Theorem dsmmlss 20088

Description: The finite hull of a product of modules is additionally closed under scalar multiplication and thus is a linear subspace of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)

Hypotheses

Ref	Expression
dsmmlss.i	|- ( ph -> I e. W )
dsmmlss.s	|- ( ph -> S e. Ring )
dsmmlss.r	|- ( ph -> R : I --> LMod )
dsmmlss.k	|- ( ( ph /\ x e. I ) -> (Scalar ` ( R ` x ) ) = S )
dsmmlss.p	|- P = ( S X_s R )
dsmmlss.u	|- U = ( LSubSp ` P )
dsmmlss.h	|- H = ( Base ` ( S (+)m R ) )

Assertion

Ref	Expression
dsmmlss	|- ( ph -> H e. U )

Distinct variable groups:   ph, x   x, S   x, R   x, I   x, P   x, H

Allowed substitution hints:   U( x)   W( x)

Proof of Theorem dsmmlss

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dsmmlss.p	. . 3 |- P = ( S X_s R )
2		dsmmlss.h	. . 3 |- H = ( Base ` ( S (+)m R ) )
3		dsmmlss.i	. . 3 |- ( ph -> I e. W )
4		dsmmlss.s	. . 3 |- ( ph -> S e. Ring )
5		dsmmlss.r	. . . 4 |- ( ph -> R : I --> LMod )
6		lmodgrp 18870	. . . . 5 |- ( a e. LMod -> a e. Grp )
7	6	ssriv 3607	. . . 4 |- LMod C_ Grp
8		fss 6056	. . . 4 |- ( ( R : I --> LMod /\ LMod C_ Grp ) -> R : I --> Grp )
9	5, 7, 8	sylancl 694	. . 3 |- ( ph -> R : I --> Grp )
10	1, 2, 3, 4, 9	dsmmsubg 20087	. 2 |- ( ph -> H e. (SubGrp ` P ) )
11		dsmmlss.k	. . . . . . 7 |- ( ( ph /\ x e. I ) -> (Scalar ` ( R ` x ) ) = S )
12	1, 4, 3, 5, 11	prdslmodd 18969	. . . . . 6 |- ( ph -> P e. LMod )
13	12	adantr 481	. . . . 5 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> P e. LMod )
14		simprl 794	. . . . 5 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> a e. ( Base ` (Scalar ` P ) ) )
15		simprr 796	. . . . . . 7 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> b e. H )
16		eqid 2622	. . . . . . . . 9 |- ( S (+)m R ) = ( S (+)m R )
17		eqid 2622	. . . . . . . . 9 |- ( Base ` P ) = ( Base ` P )
18		ffn 6045	. . . . . . . . . 10 |- ( R : I --> LMod -> R Fn I )
19	5, 18	syl 17	. . . . . . . . 9 |- ( ph -> R Fn I )
20	1, 16, 17, 2, 3, 19	dsmmelbas 20083	. . . . . . . 8 |- ( ph -> ( b e. H <-> ( b e. ( Base ` P ) /\ { x e. I | ( b ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin ) ) )
21	20	adantr 481	. . . . . . 7 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> ( b e. H <-> ( b e. ( Base ` P ) /\ { x e. I | ( b ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin ) ) )
22	15, 21	mpbid 222	. . . . . 6 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> ( b e. ( Base ` P ) /\ { x e. I | ( b ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin ) )
23	22	simpld 475	. . . . 5 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> b e. ( Base ` P ) )
24		eqid 2622	. . . . . 6 |- (Scalar ` P ) = (Scalar ` P )
25		eqid 2622	. . . . . 6 |- ( .s ` P ) = ( .s ` P )
26		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` P ) ) = ( Base ` (Scalar ` P ) )
27	17, 24, 25, 26	lmodvscl 18880	. . . . 5 |- ( ( P e. LMod /\ a e. ( Base ` (Scalar ` P ) ) /\ b e. ( Base ` P ) ) -> ( a ( .s ` P ) b ) e. ( Base ` P ) )
28	13, 14, 23, 27	syl3anc 1326	. . . 4 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> ( a ( .s ` P ) b ) e. ( Base ` P ) )
29	22	simprd 479	. . . . 5 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> { x e. I | ( b ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin )
