Theorem prdsmgp 18610

Description: The multiplicative monoid of a product is the product of the multiplicative monoids of the factors. (Contributed by Mario Carneiro, 11-Mar-2015.)

Hypotheses

Ref	Expression
prdsmgp.y	|- Y = ( S X_s R )
prdsmgp.m	|- M = (mulGrp ` Y )
prdsmgp.z	|- Z = ( S X_s (mulGrp o. R ) )
prdsmgp.i	|- ( ph -> I e. V )
prdsmgp.s	|- ( ph -> S e. W )
prdsmgp.r	|- ( ph -> R Fn I )

Assertion

Ref	Expression
prdsmgp	|- ( ph -> ( ( Base ` M ) = ( Base ` Z ) /\ ( +g ` M ) = ( +g ` Z ) ) )

Proof of Theorem prdsmgp

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- (mulGrp ` ( R ` x ) ) = (mulGrp ` ( R ` x ) )
2		eqid 2622	. . . . . 6 |- ( Base ` ( R ` x ) ) = ( Base ` ( R ` x ) )
3	1, 2	mgpbas 18495	. . . . 5 |- ( Base ` ( R ` x ) ) = ( Base ` (mulGrp ` ( R ` x ) ) )
4		prdsmgp.r	. . . . . . . 8 |- ( ph -> R Fn I )
5		fvco2 6273	. . . . . . . 8 |- ( ( R Fn I /\ x e. I ) -> ( (mulGrp o. R ) ` x ) = (mulGrp ` ( R ` x ) ) )
6	4, 5	sylan 488	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( (mulGrp o. R ) ` x ) = (mulGrp ` ( R ` x ) ) )
7	6	eqcomd 2628	. . . . . 6 |- ( ( ph /\ x e. I ) -> (mulGrp ` ( R ` x ) ) = ( (mulGrp o. R ) ` x ) )
8	7	fveq2d 6195	. . . . 5 |- ( ( ph /\ x e. I ) -> ( Base ` (mulGrp ` ( R ` x ) ) ) = ( Base ` ( (mulGrp o. R ) ` x ) ) )
9	3, 8	syl5eq 2668	. . . 4 |- ( ( ph /\ x e. I ) -> ( Base ` ( R ` x ) ) = ( Base ` ( (mulGrp o. R ) ` x ) ) )
10	9	ixpeq2dva 7923	. . 3 |- ( ph -> X_ x e. I ( Base ` ( R ` x ) ) = X_ x e. I ( Base ` ( (mulGrp o. R ) ` x ) ) )
11		prdsmgp.y	. . . 4 |- Y = ( S X_s R )
12		prdsmgp.m	. . . . . 6 |- M = (mulGrp ` Y )
13		eqid 2622	. . . . . 6 |- ( Base ` Y ) = ( Base ` Y )
14	12, 13	mgpbas 18495	. . . . 5 |- ( Base ` Y ) = ( Base ` M )
15	14	eqcomi 2631	. . . 4 |- ( Base ` M ) = ( Base ` Y )
16		prdsmgp.s	. . . 4 |- ( ph -> S e. W )
17		prdsmgp.i	. . . 4 |- ( ph -> I e. V )
18	11, 15, 16, 17, 4	prdsbas2 16129	. . 3 |- ( ph -> ( Base ` M ) = X_ x e. I ( Base ` ( R ` x ) ) )
19		prdsmgp.z	. . . 4 |- Z = ( S X_s (mulGrp o. R ) )
20		eqid 2622	. . . 4 |- ( Base ` Z ) = ( Base ` Z )
21		fnmgp 18491	. . . . . 6 |- mulGrp Fn _V
22	21	a1i 11	. . . . 5 |- ( ph -> mulGrp Fn _V )
23		ssv 3625	. . . . . 6 |- ran R C_ _V
24	23	a1i 11	. . . . 5 |- ( ph -> ran R C_ _V )
25		fnco 5999	. . . . 5 |- ( (mulGrp Fn _V /\ R Fn I /\ ran R C_ _V ) -> (mulGrp o. R ) Fn I )
26	22, 4, 24, 25	syl3anc 1326	. . . 4 |- ( ph -> (mulGrp o. R ) Fn I )
27	19, 20, 16, 17, 26	prdsbas2 16129	. . 3 |- ( ph -> ( Base ` Z ) = X_ x e. I ( Base ` ( (mulGrp o. R ) ` x ) ) )
28	10, 18, 27	3eqtr4d 2666	. 2 |- ( ph -> ( Base ` M ) = ( Base ` Z ) )
29		eqid 2622	. . . 4 |- ( .r ` Y ) = ( .r ` Y )
30	12, 29	mgpplusg 18493	. . 3 |- ( .r ` Y ) = ( +g ` M )
31		eqid 2622	. . . . . . . . 9 |- (mulGrp ` ( R ` z ) ) = (mulGrp ` ( R ` z ) )
32		eqid 2622	. . . . . . . . 9 |- ( .r ` ( R ` z ) ) = ( .r ` ( R ` z ) )
