Theorem mgpress 13430

Description: Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)

Hypotheses

Ref	Expression
mgpress.1	|- S = ( R ↾s A )
mgpress.2	|- M = (mulGrp ` R )

Assertion

Ref	Expression
mgpress	|- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )

Proof of Theorem mgpress

Step	Hyp	Ref	Expression
1		mgpress.2	. . . . 5 |- M = (mulGrp ` R )
2		eqid 2193	. . . . 5 |- ( .r ` R ) = ( .r ` R )
3	1, 2	mgpvalg 13422	. . . 4 |- ( R e. V -> M = ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) )
4	3	adantr 276	. . 3 |- ( ( R e. V /\ A e. W ) -> M = ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) )
5	4	oveq1d 5934	. 2 |- ( ( R e. V /\ A e. W ) -> ( M sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) = ( ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
6	1	mgpex 13424	. . . 4 |- ( R e. V -> M e. _V )
7		ressvalsets 12685	. . . 4 |- ( ( M e. _V /\ A e. W ) -> ( M ↾s A ) = ( M sSet <. ( Base ` ndx ) , ( A i^i ( Base ` M ) ) >. ) )
8	6, 7	sylan 283	. . 3 |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = ( M sSet <. ( Base ` ndx ) , ( A i^i ( Base ` M ) ) >. ) )
9		eqid 2193	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
10	1, 9	mgpbasg 13425	. . . . . . 7 |- ( R e. V -> ( Base ` R ) = ( Base ` M ) )
11	10	adantr 276	. . . . . 6 |- ( ( R e. V /\ A e. W ) -> ( Base ` R ) = ( Base ` M ) )
12	11	ineq2d 3361	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( A i^i ( Base ` R ) ) = ( A i^i ( Base ` M ) ) )
13	12	opeq2d 3812	. . . 4 |- ( ( R e. V /\ A e. W ) -> <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. = <. ( Base ` ndx ) , ( A i^i ( Base ` M ) ) >. )
14	13	oveq2d 5935	. . 3 |- ( ( R e. V /\ A e. W ) -> ( M sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) = ( M sSet <. ( Base ` ndx ) , ( A i^i ( Base ` M ) ) >. ) )
15	8, 14	eqtr4d 2229	. 2 |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = ( M sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
16		mgpress.1	. . . . 5 |- S = ( R ↾s A )
17		ressvalsets 12685	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( R ↾s A ) = ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
18	16, 17	eqtrid 2238	. . . 4 |- ( ( R e. V /\ A e. W ) -> S = ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
19	16, 2	ressmulrg 12765	. . . . . . 7 |- ( ( A e. W /\ R e. V ) -> ( .r ` R ) = ( .r ` S ) )
20	19	ancoms 268	. . . . . 6 |- ( ( R e. V /\ A e. W ) -> ( .r ` R ) = ( .r ` S ) )
21	20	eqcomd 2199	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( .r ` S ) = ( .r ` R ) )
22	21	opeq2d 3812	. . . 4 |- ( ( R e. V /\ A e. W ) -> <. ( +g ` ndx ) , ( .r ` S ) >. = <. ( +g ` ndx ) , ( .r ` R ) >. )
23	18, 22	oveq12d 5937	. . 3 |- ( ( R e. V /\ A e. W ) -> ( S sSet <. ( +g ` ndx ) , ( .r ` S ) >. ) = ( ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) )
24		ressex 12686	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( R ↾s A ) e. _V )
25	16, 24	eqeltrid 2280	. . . 4 |- ( ( R e. V /\ A e. W ) -> S e. _V )
26		eqid 2193	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
27		eqid 2193	. . . . 5 |- ( .r ` S ) = ( .r ` S )
28	26, 27	mgpvalg 13422	. . . 4 |- ( S e. _V -> (mulGrp ` S ) = ( S sSet <. ( +g ` ndx ) , ( .r ` S ) >. ) )
29	25, 28	syl 14	. . 3 |- ( ( R e. V /\ A e. W ) -> (mulGrp ` S ) = ( S sSet <. ( +g ` ndx ) , ( .r ` S ) >. ) )
30		plusgslid 12733	. . . . . 6 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
31	30	simpri 113	. . . . 5 |- ( +g ` ndx ) e. NN
32	31	a1i 9	. . . 4 |- ( ( R e. V /\ A e. W ) -> ( +g ` ndx ) e. NN )
33		basendxnn 12677	. . . . 5 |- ( Base ` ndx ) e. NN
34	33	a1i 9	. . . 4 |- ( ( R e. V /\ A e. W ) -> ( Base ` ndx ) e. NN )
35		simpl 109	. . . 4 |- ( ( R e. V /\ A e. W ) -> R e. V )
36		basendxnplusgndx 12745	. . . . . 6 |- ( Base ` ndx ) =/= ( +g ` ndx )
37	36	necomi 2449	. . . . 5 |- ( +g ` ndx ) =/= ( Base ` ndx )
38	37	a1i 9	. . . 4 |- ( ( R e. V /\ A e. W ) -> ( +g ` ndx ) =/= ( Base ` ndx ) )
39		mulrslid 12752	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
40	39	slotex 12648	. . . . 5 |- ( R e. V -> ( .r ` R ) e. _V )
41	40	adantr 276	. . . 4 |- ( ( R e. V /\ A e. W ) -> ( .r ` R ) e. _V )
42		inex1g 4166	. . . . 5 |- ( A e. W -> ( A i^i ( Base ` R ) ) e. _V )
43	42	adantl 277	. . . 4 |- ( ( R e. V /\ A e. W ) -> ( A i^i ( Base ` R ) ) e. _V )
44	32, 34, 35, 38, 41, 43	setscomd 12662	. . 3 |- ( ( R e. V /\ A e. W ) -> ( ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) = ( ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) )
45	23, 29, 44	3eqtr4d 2236	. 2 |- ( ( R e. V /\ A e. W ) -> (mulGrp ` S ) = ( ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
46	5, 15, 45	3eqtr4d 2236	1 |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   =/= wne 2364   _Vcvv 2760   i^i cin 3153   <.cop 3622   ` cfv 5255  (class class class)co 5919   NNcn 8984   ndxcnx 12618   sSet csts 12619  Slot cslot 12620   Basecbs 12621   ↾_s cress 12622   +g cplusg 12698   .rcmulr 12699  mulGrpcmgp 13419

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-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  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-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mulr 12712  df-mgp 13420

This theorem is referenced by:  rdivmuldivd  13643  subrgcrng  13724  subrgsubm  13733  resrhm  13747  resrhm2b  13748  zringmpg  14105