Theorem islss3 18959

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		islss3.v	. . . . 5 |- V = ( Base ` W )
2		islss3.s	. . . . 5 |- S = ( LSubSp ` W )
3	1, 2	lssss 18937	. . . 4 |- ( U e. S -> U C_ V )
4	3	adantl 482	. . 3 |- ( ( W e. LMod /\ U e. S ) -> U C_ V )
5		islss3.x	. . . . . . 7 |- X = ( W ↾s U )
6	5, 1	ressbas2 15931	. . . . . 6 |- ( U C_ V -> U = ( Base ` X ) )
7	6	adantl 482	. . . . 5 |- ( ( W e. LMod /\ U C_ V ) -> U = ( Base ` X ) )
8	3, 7	sylan2 491	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> U = ( Base ` X ) )
9		eqid 2622	. . . . . 6 |- ( +g ` W ) = ( +g ` W )
10	5, 9	ressplusg 15993	. . . . 5 |- ( U e. S -> ( +g ` W ) = ( +g ` X ) )
11	10	adantl 482	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( +g ` W ) = ( +g ` X ) )
12		eqid 2622	. . . . . 6 |- (Scalar ` W ) = (Scalar ` W )
13	5, 12	resssca 16031	. . . . 5 |- ( U e. S -> (Scalar ` W ) = (Scalar ` X ) )
14	13	adantl 482	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> (Scalar ` W ) = (Scalar ` X ) )
15		eqid 2622	. . . . . 6 |- ( .s ` W ) = ( .s ` W )
16	5, 15	ressvsca 16032	. . . . 5 |- ( U e. S -> ( .s ` W ) = ( .s ` X ) )
17	16	adantl 482	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( .s ` W ) = ( .s ` X ) )
18		eqidd 2623	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
19		eqidd 2623	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) ) )
20		eqidd 2623	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) ) )
21		eqidd 2623	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) ) )
22	12	lmodring 18871	. . . . 5 |- ( W e. LMod -> (Scalar ` W ) e. Ring )
23	22	adantr 481	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> (Scalar ` W ) e. Ring )
24	2	lsssubg 18957	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> U e. (SubGrp ` W ) )
25	5	subggrp 17597	. . . . 5 |- ( U e. (SubGrp ` W ) -> X e. Grp )
26	24, 25	syl 17	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> X e. Grp )
27		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
28	12, 15, 27, 2	lssvscl 18955	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U ) ) -> ( x ( .s ` W ) a ) e. U )
29	28	3impb 1260	. . . 4 |- ( ( ( W e. LMod /\ U e. S ) /\ x e. ( Base ` (Scalar ` W ) ) /\ a e. U ) -> ( x ( .s ` W ) a ) e. U )
30		simpll 790	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> W e. LMod )
31		simpr1 1067	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> x e. ( Base ` (Scalar ` W ) ) )
32	3	ad2antlr 763	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> U C_ V )
33		simpr2 1068	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> a e. U )
34	32, 33	sseldd 3604	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> a e. V )
35		simpr3 1069	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> b e. U )
36	32, 35	sseldd 3604	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> b e. V )
37	1, 9, 12, 15, 27	lmodvsdi 18886	. . . . 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 ) ) )
38	30, 31, 34, 36, 37	syl13anc 1328	. . . 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 ) ) )
39		simpll 790	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> W e. LMod )
40		simpr1 1067	. . . . 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 ) ) )
41		simpr2 1068	. . . . 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 ) ) )
42	3	ad2antlr 763	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> U C_ V )
43		simpr3 1069	. . . . . 6 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> b e. U )
44	42, 43	sseldd 3604	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. ( Base ` (Scalar ` W ) ) /\ b e. U ) ) -> b e. V )
45		eqid 2622	. . . . . 6 |- ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) )
46	1, 9, 12, 15, 27, 45	lmodvsdir 18887	. . . . 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 ) ) )
47	39, 40, 41, 44, 46	syl13anc 1328	. . . 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 ) ) )
48		eqid 2622	. . . . . 6 |- ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) )
49	1, 12, 15, 27, 48	lmodvsass 18888	. . . . 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 ) ) )
