Theorem mulgpropdg 13696

Description: Two structures with the same group-nature have the same group multiple function. K is expected to either be _V (when strong equality is available) or B (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
mulgpropdg.m	|- ( ph -> .x. = (.g ` G ) )
mulgpropdg.n	|- ( ph -> .X. = (.g ` H ) )
mulgpropdg.g	|- ( ph -> G e. V )
mulgpropdg.h	|- ( ph -> H e. W )
mulgpropd.b1	|- ( ph -> B = ( Base ` G ) )
mulgpropd.b2	|- ( ph -> B = ( Base ` H ) )
mulgpropd.i	|- ( ph -> B C_ K )
mulgpropd.k	|- ( ( ph /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) e. K )
mulgpropd.e	|- ( ( ph /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )

Assertion

Ref	Expression
mulgpropdg	|- ( ph -> .x. = .X. )

Distinct variable groups:   ph, x, y   x, B, y   x, G, y   x, H, y   x, K, y

Allowed substitution hints:   .x. ( x, y)   .X. ( x, y)   V( x, y)   W( x, y)

Proof of Theorem mulgpropdg

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulgpropd.b1	. . . . . . 7 |- ( ph -> B = ( Base ` G ) )
2		mulgpropd.b2	. . . . . . 7 |- ( ph -> B = ( Base ` H ) )
3		mulgpropdg.g	. . . . . . 7 |- ( ph -> G e. V )
4		mulgpropdg.h	. . . . . . 7 |- ( ph -> H e. W )
5		mulgpropd.i	. . . . . . . . . 10 |- ( ph -> B C_ K )
6		ssel 3218	. . . . . . . . . . 11 |- ( B C_ K -> ( x e. B -> x e. K ) )
7		ssel 3218	. . . . . . . . . . 11 |- ( B C_ K -> ( y e. B -> y e. K ) )
8	6, 7	anim12d 335	. . . . . . . . . 10 |- ( B C_ K -> ( ( x e. B /\ y e. B ) -> ( x e. K /\ y e. K ) ) )
9	5, 8	syl 14	. . . . . . . . 9 |- ( ph -> ( ( x e. B /\ y e. B ) -> ( x e. K /\ y e. K ) ) )
10	9	imp 124	. . . . . . . 8 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x e. K /\ y e. K ) )
11		mulgpropd.e	. . . . . . . 8 |- ( ( ph /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )
12	10, 11	syldan 282	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )
13	1, 2, 3, 4, 12	grpidpropdg 13402	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
14	13	3ad2ant1 1042	. . . . 5 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( 0g ` G ) = ( 0g ` H ) )
15		1zzd 9469	. . . . . . . 8 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> 1 e. ZZ )
16		nnuz 9754	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
17	5	3ad2ant1 1042	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> B C_ K )
18		simp3 1023	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> b e. B )
19	17, 18	sseldd 3225	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> b e. K )
20	16, 19	ialgrlemconst 12560	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ /\ b e. B ) /\ x e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { b } ) ` x ) e. K )
21		mulgpropd.k	. . . . . . . . 9 |- ( ( ph /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) e. K )
22	21	3ad2antl1 1183	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ /\ b e. B ) /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) e. K )
23	11	3ad2antl1 1183	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ /\ b e. B ) /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) )
24	15, 20, 22, 23	seqfeq3 10746	. . . . . . 7 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) )
25	24	fveq1d 5628	. . . . . 6 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) = ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) )
26	1, 2, 3, 4, 12	grpinvpropdg 13603	. . . . . . . 8 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
27	26	3ad2ant1 1042	. . . . . . 7 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( invg ` G ) = ( invg ` H ) )
28	24	fveq1d 5628	. . . . . . 7 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) = ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) )
29	27, 28	fveq12d 5633	. . . . . 6 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) )
30	25, 29	ifeq12d 3622	. . . . 5 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> if ( 0 < a , ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) ) = if ( 0 < a , ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) ) )
31	14, 30	ifeq12d 3622	. . . 4 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> if ( a = 0 , ( 0g ` G ) , if ( 0 < a , ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) ) ) = if ( a = 0 , ( 0g ` H ) , if ( 0 < a , ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) ) ) )
