Theorem mgpress 13563

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 2196	. . . . 5 |- ( .r ` R ) = ( .r ` R )
3	1, 2	mgpvalg 13555	. . . 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 5940	. 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 13557	. . . 4 |- ( R e. V -> M e. _V )
7		ressvalsets 12767	. . . 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 2196	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
10	1, 9	mgpbasg 13558	. . . . . . 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 3365	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( A i^i ( Base ` R ) ) = ( A i^i ( Base ` M ) ) )
13	12	opeq2d 3816	. . . 4 |- ( ( R e. V /\ A e. W ) -> <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. = <. ( Base ` ndx ) , ( A i^i ( Base ` M ) ) >. )
14	13	oveq2d 5941	. . 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 2232	. 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 12767	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( R ↾s A ) = ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
18	16, 17	eqtrid 2241	. . . 4 |- ( ( R e. V /\ A e. W ) -> S = ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
19	16, 2	ressmulrg 12847	. . . . . . 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 2202	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( .r ` S ) = ( .r ` R ) )
22	21	opeq2d 3816	. . . 4 |- ( ( R e. V /\ A e. W ) -> <. ( +g ` ndx ) , ( .r ` S ) >. = <. ( +g ` ndx ) , ( .r ` R ) >. )
23	18, 22	oveq12d 5943	. . 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 12768	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( R ↾s A ) e. _V )
25	16, 24	eqeltrid 2283	. . . 4 |- ( ( R e. V /\ A e. W ) -> S e. _V )
26		eqid 2196	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
27		eqid 2196	. . . . 5 |- ( .r ` S ) = ( .r ` S )
28	26, 27	mgpvalg 13555	. . . 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 12815	. . . . . 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 12759	. . . . 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 12827	. . . . . 6 |- ( Base ` ndx ) =/= ( +g ` ndx )
37	36	necomi 2452	. . . . 5 |- ( +g ` ndx ) =/= ( Base ` ndx )
38	37	a1i 9	. . . 4 |- ( ( R e. V /\ A e. W ) -> ( +g ` ndx ) =/= ( Base ` ndx ) )
39		mulrslid 12834	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
40	39	slotex 12730	. . . . 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 4170	. . . . 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 12744	. . 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 2239	. 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 2239	1 |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   =/= wne 2367   _Vcvv 2763   i^i cin 3156   <.cop 3626   ` cfv 5259  (class class class)co 5925   NNcn 9007   ndxcnx 12700   sSet csts 12701  Slot cslot 12702   Basecbs 12703   ↾_s cress 12704   +g cplusg 12780   .rcmulr 12781  mulGrpcmgp 13552

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-mgp 13553

This theorem is referenced by:  rdivmuldivd  13776  subrgcrng  13857  subrgsubm  13866  resrhm  13880  resrhm2b  13881  zringmpg  14238