Theorem islss3 14527

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 14508	. . 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 13281	. . . . 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 2233	. . . . 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 13342	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( +g ` W ) = ( +g ` X ) )
16		eqid 2232	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
17	4, 16	ressscag 13396	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> (Scalar ` W ) = (Scalar ` X ) )
18		eqid 2232	. . . . 5 |- ( .s ` W ) = ( .s ` W )
19	4, 18	ressvscag 13397	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( .s ` W ) = ( .s ` X ) )
20		eqidd 2233	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
21		eqidd 2233	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) ) )
22		eqidd 2233	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) ) )
23		eqidd 2233	. . . 4 |- ( ( W e. LMod /\ U e. S ) -> ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) ) )
24	16	lmodring 14443	. . . . 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 14525	. . . . 5 |- ( ( W e. LMod /\ U e. S ) -> U e. (SubGrp ` W ) )
27	4	subggrp 13894	. . . . 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 2232	. . . . . 6 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
30	16, 18, 29, 2	lssvscl 14523	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U ) ) -> ( x ( .s ` W ) a ) e. U )
31	30	3impb 1226	. . . 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 1030	. . . . 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 1031	. . . . . 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 3239	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> a e. V )
37		simpr3 1032	. . . . . 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 3239	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. U /\ b e. U ) ) -> b e. V )
39		eqid 2232	. . . . . 6 |- ( +g ` W ) = ( +g ` W )
40	1, 39, 16, 18, 29	lmodvsdi 14459	. . . . 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 1276	. . . 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 1030	. . . . 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 1031	. . . . 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 1032	. . . . . 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 3239	. . . . 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 2232	. . . . . 6 |- ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) )
49	1, 39, 16, 18, 29, 48	lmodvsdir 14460	. . . . 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 1276	. . . 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 2232	. . . . . 6 |- ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) )
52	1, 16, 18, 29, 51	lmodvsass 14461	. . . . 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 1276	. . . 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 3238	. . . . 5 |- ( ( ( W e. LMod /\ U e. S ) /\ x e. U ) -> x e. V )
55		eqid 2232	. . . . . . 7 |- ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) )
56	1, 16, 18, 55	lmodvs1 14464	. . . . . 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 14441	. . 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 531	. . . 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 13271	. . . . . . . 8 |- Base Fn _V
64		simprr 533	. . . . . . . . 9 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> X e. LMod )
65	64	elexd 2827	. . . . . . . 8 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> X e. _V )
66		funfvex 5687	. . . . . . . . 9 |- ( ( Fun Base /\ X e. dom Base ) -> ( Base ` X ) e. _V )
67	66	funfni 5458	. . . . . . . 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 2309	. . . . . 6 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U e. _V )
70	4, 16	ressscag 13396	. . . . . 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 2238	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> (Scalar ` X ) = (Scalar ` W ) )
73		eqidd 2233	. . . 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 2233	. . . . . 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 13342	. . . . 5 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( +g ` W ) = ( +g ` X ) )
79	78	eqcomd 2238	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( +g ` X ) = ( +g ` W ) )
80	4, 18	ressvscag 13397	. . . . . 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 2238	. . . 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 3275	. . . 4 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) C_ V )
85		lmodgrp 14442	. . . . . 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 2232	. . . . . 6 |- ( Base ` X ) = ( Base ` X )
88		eqid 2232	. . . . . 6 |- ( 0g ` X ) = ( 0g ` X )
89	87, 88	grpidcl 13742	. . . . 5 |- ( X e. Grp -> ( 0g ` X ) e. ( Base ` X ) )
90		elex2 2830	. . . . 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 2232	. . . . . . 7 |- ( LSubSp ` X ) = ( LSubSp ` X )
94	87, 93	lss1 14510	. . . . . 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 2232	. . . . . 6 |- (Scalar ` X ) = (Scalar ` X )
98		eqid 2232	. . . . . 6 |- ( Base ` (Scalar ` X ) ) = ( Base ` (Scalar ` X ) )
99		eqid 2232	. . . . . 6 |- ( +g ` X ) = ( +g ` X )
100		eqid 2232	. . . . . 6 |- ( .s ` X ) = ( .s ` X )
101	97, 98, 99, 100, 93	lssclg 14512	. . . . 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 1274	. . . 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 14507	. . 3 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> ( Base ` X ) e. S )
104	62, 103	eqeltrd 2309	. 2 |- ( ( W e. LMod /\ ( U C_ V /\ X e. LMod ) ) -> U e. S )
105	60, 104	impbida 600	1 |- ( W e. LMod -> ( U e. S <-> ( U C_ V /\ X e. LMod ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2203   _Vcvv 2813   C_ wss 3211   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   Basecbs 13212   ↾_s cress 13213   +g cplusg 13290   .rcmulr 13291  Scalarcsca 13293   .scvsca 13294   0gc0g 13469   Grpcgrp 13713  SubGrpcsubg 13884   1rcur 14103   Ringcrg 14140   LModclmod 14435   LSubSpclss 14500

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-sca 13306  df-vsca 13307  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-sbg 13718  df-subg 13887  df-mgp 14065  df-ur 14104  df-ring 14142  df-lmod 14437  df-lssm 14501

This theorem is referenced by:  lsslmod  14528  lsslss  14529