Theorem mgpress 16716

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 5802	. . . . . 6 |- (mulGrp ` R ) e. _V
4	1, 3	eqeltri 2535	. . . . 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 2451	. . . . 5 |- ( M ↾s A ) = ( M ↾s A )
8		eqid 2451	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
9	1, 8	mgpbas 16711	. . . . 5 |- ( Base ` R ) = ( Base ` M )
10	7, 9	ressid2 14337	. . . 4 |- ( ( ( Base ` R ) C_ A /\ M e. _V /\ A e. W ) -> ( M ↾s A ) = M )
11	2, 5, 6, 10	syl3anc 1219	. . 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 14337	. . . . 5 |- ( ( ( Base ` R ) C_ A /\ R e. V /\ A e. W ) -> S = R )
15	2, 12, 6, 14	syl3anc 1219	. . . 4 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> S = R )
16	15	fveq2d 5796	. . 3 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> (mulGrp ` S ) = (mulGrp ` R ) )
17	1, 11, 16	3eqtr4a 2518	. 2 |- ( ( ( R e. V /\ A e. W ) /\ ( Base ` R ) C_ A ) -> ( M ↾s A ) = (mulGrp ` S ) )
18		eqid 2451	. . . . 5 |- ( .r ` R ) = ( .r ` R )
19	1, 18	mgpval 16708	. . . 4 |- M = ( R sSet <. ( +g ` ndx ) , ( .r ` R ) >. )
20	19	oveq1i 6203	. . 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 14338	. . . 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 1219	. . 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 2451	. . . . . 6 |- (mulGrp ` S ) = (mulGrp ` S )
27		eqid 2451	. . . . . 6 |- ( .r ` S ) = ( .r ` S )
28	26, 27	mgpval 16708	. . . . 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 14338	. . . . . . 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 1219	. . . . . 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 14402	. . . . . . . . 9 |- ( A e. W -> ( .r ` R ) = ( .r ` S ) )
33	32	eqcomd 2459	. . . . . . . 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 4167	. . . . . 6 |- ( ( ( R e. V /\ A e. W ) /\ -. ( Base ` R ) C_ A ) -> <. ( +g ` ndx ) , ( .r ` S ) >. = <. ( +g ` ndx ) , ( .r ` R ) >. )
36	31, 35	oveq12d 6211	. . . . 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 2504	. . . 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 10638	. . . . . . 7 |- 1 =/= 2
39	38	necomi 2718	. . . . . 6 |- 2 =/= 1
40		plusgndx 14383	. . . . . . 7 |- ( +g ` ndx ) = 2
41		basendx 14334	. . . . . . 7 |- ( Base ` ndx ) = 1
42	40, 41	neeq12i 2737	. . . . . 6 |- ( ( +g ` ndx ) =/= ( Base ` ndx ) <-> 2 =/= 1 )
43	39, 42	mpbir 209	. . . . 5 |- ( +g ` ndx ) =/= ( Base ` ndx )
44		fvex 5802	. . . . . 6 |- ( .r ` R ) e. _V
45		fvex 5802	. . . . . . 7 |- ( Base ` R ) e. _V
46	45	inex2 4535	. . . . . 6 |- ( A i^i ( Base ` R ) ) e. _V
47		fvex 5802	. . . . . . 7 |- ( +g ` ndx ) e. _V
48		fvex 5802	. . . . . . 7 |- ( Base ` ndx ) e. _V
49	47, 48	setscom 14315	. . . . . 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 2495	. . 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 2518	. 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 1370   e. wcel 1758   =/= wne 2644   _Vcvv 3071   i^i cin 3428   C_ wss 3429   <.cop 3984   ` cfv 5519  (class class class)co 6193   1c1 9387   2c2 10475   ndxcnx 14282   sSet csts 14283   Basecbs 14285   ↾_s cress 14286   +g cplusg 14349   .rcmulr 14350  mulGrpcmgp 16705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-mgp 16706

This theorem is referenced by:  subrgcrng  16984  subrgsubm  16993  resrhm  17009  nn0srg  17999  rge0srg  18000  zringmpg  18034  rdivmuldivd  26397  xrge0iifmhm  26507  xrge0pluscn  26508  xrge0tmd  26514  m2cpmmhm  31211