Theorem mgpress 13808

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 2207	. . . . 5 |- ( .r ` R ) = ( .r ` R )
3	1, 2	mgpvalg 13800	. . . 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 5982	. 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 13802	. . . 4 |- ( R e. V -> M e. _V )
7		ressvalsets 13011	. . . 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 2207	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
10	1, 9	mgpbasg 13803	. . . . . . 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 3382	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( A i^i ( Base ` R ) ) = ( A i^i ( Base ` M ) ) )
13	12	opeq2d 3840	. . . 4 |- ( ( R e. V /\ A e. W ) -> <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. = <. ( Base ` ndx ) , ( A i^i ( Base ` M ) ) >. )
14	13	oveq2d 5983	. . 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 2243	. 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 13011	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( R ↾s A ) = ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
18	16, 17	eqtrid 2252	. . . 4 |- ( ( R e. V /\ A e. W ) -> S = ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
19	16, 2	ressmulrg 13092	. . . . . . 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 2213	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( .r ` S ) = ( .r ` R ) )
22	21	opeq2d 3840	. . . 4 |- ( ( R e. V /\ A e. W ) -> <. ( +g ` ndx ) , ( .r ` S ) >. = <. ( +g ` ndx ) , ( .r ` R ) >. )
23	18, 22	oveq12d 5985	. . 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 13012	. . . . 5 |- ( ( R e. V /\ A e. W ) -> ( R ↾s A ) e. _V )
25	16, 24	eqeltrid 2294	. . . 4 |- ( ( R e. V /\ A e. W ) -> S e. _V )
26		eqid 2207	. . . . 5 |- (mulGrp ` S ) = (mulGrp ` S )
27		eqid 2207	. . . . 5 |- ( .r ` S ) = ( .r ` S )
28	26, 27	mgpvalg 13800	. . . 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 13059	. . . . . 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 13003	. . . . 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 13072	. . . . . 6 |- ( Base ` ndx ) =/= ( +g ` ndx )
37	36	necomi 2463	. . . . 5 |- ( +g ` ndx ) =/= ( Base ` ndx )
38	37	a1i 9	. . . 4 |- ( ( R e. V /\ A e. W ) -> ( +g ` ndx ) =/= ( Base ` ndx ) )
39		mulrslid 13079	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
40	39	slotex 12974	. . . . 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 4196	. . . . 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 12988	. . 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 2250	. 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 2250	1 |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   =/= wne 2378   _Vcvv 2776   i^i cin 3173   <.cop 3646   ` cfv 5290  (class class class)co 5967   NNcn 9071   ndxcnx 12944   sSet csts 12945  Slot cslot 12946   Basecbs 12947   ↾_s cress 12948   +g cplusg 13024   .rcmulr 13025  mulGrpcmgp 13797

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-mulr 13038  df-mgp 13798

This theorem is referenced by:  rdivmuldivd  14021  subrgcrng  14102  subrgsubm  14111  resrhm  14125  resrhm2b  14126  zringmpg  14483