Theorem islss3 14011

Description: A linear subspace of a module is a subset which is a module in its own right. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
islss3.x	|- X = ( W ↾s U )
islss3.v	|- V = ( Base ` W )
islss3.s	|- S = ( LSubSp ` W )

Assertion

Ref	Expression
islss3	|- ( W e. LMod -> ( U e. S <-> ( U C_ V /\ X e. LMod ) ) )

Proof of Theorem islss3

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

Step	Hyp	Ref	Expression
1		islss3.v	. . . 4 |- V = ( Base ` W )
2		islss3.s	. . . 4 |- S = ( LSubSp ` W )
3	1, 2	lssssg 13992	. . 3 |- ( ( W e. LMod /\ U e. S ) -> U C_ V )
4		islss3.x	. . . . . . 7 |- X = ( W ↾s U )
5	4	a1i 9	. . . . . 6 |- ( ( W e. LMod /\ U C_ V ) -> X = ( W ↾s U ) )
6	1	a1i 9	. . . . . 6 |- ( ( W e. LMod /\ U C_ V ) -> V = ( Base ` W ) )
7		simpl 109	. . . . . 6 |- ( ( W e. LMod /\ U C_ V ) -> W e. LMod )
8		simpr 110	. . . . . 6 |- ( ( W e. LMod /\ U C_ V ) -> U C_ V )
9	5, 6, 7, 8	ressbas2d 12771	. . . . 5 |- ( ( W e. LMod /\ U C_ V ) -> U = ( Base ` X ) )
10	3, 9	syldan 282	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> U = ( Base ` X ) )
11	4	a1i 9	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> X = ( W ↾s U ) )
12		eqidd 2197	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> ( +g ` W ) = ( +g ` W ) )
13		simpr 110	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> U e. S )
14		simpl 109	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> W e. LMod )
15	11, 12, 13, 14	ressplusgd 12831	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( +g ` W ) = ( +g ` X ) )
16		eqid 2196	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
17	4, 16	ressscag 12885	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> (Scalar ` W ) = (Scalar ` X ) )
18		eqid 2196	. . . . 5 |- ( .s ` W ) = ( .s ` W )
19	4, 18	ressvscag 12886	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( .s ` W ) = ( .s ` X ) )
20		eqidd 2197	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
21		eqidd 2197	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) ) )
22		eqidd 2197	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) ) )
23		eqidd 2197	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) ) )
24	16	lmodring 13927	. . . . 5 |- ( W e. LMod -> (Scalar ` W ) e. Ring )
25	24	adantr 276	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> (Scalar ` W ) e. Ring )
26	2	lsssubg 14009	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> U e. (SubGrp ` W ) )
27	4	subggrp 13383	. . . . 5 |- ( U e. (SubGrp ` W ) -> X e. Grp )
28	26, 27	syl 14	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> X e. Grp )
29		eqid 2196	. . . . . 6 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
30	16, 18, 29, 2	lssvscl 14007	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U ) ) -> ( x ( .s ` W ) a ) e. U )
31	30	3impb 1201	. . . 4 |- ( ( ( W e. LMod /\ U e. S ) /\ x e. ( Base ` (Scalar ` W ) ) /\ a e. U ) -> ( x ( .s ` W ) a ) e. U )
32		simpll 527	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> W e. LMod )
33		simpr1 1005	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> x e. ( Base ` (Scalar ` W ) ) )
34	3	adantr 276	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> U C_ V )
35		simpr2 1006	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> a e. U )
36	34, 35	sseldd 3185	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> a e. V )
37		simpr3 1007	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> b e. U )
38	34, 37	sseldd 3185	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> b e. V )
39		eqid 2196	. . . . . 6 |- ( +g ` W ) = ( +g ` W )
40	1, 39, 16, 18, 29	lmodvsdi 13943	. . . . 5 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> ( x ( .s ` W ) ( a ( +g ` W ) b ) ) = ( ( x ( .s ` W ) a ) ( +g ` W ) ( x ( .s ` W ) b ) ) )
