23grpinvfval 15937 . 2  |-  ( invg `  L )  =  ( y  e.  ( Base `  L
)  |->  ( iota_ x  e.  ( Base `  L
) ( x ( +g  `  L ) y )  =  ( 0g `  L ) ) )
2514, 19, 243eqtr4g 2533 1  |-  ( ph  ->  ( invg `  K )  =  ( invg `  L
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4510   ` cfv 5593   iota_crio 6254  (class class class)co 6294   Basecbs 14502   +g cplusg 14567   0gc0g 14707   invgcminusg 15903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-0g 14709  df-minusg 15907
This theorem is referenced by:  grpsubpropd  15989  grpsubpropd2  15990  mulgpropd  16024  invrpropd  17196  rlmvneg  17701  matinvg  18766
  Copyright terms: Public domain W3C validator