Theorem mulgpropdg 13234

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 3173	. . . . . . . . . . 11 |- ( B C_ K -> ( x e. B -> x e. K ) )
7		ssel 3173	. . . . . . . . . . 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 12957	. . . . . 6 |- ( ph -> ( 0g ` G ) = ( 0g ` H ) )
14	13	3ad2ant1 1020	. . . . 5 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( 0g ` G ) = ( 0g ` H ) )
15		1zzd 9344	. . . . . . . 8 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> 1 e. ZZ )
16		nnuz 9628	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
17	5	3ad2ant1 1020	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> B C_ K )
18		simp3 1001	. . . . . . . . . 10 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> b e. B )
19	17, 18	sseldd 3180	. . . . . . . . 9 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> b e. K )
20	16, 19	ialgrlemconst 12181	. . . . . . . 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 1161	. . . . . . . 8 |- ( ( ( ph /\ a e. ZZ /\ b e. B ) /\ ( x e. K /\ y e. K ) ) -> ( x ( +g ` G ) y ) e. K )
23	11	3ad2antl1 1161	. . . . . . . 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 10600	. . . . . . 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 5556	. . . . . 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 13147	. . . . . . . 8 |- ( ph -> ( invg ` G ) = ( invg ` H ) )
27	26	3ad2ant1 1020	. . . . . . 7 |- ( ( ph /\ a e. ZZ /\ b e. B ) -> ( invg ` G ) = ( invg ` H ) )
28	24	fveq1d 5556	. . . . . . 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 5561	. . . . . 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 3576	. . . . 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 3576	. . . 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 5982	. . 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 2194	. . . 4 |- ( ph -> ZZ = ZZ )
34		eqidd 2194	. . . 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 5980	. . 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 2194	. . . 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 5980	. . 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 2234	. 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 2193	. . . . 5 |- ( Base ` G ) = ( Base ` G )
41		eqid 2193	. . . . 5 |- ( +g ` G ) = ( +g ` G )
42		eqid 2193	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
43		eqid 2193	. . . . 5 |- ( invg ` G ) = ( invg ` G )
44		eqid 2193	. . . . 5 |- (.g ` G ) = (.g ` G )
45	40, 41, 42, 43, 44	mulgfvalg 13191	. . . 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 2226	. 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 2193	. . . . 5 |- ( Base ` H ) = ( Base ` H )
50		eqid 2193	. . . . 5 |- ( +g ` H ) = ( +g ` H )
51		eqid 2193	. . . . 5 |- ( 0g ` H ) = ( 0g ` H )
52		eqid 2193	. . . . 5 |- ( invg ` H ) = ( invg ` H )
53		eqid 2193	. . . . 5 |- (.g ` H ) = (.g ` H )
54	49, 50, 51, 52, 53	mulgfvalg 13191	. . . 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 2226	. 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 2236	1 |- ( ph -> .x. = .X. )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   C_ wss 3153   ifcif 3557   {csn 3618   class class class wbr 4029   X. cxp 4657   ` cfv 5254  (class class class)co 5918   e. cmpo 5920   0cc0 7872   1c1 7873   < clt 8054   -ucneg 8191   NNcn 8982   ZZcz 9317   seqcseq 10518   Basecbs 12618   +g cplusg 12695   0gc0g 12867   invgcminusg 13073  ._gcmg 13189

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-0g 12869  df-minusg 13076  df-mulg 13190

This theorem is referenced by:  mulgass3  13581