Theorem mgpress 17733

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 462	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( Base ` R ) C_ A )
3		fvex 5891	. . . . . 6 |- (mulGrp ` R ) e. _V
4	1, 3	eqeltri 2503	. . . . 5 |- M e. _V
5	4	a1i 11	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> M e. _V )
6		simplr 760	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> A e. W )
7		eqid 2422	. . . . 5 |- ( M ↾s A ) = ( M ↾s A )
8		eqid 2422	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
9	1, 8	mgpbas 17728	. . . . 5 |- ( Base ` R ) = ( Base ` M )
10	7, 9	ressid2 15176	. . . 4 |- ( ( ( Base ` R ) C_ A /\ M e. _V /\ A e. W ) -> ( M ↾s A ) = M )
11	2, 5, 6, 10	syl3anc 1264	. . 3 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( M ↾s A ) = M )
12		simpll 758	. . . . 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 15176	. . . . 5 |- ( ( ( Base ` R ) C_ A /\ R e. V /\ A e. W ) -> S = R )
15	2, 12, 6, 14	syl3anc 1264	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> S = R )
16	15	fveq2d 5885	. . 3 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> (mulGrp ` S ) = (mulGrp ` R ) )
17	1, 11, 16	3eqtr4a 2489	. 2 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( M ↾s A ) = (mulGrp ` S ) )
18		eqid 2422	. . . . 5 |- ( .r ` R ) = ( .r ` R )
19	1, 18	mgpval 17725	. . . 4 |- M = ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. )
20	19	oveq1i 6315	. . 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 462	. . . 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 760	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> A e. W )
24	7, 9	ressval2 15177	. . . 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 1264	. . 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 2422	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
27		eqid 2422	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
28	26, 27	mgpval 17725	. . . . 5 |- (mulGrp ` S ) = ( S sSet <. ( +g ` ndx ) , ( .r ` S ) >. )
29		simpll 758	. . . . . . 7 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> R e. V )
30	13, 8	ressval2 15177	. . . . . . 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 1264	. . . . . 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 15249	. . . . . . . . 9 |- ( A e. W -> ( .r ` R ) = ( .r ` S ) )
33	32	eqcomd 2430	. . . . . . . 8 |- ( A e. W -> ( .r ` S ) = ( .r ` R ) )
34	33	ad2antlr 731	. . . . . . 7 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> ( .r ` S ) = ( .r ` R ) )
35	34	opeq2d 4194	. . . . . 6 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> <. ( +g ` ndx ) , ( .r ` S ) >. = <. ( +g ` ndx ) , ( .r ` R ) >. )
36	31, 35	oveq12d 6323	. . . . 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 2475	. . . 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 10829	. . . . . . 7 |- 1 =/= 2
39	38	necomi 2690	. . . . . 6 |- 2 =/= 1
40		plusgndx 15223	. . . . . . 7 |- ( +g ` ndx ) = 2
41		basendx 15172	. . . . . . 7 |- ( Base ` ndx ) = 1
42	40, 41	neeq12i 2709	. . . . . 6 |- ( ( +g ` ndx ) =/= ( Base ` ndx ) <-> 2 =/= 1 )
43	39, 42	mpbir 212	. . . . 5 |- ( +g ` ndx ) =/= ( Base ` ndx )
44		fvex 5891	. . . . . 6 |- ( .r ` R ) e. _V
45		fvex 5891	. . . . . . 7 |- ( Base ` R ) e. _V
46	45	inex2 4566	. . . . . 6 |- ( A i^i ( Base ` R ) ) e. _V
47		fvex 5891	. . . . . . 7 |- ( +g ` ndx ) e. _V
48		fvex 5891	. . . . . . 7 |- ( Base ` ndx ) e. _V
49	47, 48	setscom 15152	. . . . . 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 689	. . . . 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 666	. . . 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 2466	. . 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 2489	. 2 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> ( M ↾s A ) = (mulGrp ` S ) )
54	17, 53	pm2.61dan 798	1 |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1872   =/= wne 2614   _Vcvv 3080   i^i cin 3435   C_ wss 3436   <.cop 4004   ` cfv 5601  (class class class)co 6305   1c1 9547   2c2 10666   ndxcnx 15117   sSet csts 15118   Basecbs 15120   ↾_s cress 15121   +g cplusg 15189   .rcmulr 15190  mulGrpcmgp 17722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-ress 15127  df-plusg 15202  df-mulr 15203  df-mgp 17723

This theorem is referenced by:  subrgcrng  18011  subrgsubm  18020  resrhm  18036  nn0srg  19035  rge0srg  19036  zringmpg  19061  m2cpmmhm  19767  rdivmuldivd  28562  xrge0iifmhm  28753  xrge0pluscn  28754  xrge0tmd  28760