Theorem mgpress 17474

Description: Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.)

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	. . 3 |- M = (mulGrp ` R )
2		simpr 461	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( Base ` R ) C_ A )
3		fvex 5861	. . . . . 6 |- (mulGrp ` R ) e. _V
4	1, 3	eqeltri 2488	. . . . 5 |- M e. _V
5	4	a1i 11	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> M e. _V )
6		simplr 756	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> A e. W )
7		eqid 2404	. . . . 5 |- ( M ↾s A ) = ( M ↾s A )
8		eqid 2404	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
9	1, 8	mgpbas 17469	. . . . 5 |- ( Base ` R ) = ( Base ` M )
10	7, 9	ressid2 14898	. . . 4 |- ( ( ( Base ` R ) C_ A /\ M e. _V /\ A e. W ) -> ( M ↾s A ) = M )
11	2, 5, 6, 10	syl3anc 1232	. . 3 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( M ↾s A ) = M )
12		simpll 754	. . . . 5 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> R e. V )
13		mgpress.1	. . . . . 6 |- S = ( R ↾s A )
14	13, 8	ressid2 14898	. . . . 5 |- ( ( ( Base ` R ) C_ A /\ R e. V /\ A e. W ) -> S = R )
15	2, 12, 6, 14	syl3anc 1232	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> S = R )
16	15	fveq2d 5855	. . 3 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> (mulGrp ` S ) = (mulGrp ` R ) )
17	1, 11, 16	3eqtr4a 2471	. 2 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( M ↾s A ) = (mulGrp ` S ) )
18		eqid 2404	. . . . 5 |- ( .r ` R ) = ( .r ` R )
19	1, 18	mgpval 17466	. . . 4 |- M = ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. )
20	19	oveq1i 6290	. . 3 |- ( M sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) = ( ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. )
21		simpr 461	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> -. ( Base ` R ) C_ A )
22	4	a1i 11	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> M e. _V )
23		simplr 756	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> A e. W )
24	7, 9	ressval2 14899	. . . 4 |- ( ( -. ( Base ` R ) C_ A /\ M e. _V /\ A e. W ) -> ( M ↾s A ) = ( M sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
25	21, 22, 23, 24	syl3anc 1232	. . 3 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> ( M ↾s A ) = ( M sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
26		eqid 2404	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
27		eqid 2404	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
28	26, 27	mgpval 17466	. . . . 5 |- (mulGrp ` S ) = ( S sSet <. ( +g ` ndx ) , ( .r ` S ) >. )
29		simpll 754	. . . . . . 7 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> R e. V )
30	13, 8	ressval2 14899	. . . . . . 7 |- ( ( -. ( Base ` R ) C_ A /\ R e. V /\ A e. W ) -> S = ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
31	21, 29, 23, 30	syl3anc 1232	. . . . . 6 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> S = ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
32	13, 18	ressmulr 14968	. . . . . . . . 9 |- ( A e. W -> ( .r ` R ) = ( .r ` S ) )
33	32	eqcomd 2412	. . . . . . . 8 |- ( A e. W -> ( .r ` S ) = ( .r ` R ) )
34	33	ad2antlr 727	. . . . . . 7 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> ( .r ` S ) = ( .r ` R ) )
35	34	opeq2d 4168	. . . . . 6 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> <. ( +g ` ndx ) , ( .r ` S ) >. = <. ( +g ` ndx ) , ( .r ` R ) >. )
36	31, 35	oveq12d 6298	. . . . 5 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> ( S sSet <. ( +g ` ndx ) , ( .r ` S ) >. ) = ( ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) )
37	28, 36	syl5eq 2457	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> (mulGrp ` S ) = ( ( R sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) )
38		1ne2 10791	. . . . . . 7 |- 1 =/= 2
39	38	necomi 2675	. . . . . 6 |- 2 =/= 1
40		plusgndx 14945	. . . . . . 7 |- ( +g ` ndx ) = 2
41		basendx 14895	. . . . . . 7 |- ( Base ` ndx ) = 1
42	40, 41	neeq12i 2694	. . . . . 6 |- ( ( +g ` ndx ) =/= ( Base ` ndx ) <-> 2 =/= 1 )
43	39, 42	mpbir 211	. . . . 5 |- ( +g ` ndx ) =/= ( Base ` ndx )
44		fvex 5861	. . . . . 6 |- ( .r ` R ) e. _V
45		fvex 5861	. . . . . . 7 |- ( Base ` R ) e. _V
46	45	inex2 4538	. . . . . 6 |- ( A i^i ( Base ` R ) ) e. _V
47		fvex 5861	. . . . . . 7 |- ( +g ` ndx ) e. _V
48		fvex 5861	. . . . . . 7 |- ( Base ` ndx ) e. _V
49	47, 48	setscom 14875	. . . . . 6 |- ( ( ( R e. V /\ ( +g ` ndx ) =/= ( Base ` ndx ) ) /\ ( ( .r ` R ) e. _V /\ ( A i^i ( Base ` R ) ) e. _V ) ) -> ( ( 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 ) >. ) )
50	44, 46, 49	mpanr12 685	. . . . 5 |- ( ( R e. V /\ ( +g ` ndx ) =/= ( Base ` ndx ) ) -> ( ( 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 ) >. ) )
51	29, 43, 50	sylancl 662	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> ( ( 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 ) >. ) )
52	37, 51	eqtr4d 2448	. . 3 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> (mulGrp ` S ) = ( ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. ) sSet <. ( Base ` ndx ) , ( A i^i ( Base ` R ) ) >. ) )
53	20, 25, 52	3eqtr4a 2471	. 2 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> ( M ↾s A ) = (mulGrp ` S ) )
54	17, 53	pm2.61dan 794	1 |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1407   e. wcel 1844   =/= wne 2600   _Vcvv 3061   i^i cin 3415   C_ wss 3416   <.cop 3980   ` cfv 5571  (class class class)co 6280   1c1 9525   2c2 10628   ndxcnx 14840   sSet csts 14841   Basecbs 14843   ↾_s cress 14844   +g cplusg 14911   .rcmulr 14912  mulGrpcmgp 17463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-mgp 17464

This theorem is referenced by:  subrgcrng  17755  subrgsubm  17764  resrhm  17780  nn0srg  18808  rge0srg  18809  zringmpg  18831  m2cpmmhm  19540  rdivmuldivd  28247  xrge0iifmhm  28387  xrge0pluscn  28388  xrge0tmd  28394