Theorem islss3 13875

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 13856	. . 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 12686	. . . . 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 2194	. . . . 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 12746	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( +g ` W ) = ( +g ` X ) )
16		eqid 2193	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
17	4, 16	ressscag 12800	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> (Scalar ` W ) = (Scalar ` X ) )
18		eqid 2193	. . . . 5 |- ( .s ` W ) = ( .s ` W )
19	4, 18	ressvscag 12801	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( .s ` W ) = ( .s ` X ) )
20		eqidd 2194	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
21		eqidd 2194	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) ) )
22		eqidd 2194	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) ) )
23		eqidd 2194	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) ) )
24	16	lmodring 13791	. . . . 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 13873	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> U e. (SubGrp ` W ) )
27	4	subggrp 13247	. . . . 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 2193	. . . . . 6 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
30	16, 18, 29, 2	lssvscl 13871	. . . . 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 3180	. . . . 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 3180	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> b e. V )
39		eqid 2193	. . . . . 6 |- ( +g ` W ) = ( +g ` W )
40	1, 39, 16, 18, 29	lmodvsdi 13807	. . . . 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 3180	. . . . 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 2193	. . . . . 6 |- ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) )
49	1, 39, 16, 18, 29, 48	lmodvsdir 13808	. . . . 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 2193	. . . . . 6 |- ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) )
52	1, 16, 18, 29, 51	lmodvsass 13809	. . . . 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 3179	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ x e. U ) -> x e. V )
55		eqid 2193	. . . . . . 7 |- ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) )
56	1, 16, 18, 55	lmodvs1 13812	. . . . . 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 13789	. . 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 12676	. . . . . . . 8 |- Base Fn _V
64		simprr 531	. . . . . . . . 9 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> X e. LMod )
65	64	elexd 2773	. . . . . . . 8 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> X e. _V )
66		funfvex 5571	. . . . . . . . 9 |- ( ( Fun Base /\ X e. dom Base ) -> ( Base ` X ) e. _V )
67	66	funfni 5354	. . . . . . . 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 2270	. . . . . 6 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U e. _V )
70	4, 16	ressscag 12800	. . . . . 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 2199	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> (Scalar ` X ) = (Scalar ` W ) )
73		eqidd 2194	. . . 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 2194	. . . . . 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 12746	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( +g ` W ) = ( +g ` X ) )
79	78	eqcomd 2199	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( +g ` X ) = ( +g ` W ) )
80	4, 18	ressvscag 12801	. . . . . 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 2199	. . . 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 3216	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) C_ V )
85		lmodgrp 13790	. . . . . 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 2193	. . . . . 6 |- ( Base ` X ) = ( Base ` X )
88		eqid 2193	. . . . . 6 |- ( 0g ` X ) = ( 0g ` X )
89	87, 88	grpidcl 13101	. . . . 5 |- ( X e. Grp -> ( 0g ` X ) e. ( Base ` X ) )
90		elex2 2776	. . . . 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 2193	. . . . . . 7 |- ( LSubSp ` X ) = ( LSubSp ` X )
94	87, 93	lss1 13858	. . . . . 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 2193	. . . . . 6 |- (Scalar ` X ) = (Scalar ` X )
98		eqid 2193	. . . . . 6 |- ( Base ` (Scalar ` X ) ) = ( Base ` (Scalar ` X ) )
99		eqid 2193	. . . . . 6 |- ( +g ` X ) = ( +g ` X )
100		eqid 2193	. . . . . 6 |- ( .s ` X ) = ( .s ` X )
101	97, 98, 99, 100, 93	lssclg 13860	. . . . 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 13855	. . 3 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) e. S )
104	62, 103	eqeltrd 2270	. 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 1503   e. wcel 2164   _Vcvv 2760   C_ wss 3153   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   Basecbs 12618   ↾_s cress 12619   +g cplusg 12695   .rcmulr 12696  Scalarcsca 12698   .scvsca 12699   0gc0g 12867   Grpcgrp 13072  SubGrpcsubg 13237   1rcur 13455   Ringcrg 13492   LModclmod 13783   LSubSpclss 13848

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-sbg 13077  df-subg 13240  df-mgp 13417  df-ur 13456  df-ring 13494  df-lmod 13785  df-lssm 13849

This theorem is referenced by:  lsslmod  13876  lsslss  13877