50	39, 40, 41, 44, 49	syl13anc 1328	. . . 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 ) ) )
51	4	sselda 3603	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ x e. U ) -> x e. V )
52		eqid 2622	. . . . . . 7 |- ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) )
53	1, 12, 15, 52	lmodvs1 18891	. . . . . 6 |- ( ( W e. LMod /\ x e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) x ) = x )
54	53	adantlr 751	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ x e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) x ) = x )
55	51, 54	syldan 487	. . . 4 |- ( ( ( W e. LMod /\ U e. S ) /\ x e. U ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) x ) = x )
56	8, 11, 14, 17, 18, 19, 20, 21, 23, 26, 29, 38, 47, 50, 55	islmodd 18869	. . 3 |- ( ( W e. LMod /\ U e. S ) -> X e. LMod )
57	4, 56	jca 554	. 2 |- ( ( W e. LMod /\ U e. S ) -> ( U C_ V /\ X e. LMod ) )
58		simprl 794	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U C_ V )
59	58, 6	syl 17	. . 3 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U = ( Base ` X ) )
60		fvex 6201	. . . . . . 7 |- ( Base ` X ) e. _V
61	59, 60	syl6eqel 2709	. . . . . 6 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U e. _V )
62	5, 12	resssca 16031	. . . . . 6 |- ( U e. _V -> (Scalar ` W ) = (Scalar ` X ) )
63	61, 62	syl 17	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> (Scalar ` W ) = (Scalar ` X ) )
64	63	eqcomd 2628	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> (Scalar ` X ) = (Scalar ` W ) )
65		eqidd 2623	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` (Scalar ` X ) ) = ( Base ` (Scalar ` X ) ) )
66	1	a1i 11	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> V = ( Base ` W ) )
67	5, 9	ressplusg 15993	. . . . . 6 |- ( U e. _V -> ( +g ` W ) = ( +g ` X ) )
68	61, 67	syl 17	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( +g ` W ) = ( +g ` X ) )
69	68	eqcomd 2628	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( +g ` X ) = ( +g ` W ) )
70	5, 15	ressvsca 16032	. . . . . 6 |- ( U e. _V -> ( .s ` W ) = ( .s ` X ) )
71	61, 70	syl 17	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( .s ` W ) = ( .s ` X ) )
72	71	eqcomd 2628	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( .s ` X ) = ( .s ` W ) )
73	2	a1i 11	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> S = ( LSubSp ` W ) )
74	59, 58	eqsstr3d 3640	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) C_ V )
75		lmodgrp 18870	. . . . . 6 |- ( X e. LMod -> X e. Grp )
76	75	ad2antll 765	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> X e. Grp )
77		eqid 2622	. . . . . 6 |- ( Base ` X ) = ( Base ` X )
78	77	grpbn0 17451	. . . . 5 |- ( X e. Grp -> ( Base ` X ) =/= (/) )
79	76, 78	syl 17	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) =/= (/) )
80		eqid 2622	. . . . . . 7 |- ( LSubSp ` X ) = ( LSubSp ` X )
81	77, 80	lss1 18939	. . . . . 6 |- ( X e. LMod -> ( Base ` X ) e. ( LSubSp ` X ) )
82	81	ad2antll 765	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) e. ( LSubSp ` X ) )
83		eqid 2622	. . . . . 6 |- (Scalar ` X ) = (Scalar ` X )
84		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` X ) ) = ( Base ` (Scalar ` X ) )
85		eqid 2622	. . . . . 6 |- ( +g ` X ) = ( +g ` X )
86		eqid 2622	. . . . . 6 |- ( .s ` X ) = ( .s ` X )
87	83, 84, 85, 86, 80	lsscl 18943	. . . . 5 |- ( ( ( 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 ) )
88	82, 87	sylan 488	. . . 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 ) )
89	64, 65, 66, 69, 72, 73, 74, 79, 88	islssd 18936	. . 3 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) e. S )
90	59, 89	eqeltrd 2701	. 2 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U e. S )
91	57, 90	impbida 877	1 |- ( W e. LMod -> ( U e. S <-> ( U C_ V /\ X e. LMod ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   Grpcgrp 17422  SubGrpcsubg 17588   1rcur 18501   Ringcrg 18547   LModclmod 18863   LSubSpclss 18932

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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-sca 15957  df-vsca 15958  df-0g 16102  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

This theorem is referenced by:  lsslmod  18960  lsslss  18961  issubassa  19324