Theorem mgpress 13487

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 13479	. . . 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 5937	. 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 13481	. . . 4 |- ( R e. V -> M e. _V )
7		ressvalsets 12742	. . . 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 13482	. . . . . . 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 3364	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( A i^i ( Base ` R ) ) = ( A i^i ( Base ` M ) ) )
13	12	opeq2d 3815	. . . 4 |- ( ( R e. V /\ A e. W ) -> <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. = <. ( Base ` ndx ) , ( A i^i ( Base ` M ) ) >. )
14	13	oveq2d 5938	. . 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 12742	. . . . 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 12822	. . . . . . 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 3815	. . . 4 |- ( ( R e. V /\ A e. W ) -> <. ( +g ` ndx ) , ( .r ` S ) >. = <. ( +g ` ndx ) , ( .r ` R ) >. )
23	18, 22	oveq12d 5940	. . 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 12743	. . . . 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 13479	. . . 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 12790	. . . . . 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 12734	. . . . 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 12802	. . . . . 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 12809	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
40	39	slotex 12705	. . . . 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 4169	. . . . 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 12719	. . 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 3625   ` cfv 5258  (class class class)co 5922   NNcn 8990   ndxcnx 12675   sSet csts 12676  Slot cslot 12677   Basecbs 12678   ↾_s cress 12679   +g cplusg 12755   .rcmulr 12756  mulGrpcmgp 13476

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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

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 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-mgp 13477

This theorem is referenced by:  rdivmuldivd  13700  subrgcrng  13781  subrgsubm  13790  resrhm  13804  resrhm2b  13805  zringmpg  14162