30		eqid 2622	. . . . . . . . . . 11 |- ( Base ` S ) = ( Base ` S )
31	4	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> S e. Ring )
32	3	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> I e. W )
33	19	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> R Fn I )
34		fex 6490	. . . . . . . . . . . . . . . . . 18 |- ( ( R : I --> LMod /\ I e. W ) -> R e. _V )
35	5, 3, 34	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ph -> R e. _V )
36	1, 4, 35	prdssca 16116	. . . . . . . . . . . . . . . 16 |- ( ph -> S = (Scalar ` P ) )
37	36	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ph -> ( Base ` S ) = ( Base ` (Scalar ` P ) ) )
38	37	eleq2d 2687	. . . . . . . . . . . . . 14 |- ( ph -> ( a e. ( Base ` S ) <-> a e. ( Base ` (Scalar ` P ) ) ) )
39	38	biimpar 502	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. ( Base ` (Scalar ` P ) ) ) -> a e. ( Base ` S ) )
40	39	adantrr 753	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> a e. ( Base ` S ) )
41	40	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> a e. ( Base ` S ) )
42	23	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> b e. ( Base ` P ) )
43		simpr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> x e. I )
44	1, 17, 25, 30, 31, 32, 33, 41, 42, 43	prdsvscafval 16140	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> ( ( a ( .s ` P ) b ) ` x ) = ( a ( .s ` ( R ` x ) ) ( b ` x ) ) )
45	44	adantrr 753	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ ( x e. I /\ ( b ` x ) = ( 0g ` ( R ` x ) ) ) ) -> ( ( a ( .s ` P ) b ) ` x ) = ( a ( .s ` ( R ` x ) ) ( b ` x ) ) )
46	5	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. I ) -> ( R ` x ) e. LMod )
47	46	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> ( R ` x ) e. LMod )
48		simplrl 800	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> a e. ( Base ` (Scalar ` P ) ) )
49	36	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. I ) -> S = (Scalar ` P ) )
50	11, 49	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. I ) -> (Scalar ` ( R ` x ) ) = (Scalar ` P ) )
51	50	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. I ) -> ( Base ` (Scalar ` ( R ` x ) ) ) = ( Base ` (Scalar ` P ) ) )
52	51	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> ( Base ` (Scalar ` ( R ` x ) ) ) = ( Base ` (Scalar ` P ) ) )
53	48, 52	eleqtrrd 2704	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> a e. ( Base ` (Scalar ` ( R ` x ) ) ) )
54		eqid 2622	. . . . . . . . . . . . 13 |- (Scalar ` ( R ` x ) ) = (Scalar ` ( R ` x ) )
55		eqid 2622	. . . . . . . . . . . . 13 |- ( .s ` ( R ` x ) ) = ( .s ` ( R ` x ) )
56		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` (Scalar ` ( R ` x ) ) ) = ( Base ` (Scalar ` ( R ` x ) ) )
57		eqid 2622	. . . . . . . . . . . . 13 |- ( 0g ` ( R ` x ) ) = ( 0g ` ( R ` x ) )
58	54, 55, 56, 57	lmodvs0 18897	. . . . . . . . . . . 12 |- ( ( ( R ` x ) e. LMod /\ a e. ( Base ` (Scalar ` ( R ` x ) ) ) ) -> ( a ( .s ` ( R ` x ) ) ( 0g ` ( R ` x ) ) ) = ( 0g ` ( R ` x ) ) )
59	47, 53, 58	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> ( a ( .s ` ( R ` x ) ) ( 0g ` ( R ` x ) ) ) = ( 0g ` ( R ` x ) ) )
60		oveq2 6658	. . . . . . . . . . . 12 |- ( ( b ` x ) = ( 0g ` ( R ` x ) ) -> ( a ( .s ` ( R ` x ) ) ( b ` x ) ) = ( a ( .s ` ( R ` x ) ) ( 0g ` ( R ` x ) ) ) )
