Theorem grpinvpropd 17490

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

Step	Hyp	Ref	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 ) )
4	2, 3, 1	grpidpropd 17261	. . . . . . . 8 |- ( ph -> ( 0g ` K ) = ( 0g ` L ) )
5	4	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( 0g ` K ) = ( 0g ` L ) )
6	1, 5	eqeq12d 2637	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( ( x ( +g ` K ) y ) = ( 0g ` K ) <-> ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
7	6	anass1rs 849	. . . . 5 |- ( ( ( ph /\ y e. B ) /\ x e. B ) -> ( ( x ( +g ` K ) y ) = ( 0g ` K ) <-> ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
8	7	riotabidva 6627	. . . 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 ) ) )
9	8	mpteq2dva 4744	. . 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 ) ) ) )
10	2	riotaeqdv 6612	. . . 4 |- ( ph -> ( iota_ x e. B ( x ( +g ` K ) y ) = ( 0g ` K ) ) = ( iota_ x e. ( Base ` K ) ( x ( +g ` K ) y ) = ( 0g ` K ) ) )
11	2, 10	mpteq12dv 4733	. . 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 ) ) ) )
12	3	riotaeqdv 6612	. . . 4 |- ( ph -> ( iota_ x e. B ( x ( +g ` L ) y ) = ( 0g ` L ) ) = ( iota_ x e. ( Base ` L ) ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
13	3, 12	mpteq12dv 4733	. . 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 ) ) ) )
14	9, 11, 13	3eqtr3d 2664	. 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 2622	. . 3 |- ( Base ` K ) = ( Base ` K )
16		eqid 2622	. . 3 |- ( +g ` K ) = ( +g ` K )
17		eqid 2622	. . 3 |- ( 0g ` K ) = ( 0g ` K )
18		eqid 2622	. . 3 |- ( invg ` K ) = ( invg ` K )
19	15, 16, 17, 18	grpinvfval 17460	. 2 |- ( invg ` K ) = ( y e. ( Base ` K ) |-> ( iota_ x e. ( Base ` K ) ( x ( +g ` K ) y ) = ( 0g ` K ) ) )
20		eqid 2622	. . 3 |- ( Base ` L ) = ( Base ` L )
21		eqid 2622	. . 3 |- ( +g ` L ) = ( +g ` L )
22		eqid 2622	. . 3 |- ( 0g ` L ) = ( 0g ` L )
23		eqid 2622	. . 3 |- ( invg ` L ) = ( invg ` L )
24	20, 21, 22, 23	grpinvfval 17460	. 2 |- ( invg ` L ) = ( y e. ( Base ` L ) |-> ( iota_ x e. ( Base ` L ) ( x ( +g ` L ) y ) = ( 0g ` L ) ) )
25	14, 19, 24	3eqtr4g 2681	1 |- ( ph -> ( invg ` K ) = ( invg ` L ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   invgcminusg 17423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-minusg 17426

This theorem is referenced by:  grpsubpropd  17520  grpsubpropd2  17521  mulgpropd  17584  invrpropd  18698  rlmvneg  19207  matinvg  20224  tngngp3  22460