Theorem mgpress 17024

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 5882	. . . . . 6 |- (mulGrp ` R ) e. _V
4	1, 3	eqeltri 2551	. . . . 5 |- M e. _V
5	4	a1i 11	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> M e. _V )
6		simplr 754	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> A e. W )
7		eqid 2467	. . . . 5 |- ( M ↾s A ) = ( M ↾s A )
8		eqid 2467	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
9	1, 8	mgpbas 17019	. . . . 5 |- ( Base ` R ) = ( Base ` M )
10	7, 9	ressid2 14560	. . . 4 |- ( ( ( Base ` R ) C_ A /\ M e. _V /\ A e. W ) -> ( M ↾s A ) = M )
11	2, 5, 6, 10	syl3anc 1228	. . 3 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( M ↾s A ) = M )
12		simpll 753	. . . . 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 14560	. . . . 5 |- ( ( ( Base ` R ) C_ A /\ R e. V /\ A e. W ) -> S = R )
15	2, 12, 6, 14	syl3anc 1228	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> S = R )
16	15	fveq2d 5876	. . 3 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> (mulGrp ` S ) = (mulGrp ` R ) )
17	1, 11, 16	3eqtr4a 2534	. 2 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( M ↾s A ) = (mulGrp ` S ) )
18		eqid 2467	. . . . 5 |- ( .r ` R ) = ( .r ` R )
19	1, 18	mgpval 17016	. . . 4 |- M = ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. )
20	19	oveq1i 6305	. . 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 754	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> A e. W )
24	7, 9	ressval2 14561	. . . 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 1228	. . 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 2467	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
27		eqid 2467	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
28	26, 27	mgpval 17016	. . . . 5 |- (mulGrp ` S ) = ( S sSet <. ( +g ` ndx ) , ( .r ` S ) >. )
29		simpll 753	. . . . . . 7 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> R e. V )
30	13, 8	ressval2 14561	. . . . . . 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 1228	. . . . . 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 14625	. . . . . . . . 9 |- ( A e. W -> ( .r ` R ) = ( .r ` S ) )
33	32	eqcomd 2475	. . . . . . . 8 |- ( A e. W -> ( .r ` S ) = ( .r ` R ) )
34	33	ad2antlr 726	. . . . . . 7 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> ( .r ` S ) = ( .r ` R ) )
35	34	opeq2d 4226	. . . . . 6 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> <. ( +g ` ndx ) , ( .r ` S ) >. = <. ( +g ` ndx ) , ( .r ` R ) >. )
36	31, 35	oveq12d 6313	. . . . 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 2520	. . . 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 10760	. . . . . . 7 |- 1 =/= 2
39	38	necomi 2737	. . . . . 6 |- 2 =/= 1
40		plusgndx 14606	. . . . . . 7 |- ( +g ` ndx ) = 2
41		basendx 14557	. . . . . . 7 |- ( Base ` ndx ) = 1
42	40, 41	neeq12i 2756	. . . . . 6 |- ( ( +g ` ndx ) =/= ( Base ` ndx ) <-> 2 =/= 1 )
43	39, 42	mpbir 209	. . . . 5 |- ( +g ` ndx ) =/= ( Base ` ndx )
44		fvex 5882	. . . . . 6 |- ( .r ` R ) e. _V
45		fvex 5882	. . . . . . 7 |- ( Base ` R ) e. _V
46	45	inex2 4595	. . . . . 6 |- ( A i^i ( Base ` R ) ) e. _V
47		fvex 5882	. . . . . . 7 |- ( +g ` ndx ) e. _V
48		fvex 5882	. . . . . . 7 |- ( Base ` ndx ) e. _V
49	47, 48	setscom 14537	. . . . . 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 2511	. . 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 2534	. 2 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> ( M ↾s A ) = (mulGrp ` S ) )
54	17, 53	pm2.61dan 789	1 |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   _Vcvv 3118   i^i cin 3480   C_ wss 3481   <.cop 4039   ` cfv 5594  (class class class)co 6295   1c1 9505   2c2 10597   ndxcnx 14504   sSet csts 14505   Basecbs 14507   ↾_s cress 14508   +g cplusg 14572   .rcmulr 14573  mulGrpcmgp 17013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-mgp 17014

This theorem is referenced by:  subrgcrng  17304  subrgsubm  17313  resrhm  17329  nn0srg  18356  rge0srg  18357  zringmpg  18391  m2cpmmhm  19115  rdivmuldivd  27606  xrge0iifmhm  27746  xrge0pluscn  27747  xrge0tmd  27753