61	60	eqeq1d 2624	. . . . . . . . . . 11 |- ( ( b ` x ) = ( 0g ` ( R ` x ) ) -> ( ( a ( .s ` ( R ` x ) ) ( b ` x ) ) = ( 0g ` ( R ` x ) ) <-> ( a ( .s ` ( R ` x ) ) ( 0g ` ( R ` x ) ) ) = ( 0g ` ( R ` x ) ) ) )
62	59, 61	syl5ibrcom 237	. . . . . . . . . 10 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> ( ( b ` x ) = ( 0g ` ( R ` x ) ) -> ( a ( .s ` ( R ` x ) ) ( b ` x ) ) = ( 0g ` ( R ` x ) ) ) )
63	62	impr 649	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ ( x e. I /\ ( b ` x ) = ( 0g ` ( R ` x ) ) ) ) -> ( a ( .s ` ( R ` x ) ) ( b ` x ) ) = ( 0g ` ( R ` x ) ) )
64	45, 63	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ ( x e. I /\ ( b ` x ) = ( 0g ` ( R ` x ) ) ) ) -> ( ( a ( .s ` P ) b ) ` x ) = ( 0g ` ( R ` x ) ) )
65	64	expr 643	. . . . . . 7 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> ( ( b ` x ) = ( 0g ` ( R ` x ) ) -> ( ( a ( .s ` P ) b ) ` x ) = ( 0g ` ( R ` x ) ) ) )
66	65	necon3d 2815	. . . . . 6 |- ( ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) /\ x e. I ) -> ( ( ( a ( .s ` P ) b ) ` x ) =/= ( 0g ` ( R ` x ) ) -> ( b ` x ) =/= ( 0g ` ( R ` x ) ) ) )
67	66	ss2rabdv 3683	. . . . 5 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> { x e. I | ( ( a ( .s ` P ) b ) ` x ) =/= ( 0g ` ( R ` x ) ) } C_ { x e. I | ( b ` x ) =/= ( 0g ` ( R ` x ) ) } )
68		ssfi 8180	. . . . 5 |- ( ( { x e. I | ( b ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin /\ { x e. I | ( ( a ( .s ` P ) b ) ` x ) =/= ( 0g ` ( R ` x ) ) } C_ { x e. I | ( b ` x ) =/= ( 0g ` ( R ` x ) ) } ) -> { x e. I | ( ( a ( .s ` P ) b ) ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin )
69	29, 67, 68	syl2anc 693	. . . 4 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> { x e. I | ( ( a ( .s ` P ) b ) ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin )
70	1, 16, 17, 2, 3, 19	dsmmelbas 20083	. . . . 5 |- ( ph -> ( ( a ( .s ` P ) b ) e. H <-> ( ( a ( .s ` P ) b ) e. ( Base ` P ) /\ { x e. I | ( ( a ( .s ` P ) b ) ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin ) ) )
71	70	adantr 481	. . . 4 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> ( ( a ( .s ` P ) b ) e. H <-> ( ( a ( .s ` P ) b ) e. ( Base ` P ) /\ { x e. I | ( ( a ( .s ` P ) b ) ` x ) =/= ( 0g ` ( R ` x ) ) } e. Fin ) ) )
72	28, 69, 71	mpbir2and 957	. . 3 |- ( ( ph /\ ( a e. ( Base ` (Scalar ` P ) ) /\ b e. H ) ) -> ( a ( .s ` P ) b ) e. H )
73	72	ralrimivva 2971	. 2 |- ( ph -> A. a e. ( Base ` (Scalar ` P ) ) A. b e. H ( a ( .s ` P ) b ) e. H )
74		dsmmlss.u	. . . 4 |- U = ( LSubSp ` P )
75	24, 26, 17, 25, 74	islss4 18962	. . 3 |- ( P e. LMod -> ( H e. U <-> ( H e. (SubGrp ` P ) /\ A. a e. ( Base ` (Scalar ` P ) ) A. b e. H ( a ( .s ` P ) b ) e. H ) ) )
76	12, 75	syl 17	. 2 |- ( ph -> ( H e. U <-> ( H e. (SubGrp ` P ) /\ A. a e. ( Base ` (Scalar ` P ) ) A. b e. H ( a ( .s ` P ) b ) e. H ) ) )
77	10, 73, 76	mpbir2and 957	1 |- ( ph -> H e. U )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   X__scprds 16106   Grpcgrp 17422  SubGrpcsubg 17588   Ringcrg 18547   LModclmod 18863   LSubSpclss 18932   (+)_m cdsmm 20075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-dsmm 20076

This theorem is referenced by:  dsmmlmod  20089  frlmlss  20095