41	32, 33, 36, 38, 40	syl13anc 1251	. . . 4 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> ( x ( .s ` W ) ( a ( +g ` W ) b ) ) = ( ( x ( .s ` W ) a ) ( +g ` W ) ( x ( .s ` W ) b ) ) )
42		simpll 527	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> W e. LMod )
43		simpr1 1005	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> x e. ( Base ` (Scalar ` W ) ) )
44		simpr2 1006	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> a e. ( Base ` (Scalar ` W ) ) )
45	3	adantr 276	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> U C_ V )
46		simpr3 1007	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> b e. U )
47	45, 46	sseldd 3185	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> b e. V )
48		eqid 2196	. . . . . 6 |- ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) )
49	1, 39, 16, 18, 29, 48	lmodvsdir 13944	. . . . 5 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. V ) ) -> ( ( x ( +g ` (Scalar ` W ) ) a ) ( .s ` W ) b ) = ( ( x ( .s ` W ) b ) ( +g ` W ) ( a ( .s ` W ) b ) ) )
50	42, 43, 44, 47, 49	syl13anc 1251	. . . 4 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> ( ( x ( +g ` (Scalar ` W ) ) a ) ( .s ` W ) b ) = ( ( x ( .s ` W ) b ) ( +g ` W ) ( a ( .s ` W ) b ) ) )
51		eqid 2196	. . . . . 6 |- ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) )
52	1, 16, 18, 29, 51	lmodvsass 13945	. . . . 5 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. V ) ) -> ( ( x ( .r ` (Scalar ` W ) ) a ) ( .s ` W ) b ) = ( x ( .s ` W ) ( a ( .s ` W ) b ) ) )
53	42, 43, 44, 47, 52	syl13anc 1251	. . . 4 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> ( ( x ( .r ` (Scalar ` W ) ) a ) ( .s ` W ) b ) = ( x ( .s ` W ) ( a ( .s ` W ) b ) ) )
54	3	sselda 3184	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ x e. U ) -> x e. V )
55		eqid 2196	. . . . . . 7 |- ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) )
56	1, 16, 18, 55	lmodvs1 13948	. . . . . 6 |- ( ( W e. LMod /\ x e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) x ) = x )
57	56	adantlr 477	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ x e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) x ) = x )
58	54, 57	syldan 282	. . . 4 |- ( ( ( W e. LMod /\ U e. S ) /\ x e. U ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) x ) = x )
59	10, 15, 17, 19, 20, 21, 22, 23, 25, 28, 31, 41, 50, 53, 58	islmodd 13925	. . 3 |- ( ( W e. LMod /\ U e. S ) -> X e. LMod )
60	3, 59	jca 306	. 2 |- ( ( W e. LMod /\ U e. S ) -> ( U C_ V /\ X e. LMod ) )
61		simprl 529	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U C_ V )
62	61, 9	syldan 282	. . 3 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U = ( Base ` X ) )
63		basfn 12761	. . . . . . . 8 |- Base Fn _V
64		simprr 531	. . . . . . . . 9 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> X e. LMod )
65	64	elexd 2776	. . . . . . . 8 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> X e. _V )
66		funfvex 5578	. . . . . . . . 9 |- ( ( Fun Base /\ X e. dom Base ) -> ( Base ` X ) e. _V )
67	66	funfni 5361	. . . . . . . 8 |- ( ( Base Fn _V /\ X e. _V ) -> ( Base ` X ) e. _V )
68	63, 65, 67	sylancr 414	. . . . . . 7 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) e. _V )
69	62, 68	eqeltrd 2273	. . . . . 6 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U e. _V )
70	4, 16	ressscag 12885	. . . . . 6 |- ( ( W e. LMod /\ U e. _V ) -> (Scalar ` W ) = (Scalar ` X ) )
71	69, 70	syldan 282	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> (Scalar ` W ) = (Scalar ` X ) )
72	71	eqcomd 2202	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> (Scalar ` X ) = (Scalar ` W ) )
73		eqidd 2197	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` (Scalar ` X ) ) = ( Base ` (Scalar ` X ) ) )
74	1	a1i 9	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> V = ( Base ` W ) )