33	31, 32	mgpplusg 18493	. . . . . . . 8 |- ( .r ` ( R ` z ) ) = ( +g ` (mulGrp ` ( R ` z ) ) )
34		fvco2 6273	. . . . . . . . . . 11 |- ( ( R Fn I /\ z e. I ) -> ( (mulGrp o. R ) ` z ) = (mulGrp ` ( R ` z ) ) )
35	4, 34	sylan 488	. . . . . . . . . 10 |- ( ( ph /\ z e. I ) -> ( (mulGrp o. R ) ` z ) = (mulGrp ` ( R ` z ) ) )
36	35	eqcomd 2628	. . . . . . . . 9 |- ( ( ph /\ z e. I ) -> (mulGrp ` ( R ` z ) ) = ( (mulGrp o. R ) ` z ) )
37	36	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ z e. I ) -> ( +g ` (mulGrp ` ( R ` z ) ) ) = ( +g ` ( (mulGrp o. R ) ` z ) ) )
38	33, 37	syl5eq 2668	. . . . . . 7 |- ( ( ph /\ z e. I ) -> ( .r ` ( R ` z ) ) = ( +g ` ( (mulGrp o. R ) ` z ) ) )
39	38	oveqd 6667	. . . . . 6 |- ( ( ph /\ z e. I ) -> ( ( x ` z ) ( .r ` ( R ` z ) ) ( y ` z ) ) = ( ( x ` z ) ( +g ` ( (mulGrp o. R ) ` z ) ) ( y ` z ) ) )
40	39	mpteq2dva 4744	. . . . 5 |- ( ph -> ( z e. I |-> ( ( x ` z ) ( .r ` ( R ` z ) ) ( y ` z ) ) ) = ( z e. I |-> ( ( x ` z ) ( +g ` ( (mulGrp o. R ) ` z ) ) ( y ` z ) ) ) )
41	28, 28, 40	mpt2eq123dv 6717	. . . 4 |- ( ph -> ( x e. ( Base ` M ) , y e. ( Base ` M ) |-> ( z e. I |-> ( ( x ` z ) ( .r ` ( R ` z ) ) ( y ` z ) ) ) ) = ( x e. ( Base ` Z ) , y e. ( Base ` Z ) |-> ( z e. I |-> ( ( x ` z ) ( +g ` ( (mulGrp o. R ) ` z ) ) ( y ` z ) ) ) ) )
42		fnex 6481	. . . . . 6 |- ( ( R Fn I /\ I e. V ) -> R e. _V )
43	4, 17, 42	syl2anc 693	. . . . 5 |- ( ph -> R e. _V )
44		fndm 5990	. . . . . 6 |- ( R Fn I -> dom R = I )
45	4, 44	syl 17	. . . . 5 |- ( ph -> dom R = I )
46	11, 16, 43, 15, 45, 29	prdsmulr 16119	. . . 4 |- ( ph -> ( .r ` Y ) = ( x e. ( Base ` M ) , y e. ( Base ` M ) |-> ( z e. I |-> ( ( x ` z ) ( .r ` ( R ` z ) ) ( y ` z ) ) ) ) )
47		fnex 6481	. . . . . 6 |- ( ( (mulGrp o. R ) Fn I /\ I e. V ) -> (mulGrp o. R ) e. _V )
48	26, 17, 47	syl2anc 693	. . . . 5 |- ( ph -> (mulGrp o. R ) e. _V )
49		fndm 5990	. . . . . 6 |- ( (mulGrp o. R ) Fn I -> dom (mulGrp o. R ) = I )
50	26, 49	syl 17	. . . . 5 |- ( ph -> dom (mulGrp o. R ) = I )
51		eqid 2622	. . . . 5 |- ( +g ` Z ) = ( +g ` Z )
52	19, 16, 48, 20, 50, 51	prdsplusg 16118	. . . 4 |- ( ph -> ( +g ` Z ) = ( x e. ( Base ` Z ) , y e. ( Base ` Z ) |-> ( z e. I |-> ( ( x ` z ) ( +g ` ( (mulGrp o. R ) ` z ) ) ( y ` z ) ) ) ) )
53	41, 46, 52	3eqtr4d 2666	. . 3 |- ( ph -> ( .r ` Y ) = ( +g ` Z ) )
54	30, 53	syl5eqr 2670	. 2 |- ( ph -> ( +g ` M ) = ( +g ` Z ) )
55	28, 54	jca 554	1 |- ( ph -> ( ( Base ` M ) = ( Base ` Z ) /\ ( +g ` M ) = ( +g ` Z ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   dom cdm 5114   ran crn 5115   o. ccom 5118   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   X_cixp 7908   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   X__scprds 16106  mulGrpcmgp 18489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-prds 16108  df-mgp 18490

This theorem is referenced by:  prdsringd  18612  prdscrngd  18613  prds1  18614  pwsmgp  18618