Theorem islss3 13563

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 13544	. . 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 12541	. . . . 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 2188	. . . . 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 12601	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( +g ` W ) = ( +g ` X ) )
16		eqid 2187	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
17	4, 16	ressscag 12655	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> (Scalar ` W ) = (Scalar ` X ) )
18		eqid 2187	. . . . 5 |- ( .s ` W ) = ( .s ` W )
19	4, 18	ressvscag 12656	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( .s ` W ) = ( .s ` X ) )
20		eqidd 2188	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
21		eqidd 2188	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) ) )
22		eqidd 2188	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) ) )
23		eqidd 2188	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) ) )
24	16	lmodring 13479	. . . . 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 13561	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> U e. (SubGrp ` W ) )
27	4	subggrp 13068	. . . . 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 2187	. . . . . 6 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
30	16, 18, 29, 2	lssvscl 13559	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U ) ) -> ( x ( .s ` W ) a ) e. U )
31	30	3impb 1200	. . . 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 1004	. . . . 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 1005	. . . . . 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 3168	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> a e. V )
37		simpr3 1006	. . . . . 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 3168	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> b e. V )
39		eqid 2187	. . . . . 6 |- ( +g ` W ) = ( +g ` W )
40	1, 39, 16, 18, 29	lmodvsdi 13495	. . . . 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 1250	. . . 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 1004	. . . . 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 1005	. . . . 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 1006	. . . . . 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 3168	. . . . 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 2187	. . . . . 6 |- ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) )
49	1, 39, 16, 18, 29, 48	lmodvsdir 13496	. . . . 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 1250	. . . 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 2187	. . . . . 6 |- ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) )
52	1, 16, 18, 29, 51	lmodvsass 13497	. . . . 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 1250	. . . 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 3167	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ x e. U ) -> x e. V )
55		eqid 2187	. . . . . . 7 |- ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) )
56	1, 16, 18, 55	lmodvs1 13500	. . . . . 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 13477	. . 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 12533	. . . . . . . 8 |- Base Fn _V
64		simprr 531	. . . . . . . . 9 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> X e. LMod )
65	64	elexd 2762	. . . . . . . 8 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> X e. _V )
66		funfvex 5544	. . . . . . . . 9 |- ( ( Fun Base /\ X e. dom Base ) -> ( Base ` X ) e. _V )
67	66	funfni 5328	. . . . . . . 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 2264	. . . . . 6 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U e. _V )
70	4, 16	ressscag 12655	. . . . . 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 2193	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> (Scalar ` X ) = (Scalar ` W ) )
73		eqidd 2188	. . . 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 2188	. . . . . 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 12601	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( +g ` W ) = ( +g ` X ) )
79	78	eqcomd 2193	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( +g ` X ) = ( +g ` W ) )
80	4, 18	ressvscag 12656	. . . . . 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 2193	. . . 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 3204	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) C_ V )
85		lmodgrp 13478	. . . . . 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 2187	. . . . . 6 |- ( Base ` X ) = ( Base ` X )
88		eqid 2187	. . . . . 6 |- ( 0g ` X ) = ( 0g ` X )
89	87, 88	grpidcl 12925	. . . . 5 |- ( X e. Grp -> ( 0g ` X ) e. ( Base ` X ) )
90		elex2 2765	. . . . 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 2187	. . . . . . 7 |- ( LSubSp ` X ) = ( LSubSp ` X )
94	87, 93	lss1 13546	. . . . . 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 2187	. . . . . 6 |- (Scalar ` X ) = (Scalar ` X )
98		eqid 2187	. . . . . 6 |- ( Base ` (Scalar ` X ) ) = ( Base ` (Scalar ` X ) )
99		eqid 2187	. . . . . 6 |- ( +g ` X ) = ( +g ` X )
100		eqid 2187	. . . . . 6 |- ( .s ` X ) = ( .s ` X )
101	97, 98, 99, 100, 93	lssclg 13548	. . . . 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 1248	. . . 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 13543	. . 3 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) e. S )
104	62, 103	eqeltrd 2264	. 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 979   = wceq 1363   E.wex 1502   e. wcel 2158   _Vcvv 2749   C_ wss 3141   Fn wfn 5223   ` cfv 5228  (class class class)co 5888   Basecbs 12475   ↾_s cress 12476   +g cplusg 12550   .rcmulr 12551  Scalarcsca 12553   .scvsca 12554   0gc0g 12722   Grpcgrp 12898  SubGrpcsubg 13058   1rcur 13206   Ringcrg 13243   LModclmod 13471   LSubSpclss 13536

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-5 8994  df-6 8995  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483  df-plusg 12563  df-mulr 12564  df-sca 12566  df-vsca 12567  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-minusg 12902  df-sbg 12903  df-subg 13061  df-mgp 13171  df-ur 13207  df-ring 13245  df-lmod 13473  df-lssm 13537

This theorem is referenced by:  lsslmod  13564  lsslss  13565