75	4	a1i 9	. . . . . 6 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> X = ( W ↾s U ) )
76		eqidd 2197	. . . . . 6 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( +g ` W ) = ( +g ` W ) )
77		simpl 109	. . . . . 6 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> W e. LMod )
78	75, 76, 69, 77	ressplusgd 12831	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( +g ` W ) = ( +g ` X ) )
79	78	eqcomd 2202	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( +g ` X ) = ( +g ` W ) )
80	4, 18	ressvscag 12886	. . . . . 6 |- ( ( W e. LMod /\ U e. _V ) -> ( .s ` W ) = ( .s ` X ) )
81	69, 80	syldan 282	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( .s ` W ) = ( .s ` X ) )
82	81	eqcomd 2202	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( .s ` X ) = ( .s ` W ) )
83	2	a1i 9	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> S = ( LSubSp ` W ) )
84	62, 61	eqsstrrd 3221	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) C_ V )
85		lmodgrp 13926	. . . . . 6 |- ( X e. LMod -> X e. Grp )
86	85	ad2antll 491	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> X e. Grp )
87		eqid 2196	. . . . . 6 |- ( Base ` X ) = ( Base ` X )
88		eqid 2196	. . . . . 6 |- ( 0g ` X ) = ( 0g ` X )
89	87, 88	grpidcl 13231	. . . . 5 |- ( X e. Grp -> ( 0g ` X ) e. ( Base ` X ) )
90		elex2 2779	. . . . 5 |- ( ( 0g ` X ) e. ( Base ` X ) -> E. j j e. ( Base ` X ) )
91	86, 89, 90	3syl 17	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> E. j j e. ( Base ` X ) )
92	64	adantr 276	. . . . 5 |- ( ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ a e. ( Base ` X ) /\ b e. ( Base ` X ) ) ) -> X e. LMod )
93		eqid 2196	. . . . . . 7 |- ( LSubSp ` X ) = ( LSubSp ` X )
94	87, 93	lss1 13994	. . . . . 6 |- ( X e. LMod -> ( Base ` X ) e. ( LSubSp ` X ) )
95	92, 94	syl 14	. . . . 5 |- ( ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ a e. ( Base ` X ) /\ b e. ( Base ` X ) ) ) -> ( Base ` X ) e. ( LSubSp ` X ) )
96		simpr 110	. . . . 5 |- ( ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ a e. ( Base ` X ) /\ b e. ( Base ` X ) ) ) -> ( x e. ( Base ` (Scalar ` X ) ) /\ a e. ( Base ` X ) /\ b e. ( Base ` X ) ) )
97		eqid 2196	. . . . . 6 |- (Scalar ` X ) = (Scalar ` X )
98		eqid 2196	. . . . . 6 |- ( Base ` (Scalar ` X ) ) = ( Base ` (Scalar ` X ) )
99		eqid 2196	. . . . . 6 |- ( +g ` X ) = ( +g ` X )
100		eqid 2196	. . . . . 6 |- ( .s ` X ) = ( .s ` X )
101	97, 98, 99, 100, 93	lssclg 13996	. . . . 5 |- ( ( X e. LMod /\ ( Base ` X ) e. ( LSubSp ` X ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ a e. ( Base ` X ) /\ b e. ( Base ` X ) ) ) -> ( ( x ( .s ` X ) a ) ( +g ` X ) b ) e. ( Base ` X ) )
102	92, 95, 96, 101	syl3anc 1249	. . . 4 |- ( ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) /\ ( x e. ( Base ` (Scalar ` X ) ) /\ a e. ( Base ` X ) /\ b e. ( Base ` X ) ) ) -> ( ( x ( .s ` X ) a ) ( +g ` X ) b ) e. ( Base ` X ) )
103	72, 73, 74, 79, 82, 83, 84, 91, 102, 77	islssmd 13991	. . 3 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) e. S )
104	62, 103	eqeltrd 2273	. 2 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U e. S )
105	60, 104	impbida 596	1 |- ( W e. LMod -> ( U e. S <-> ( U C_ V /\ X e. LMod ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   C_ wss 3157   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   Basecbs 12703   ↾_s cress 12704   +g cplusg 12780   .rcmulr 12781  Scalarcsca 12783   .scvsca 12784   0gc0g 12958   Grpcgrp 13202  SubGrpcsubg 13373   1rcur 13591   Ringcrg 13628   LModclmod 13919   LSubSpclss 13984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-sca 12796  df-vsca 12797  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-sbg 13207  df-subg 13376  df-mgp 13553  df-ur 13592  df-ring 13630  df-lmod 13921  df-lssm 13985

This theorem is referenced by:  lsslmod  14012  lsslss  14013