Theorem mulgpropdg 13035

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 3151	. . . . . . . . . . 11 |- ( B C_ K -> ( x e. B -> x e. K ) )
7		ssel 3151	. . . . . . . . . . 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 12800	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
14	13	3ad2ant1 1018	. . . . 5 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( 0g ` G ) = ( 0g ` H ) )
15		1zzd 9283	. . . . . . . 8 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> 1 e. ZZ )
16		nnuz 9566	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
17	5	3ad2ant1 1018	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> B C_ K )
18		simp3 999	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> b e. B )
19	17, 18	sseldd 3158	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> b e. K )
20	16, 19	ialgrlemconst 12046	. . . . . . . 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 1159	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ /\ b e. B ) /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) e. K )
23	11	3ad2antl1 1159	. . . . . . . 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 10515	. . . . . . 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 5519	. . . . . 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 12952	. . . . . . . 8 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
27	26	3ad2ant1 1018	. . . . . . 7 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( invg ` G ) = ( invg ` H ) )
28	24	fveq1d 5519	. . . . . . 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 5524	. . . . . 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 3555	. . . . 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 3555	. . . 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 5942	. . 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 2178	. . . 4 |- ( ph -> ZZ = ZZ )
34		eqidd 2178	. . . 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 5940	. . 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 2178	. . . 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 5940	. . 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 2218	. 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 2177	. . . . 5 |- ( Base ` G ) = ( Base ` G )
41		eqid 2177	. . . . 5 |- ( +g ` G ) = ( +g ` G )
42		eqid 2177	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
43		eqid 2177	. . . . 5 |- ( invg ` G ) = ( invg ` G )
44		eqid 2177	. . . . 5 |- (.g ` G ) = (.g ` G )
45	40, 41, 42, 43, 44	mulgfvalg 12994	. . . 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 2210	. 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 2177	. . . . 5 |- ( Base ` H ) = ( Base ` H )
50		eqid 2177	. . . . 5 |- ( +g ` H ) = ( +g ` H )
51		eqid 2177	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
52		eqid 2177	. . . . 5 |- ( invg ` H ) = ( invg ` H )
53		eqid 2177	. . . . 5 |- (.g ` H ) = (.g ` H )
54	49, 50, 51, 52, 53	mulgfvalg 12994	. . . 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 2210	. 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 2220	1 |- ( ph -> .x. = .X. )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   C_ wss 3131   ifcif 3536   {csn 3594   class class class wbr 4005   X. cxp 4626   ` cfv 5218  (class class class)co 5878   e. cmpo 5880   0cc0 7814   1c1 7815   < clt 7995   -ucneg 8132   NNcn 8922   ZZcz 9256   seqcseq 10448   Basecbs 12465   +g cplusg 12539   0gc0g 12711   invgcminusg 12885  ._gcmg 12992

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-ltadd 7930

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-inn 8923  df-n0 9180  df-z 9257  df-uz 9532  df-seqfrec 10449  df-ndx 12468  df-slot 12469  df-base 12471  df-0g 12713  df-minusg 12888  df-mulg 12993

This theorem is referenced by:  mulgass3  13265