Theorem mulgpropdg 13881

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 3232	. . . . . . . . . . 11 |- ( B C_ K -> ( x e. B -> x e. K ) )
7		ssel 3232	. . . . . . . . . . 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 13587	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
14	13	3ad2ant1 1045	. . . . 5 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( 0g ` G ) = ( 0g ` H ) )
15		1zzd 9604	. . . . . . . 8 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> 1 e. ZZ )
16		nnuz 9890	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
17	5	3ad2ant1 1045	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> B C_ K )
18		simp3 1026	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> b e. B )
19	17, 18	sseldd 3239	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> b e. K )
20	16, 19	ialgrlemconst 12740	. . . . . . . 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 1186	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ /\ b e. B ) /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) e. K )
23	11	3ad2antl1 1186	. . . . . . . 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 10891	. . . . . . 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 5672	. . . . . 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 13788	. . . . . . . 8 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
27	26	3ad2ant1 1045	. . . . . . 7 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( invg ` G ) = ( invg ` H ) )
28	24	fveq1d 5672	. . . . . . 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 5677	. . . . . 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 3642	. . . . 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 3642	. . . 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 6117	. . 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 2233	. . . 4 |- ( ph -> ZZ = ZZ )
34		eqidd 2233	. . . 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 6115	. . 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 2233	. . . 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 6115	. . 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 2273	. 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 2232	. . . . 5 |- ( Base ` G ) = ( Base ` G )
41		eqid 2232	. . . . 5 |- ( +g ` G ) = ( +g ` G )
42		eqid 2232	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
43		eqid 2232	. . . . 5 |- ( invg ` G ) = ( invg ` G )
44		eqid 2232	. . . . 5 |- (.g ` G ) = (.g ` G )
45	40, 41, 42, 43, 44	mulgfvalg 13838	. . . 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 2265	. 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 2232	. . . . 5 |- ( Base ` H ) = ( Base ` H )
50		eqid 2232	. . . . 5 |- ( +g ` H ) = ( +g ` H )
51		eqid 2232	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
52		eqid 2232	. . . . 5 |- ( invg ` H ) = ( invg ` H )
53		eqid 2232	. . . . 5 |- (.g ` H ) = (.g ` H )
54	49, 50, 51, 52, 53	mulgfvalg 13838	. . . 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 2265	. 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 2275	1 |- ( ph -> .x. = .X. )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   C_ wss 3211   ifcif 3620   {csn 3689   class class class wbr 4109   X. cxp 4747   ` cfv 5352  (class class class)co 6050   e. cmpo 6052   0cc0 8127   1c1 8128   < clt 8308   -ucneg 8445   NNcn 9237   ZZcz 9577   seqcseq 10809   Basecbs 13212   +g cplusg 13290   0gc0g 13469   invgcminusg 13714  ._gcmg 13836

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-seqfrec 10810  df-ndx 13215  df-slot 13216  df-base 13218  df-0g 13471  df-minusg 13717  df-mulg 13837

This theorem is referenced by:  mulgass3  14229