32	31	mpoeq3dva 6067	. . 3 |- ( ph -> ( a e. ZZ , b e. B |-> if ( a = 0 , ( 0g ` G ) , if ( 0 < a , ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) = ( a e. ZZ , b e. B |-> if ( a = 0 , ( 0g ` H ) , if ( 0 < a , ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) )
33		eqidd 2230	. . . 4 |- ( ph -> ZZ = ZZ )
34		eqidd 2230	. . . 4 |- ( ph -> if ( a = 0 , ( 0g ` G ) , if ( 0 < a , ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) ) ) = if ( a = 0 , ( 0g ` G ) , if ( 0 < a , ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) ) ) )
35	33, 1, 34	mpoeq123dv 6065	. . 3 |- ( ph -> ( a e. ZZ , b e. B |-> if ( a = 0 , ( 0g ` G ) , if ( 0 < a , ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) = ( a e. ZZ , b e. ( Base ` G ) |-> if ( a = 0 , ( 0g ` G ) , if ( 0 < a , ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) )
36		eqidd 2230	. . . 4 |- ( ph -> if ( a = 0 , ( 0g ` H ) , if ( 0 < a , ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) ) ) = if ( a = 0 , ( 0g ` H ) , if ( 0 < a , ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) ) ) )
37	33, 2, 36	mpoeq123dv 6065	. . 3 |- ( ph -> ( a e. ZZ , b e. B |-> if ( a = 0 , ( 0g ` H ) , if ( 0 < a , ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) = ( a e. ZZ , b e. ( Base ` H ) |-> if ( a = 0 , ( 0g ` H ) , if ( 0 < a , ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) )
38	32, 35, 37	3eqtr3d 2270	. 2 |- ( ph -> ( a e. ZZ , b e. ( Base ` G ) |-> if ( a = 0 , ( 0g ` G ) , if ( 0 < a , ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) = ( a e. ZZ , b e. ( Base ` H ) |-> if ( a = 0 , ( 0g ` H ) , if ( 0 < a , ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) )
39		mulgpropdg.m	. . 3 |- ( ph -> .x. = (.g ` G ) )
40		eqid 2229	. . . . 5 |- ( Base ` G ) = ( Base ` G )
41		eqid 2229	. . . . 5 |- ( +g ` G ) = ( +g ` G )
42		eqid 2229	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
43		eqid 2229	. . . . 5 |- ( invg ` G ) = ( invg ` G )
44		eqid 2229	. . . . 5 |- (.g ` G ) = (.g ` G )
45	40, 41, 42, 43, 44	mulgfvalg 13653	. . . 4 |- ( G e. V -> (.g ` G ) = ( a e. ZZ , b e. ( Base ` G ) |-> if ( a = 0 , ( 0g ` G ) , if ( 0 < a , ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) )
46	3, 45	syl 14	. . 3 |- ( ph -> (.g ` G ) = ( a e. ZZ , b e. ( Base ` G ) |-> if ( a = 0 , ( 0g ` G ) , if ( 0 < a , ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) )
47	39, 46	eqtrd 2262	. 2 |- ( ph -> .x. = ( a e. ZZ , b e. ( Base ` G ) |-> if ( a = 0 , ( 0g ` G ) , if ( 0 < a , ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) )
48		mulgpropdg.n	. . 3 |- ( ph -> .X. = (.g ` H ) )
49		eqid 2229	. . . . 5 |- ( Base ` H ) = ( Base ` H )
50		eqid 2229	. . . . 5 |- ( +g ` H ) = ( +g ` H )
51		eqid 2229	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
52		eqid 2229	. . . . 5 |- ( invg ` H ) = ( invg ` H )
53		eqid 2229	. . . . 5 |- (.g ` H ) = (.g ` H )
54	49, 50, 51, 52, 53	mulgfvalg 13653	. . . 4 |- ( H e. W -> (.g ` H ) = ( a e. ZZ , b e. ( Base ` H ) |-> if ( a = 0 , ( 0g ` H ) , if ( 0 < a , ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) )
55	4, 54	syl 14	. . 3 |- ( ph -> (.g ` H ) = ( a e. ZZ , b e. ( Base ` H ) |-> if ( a = 0 , ( 0g ` H ) , if ( 0 < a , ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) )
56	48, 55	eqtrd 2262	. 2 |- ( ph -> .X. = ( a e. ZZ , b e. ( Base ` H ) |-> if ( a = 0 , ( 0g ` H ) , if ( 0 < a , ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` a ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { b } ) ) ` -u a ) ) ) ) ) )
57	38, 47, 56	3eqtr4d 2272	1 |- ( ph -> .x. = .X. )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   C_ wss 3197   ifcif 3602   {csn 3666   class class class wbr 4082   X. cxp 4716   ` cfv 5317  (class class class)co 6000   e. cmpo 6002   0cc0 7995   1c1 7996   < clt 8177   -ucneg 8314   NNcn 9106   ZZcz 9442   seqcseq 10664   Basecbs 13027   +g cplusg 13105   0gc0g 13284   invgcminusg 13529  ._gcmg 13651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-seqfrec 10665  df-ndx 13030  df-slot 13031  df-base 13033  df-0g 13286  df-minusg 13532  df-mulg 13652

This theorem is referenced by:  mulgass3  14043