Theorem mgpress 17812

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 468	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( Base ` R ) C_ A )
3		fvex 5889	. . . . . 6 |- (mulGrp ` R ) e. _V
4	1, 3	eqeltri 2545	. . . . 5 |- M e. _V
5	4	a1i 11	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> M e. _V )
6		simplr 770	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> A e. W )
7		eqid 2471	. . . . 5 |- ( M ↾s A ) = ( M ↾s A )
8		eqid 2471	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
9	1, 8	mgpbas 17807	. . . . 5 |- ( Base ` R ) = ( Base ` M )
10	7, 9	ressid2 15255	. . . 4 |- ( ( ( Base ` R ) C_ A /\ M e. _V /\ A e. W ) -> ( M ↾s A ) = M )
11	2, 5, 6, 10	syl3anc 1292	. . 3 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( M ↾s A ) = M )
12		simpll 768	. . . . 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 15255	. . . . 5 |- ( ( ( Base ` R ) C_ A /\ R e. V /\ A e. W ) -> S = R )
15	2, 12, 6, 14	syl3anc 1292	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> S = R )
16	15	fveq2d 5883	. . 3 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> (mulGrp ` S ) = (mulGrp ` R ) )
17	1, 11, 16	3eqtr4a 2531	. 2 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( M ↾s A ) = (mulGrp ` S ) )
18		eqid 2471	. . . . 5 |- ( .r ` R ) = ( .r ` R )
19	1, 18	mgpval 17804	. . . 4 |- M = ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. )
20	19	oveq1i 6318	. . 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 468	. . . 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 770	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> A e. W )
24	7, 9	ressval2 15256	. . . 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 1292	. . 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 2471	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
27		eqid 2471	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
28	26, 27	mgpval 17804	. . . . 5 |- (mulGrp ` S ) = ( S sSet <. ( +g ` ndx ) , ( .r ` S ) >. )
29		simpll 768	. . . . . . 7 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> R e. V )
30	13, 8	ressval2 15256	. . . . . . 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 1292	. . . . . 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 15328	. . . . . . . . 9 |- ( A e. W -> ( .r ` R ) = ( .r ` S ) )
33	32	eqcomd 2477	. . . . . . . 8 |- ( A e. W -> ( .r ` S ) = ( .r ` R ) )
34	33	ad2antlr 741	. . . . . . 7 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> ( .r ` S ) = ( .r ` R ) )
35	34	opeq2d 4165	. . . . . 6 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> <. ( +g ` ndx ) , ( .r ` S ) >. = <. ( +g ` ndx ) , ( .r ` R ) >. )
36	31, 35	oveq12d 6326	. . . . 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 2517	. . . 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 10845	. . . . . . 7 |- 1 =/= 2
39	38	necomi 2697	. . . . . 6 |- 2 =/= 1
40		plusgndx 15302	. . . . . . 7 |- ( +g ` ndx ) = 2
41		basendx 15251	. . . . . . 7 |- ( Base ` ndx ) = 1
42	40, 41	neeq12i 2709	. . . . . 6 |- ( ( +g ` ndx ) =/= ( Base ` ndx ) <-> 2 =/= 1 )
43	39, 42	mpbir 214	. . . . 5 |- ( +g ` ndx ) =/= ( Base ` ndx )
44		fvex 5889	. . . . . 6 |- ( .r ` R ) e. _V
45		fvex 5889	. . . . . . 7 |- ( Base ` R ) e. _V
46	45	inex2 4538	. . . . . 6 |- ( A i^i ( Base ` R ) ) e. _V
47		fvex 5889	. . . . . . 7 |- ( +g ` ndx ) e. _V
48		fvex 5889	. . . . . . 7 |- ( Base ` ndx ) e. _V
49	47, 48	setscom 15231	. . . . . 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 699	. . . . 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 675	. . . 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 2508	. . 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 2531	. 2 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> ( M ↾s A ) = (mulGrp ` S ) )
54	17, 53	pm2.61dan 808	1 |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   _Vcvv 3031   i^i cin 3389   C_ wss 3390   <.cop 3965   ` cfv 5589  (class class class)co 6308   1c1 9558   2c2 10681   ndxcnx 15196   sSet csts 15197   Basecbs 15199   ↾_s cress 15200   +g cplusg 15268   .rcmulr 15269  mulGrpcmgp 17801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-mgp 17802

This theorem is referenced by:  subrgcrng  18090  subrgsubm  18099  resrhm  18115  nn0srg  19114  rge0srg  19115  zringmpg  19140  m2cpmmhm  19846  rdivmuldivd  28628  xrge0iifmhm  28819  xrge0pluscn  28820  xrge0tmd  28826