Theorem mgpress 14025

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 2231	. . . . 5 |- ( .r ` R ) = ( .r ` R )
3	1, 2	mgpvalg 14017	. . . 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 6043	. 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 14019	. . . 4 |- ( R e. V -> M e. _V )
7		ressvalsets 13227	. . . 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 2231	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
10	1, 9	mgpbasg 14020	. . . . . . 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 3410	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( A i^i ( Base ` R ) ) = ( A i^i ( Base ` M ) ) )
13	12	opeq2d 3874	. . . 4 |- ( ( R e. V /\ A e. W ) -> <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. = <. ( Base ` ndx ) , ( A i^i ( Base ` M ) ) >. )
14	13	oveq2d 6044	. . 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 2267	. 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 13227	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( R ↾s A ) = ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
18	16, 17	eqtrid 2276	. . . 4 |- ( ( R e. V /\ A e. W ) -> S = ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
19	16, 2	ressmulrg 13308	. . . . . . 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 2237	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( .r ` S ) = ( .r ` R ) )
22	21	opeq2d 3874	. . . 4 |- ( ( R e. V /\ A e. W ) -> <. ( +g ` ndx ) , ( .r ` S ) >. = <. ( +g ` ndx ) , ( .r ` R ) >. )
23	18, 22	oveq12d 6046	. . 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 13228	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( R ↾s A ) e. _V )
25	16, 24	eqeltrid 2318	. . . 4 |- ( ( R e. V /\ A e. W ) -> S e. _V )
26		eqid 2231	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
27		eqid 2231	. . . . 5 |- ( .r ` S ) = ( .r ` S )
28	26, 27	mgpvalg 14017	. . . 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 13275	. . . . . 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 13218	. . . . 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 13288	. . . . . 6 |- ( Base ` ndx ) =/= ( +g ` ndx )
37	36	necomi 2488	. . . . 5 |- ( +g ` ndx ) =/= ( Base ` ndx )
38	37	a1i 9	. . . 4 |- ( ( R e. V /\ A e. W ) -> ( +g ` ndx ) =/= ( Base ` ndx ) )
39		mulrslid 13295	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
40	39	slotex 13189	. . . . 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 4230	. . . . 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 13203	. . 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 2274	. 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 2274	1 |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   =/= wne 2403   _Vcvv 2803   i^i cin 3200   <.cop 3676   ` cfv 5333  (class class class)co 6028   NNcn 9202   ndxcnx 13159   sSet csts 13160  Slot cslot 13161   Basecbs 13162   ↾_s cress 13163   +g cplusg 13240   .rcmulr 13241  mulGrpcmgp 14014

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-mgp 14015

This theorem is referenced by:  rdivmuldivd  14239  subrgcrng  14320  subrgsubm  14329  resrhm  14343  resrhm2b  14344  zringmpg  14702