Theorem mulgpropdg 13750

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 3221	. . . . . . . . . . 11 |- ( B C_ K -> ( x e. B -> x e. K ) )
7		ssel 3221	. . . . . . . . . . 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 13456	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
14	13	3ad2ant1 1044	. . . . 5 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( 0g ` G ) = ( 0g ` H ) )
15		1zzd 9505	. . . . . . . 8 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> 1 e. ZZ )
16		nnuz 9791	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
17	5	3ad2ant1 1044	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> B C_ K )
18		simp3 1025	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> b e. B )
19	17, 18	sseldd 3228	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> b e. K )
20	16, 19	ialgrlemconst 12614	. . . . . . . 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 1185	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ /\ b e. B ) /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) e. K )
23	11	3ad2antl1 1185	. . . . . . . 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 10790	. . . . . . 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 5641	. . . . . 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 13657	. . . . . . . 8 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
27	26	3ad2ant1 1044	. . . . . . 7 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( invg ` G ) = ( invg ` H ) )
28	24	fveq1d 5641	. . . . . . 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 5646	. . . . . 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 3625	. . . . 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 3625	. . . 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 6084	. . 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 2232	. . . 4 |- ( ph -> ZZ = ZZ )
34		eqidd 2232	. . . 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 6082	. . 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 2232	. . . 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 6082	. . 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 2272	. 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 2231	. . . . 5 |- ( Base ` G ) = ( Base ` G )
41		eqid 2231	. . . . 5 |- ( +g ` G ) = ( +g ` G )
42		eqid 2231	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
43		eqid 2231	. . . . 5 |- ( invg ` G ) = ( invg ` G )
44		eqid 2231	. . . . 5 |- (.g ` G ) = (.g ` G )
45	40, 41, 42, 43, 44	mulgfvalg 13707	. . . 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 2264	. 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 2231	. . . . 5 |- ( Base ` H ) = ( Base ` H )
50		eqid 2231	. . . . 5 |- ( +g ` H ) = ( +g ` H )
51		eqid 2231	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
52		eqid 2231	. . . . 5 |- ( invg ` H ) = ( invg ` H )
53		eqid 2231	. . . . 5 |- (.g ` H ) = (.g ` H )
54	49, 50, 51, 52, 53	mulgfvalg 13707	. . . 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 2264	. 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 2274	1 |- ( ph -> .x. = .X. )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   C_ wss 3200   ifcif 3605   {csn 3669   class class class wbr 4088   X. cxp 4723   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   0cc0 8031   1c1 8032   < clt 8213   -ucneg 8350   NNcn 9142   ZZcz 9478   seqcseq 10708   Basecbs 13081   +g cplusg 13159   0gc0g 13338   invgcminusg 13583  ._gcmg 13705

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-seqfrec 10709  df-ndx 13084  df-slot 13085  df-base 13087  df-0g 13340  df-minusg 13586  df-mulg 13706

This theorem is referenced by:  mulgass3  14097