Theorem lsmsat 34295

Description: Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 35091 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)

Hypotheses

Ref	Expression
lsmsat.o	|- .0. = ( 0g ` W )
lsmsat.s	|- S = ( LSubSp ` W )
lsmsat.p	|- .(+) = ( LSSum ` W )
lsmsat.a	|- A = (LSAtoms ` W )
lsmsat.w	|- ( ph -> W e. LMod )
lsmsat.t	|- ( ph -> T e. S )
lsmsat.u	|- ( ph -> U e. S )
lsmsat.q	|- ( ph -> Q e. A )
lsmsat.n	|- ( ph -> T =/= { .0. } )
lsmsat.l	|- ( ph -> Q C_ ( T .(+) U ) )

Assertion

Ref	Expression
lsmsat	|- ( ph -> E. p e. A ( p C_ T /\ Q C_ ( p .(+) U ) ) )

Distinct variable groups:   A, p   .(+) , p   Q, p   T, p   U, p   W, p

Allowed substitution hints:   ph( p)   S( p)   .0. ( p)

Proof of Theorem lsmsat

Dummy variables q r y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lsmsat.q	. . 3 |- ( ph -> Q e. A )
2		lsmsat.w	. . . 4 |- ( ph -> W e. LMod )
3		eqid 2622	. . . . 5 |- ( Base ` W ) = ( Base ` W )
4		eqid 2622	. . . . 5 |- ( LSpan ` W ) = ( LSpan ` W )
5		lsmsat.o	. . . . 5 |- .0. = ( 0g ` W )
6		lsmsat.a	. . . . 5 |- A = (LSAtoms ` W )
7	3, 4, 5, 6	islsat 34278	. . . 4 |- ( W e. LMod -> ( Q e. A <-> E. r e. ( ( Base ` W ) \ { .0. } ) Q = ( ( LSpan ` W ) ` { r } ) ) )
8	2, 7	syl 17	. . 3 |- ( ph -> ( Q e. A <-> E. r e. ( ( Base ` W ) \ { .0. } ) Q = ( ( LSpan ` W ) ` { r } ) ) )
9	1, 8	mpbid 222	. 2 |- ( ph -> E. r e. ( ( Base ` W ) \ { .0. } ) Q = ( ( LSpan ` W ) ` { r } ) )
10		simp3 1063	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> Q = ( ( LSpan ` W ) ` { r } ) )
11		lsmsat.l	. . . . . . . . . 10 |- ( ph -> Q C_ ( T .(+) U ) )
12	11	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> Q C_ ( T .(+) U ) )
13	10, 12	eqsstr3d 3640	. . . . . . . 8 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> ( ( LSpan ` W ) ` { r } ) C_ ( T .(+) U ) )
14		lsmsat.s	. . . . . . . . 9 |- S = ( LSubSp ` W )
15	2	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> W e. LMod )
16		lsmsat.t	. . . . . . . . . . 11 |- ( ph -> T e. S )
17		lsmsat.u	. . . . . . . . . . 11 |- ( ph -> U e. S )
18		lsmsat.p	. . . . . . . . . . . 12 |- .(+) = ( LSSum ` W )
19	14, 18	lsmcl 19083	. . . . . . . . . . 11 |- ( ( W e. LMod /\ T e. S /\ U e. S ) -> ( T .(+) U ) e. S )
20	2, 16, 17, 19	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> ( T .(+) U ) e. S )
21	20	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> ( T .(+) U ) e. S )
22		eldifi 3732	. . . . . . . . . 10 |- ( r e. ( ( Base ` W ) \ { .0. } ) -> r e. ( Base ` W ) )
23	22	3ad2ant2 1083	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> r e. ( Base ` W ) )
24	3, 14, 4, 15, 21, 23	lspsnel5 18995	. . . . . . . 8 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> ( r e. ( T .(+) U ) <-> ( ( LSpan ` W ) ` { r } ) C_ ( T .(+) U ) ) )
25	13, 24	mpbird 247	. . . . . . 7 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> r e. ( T .(+) U ) )
26	14	lsssssubg 18958	. . . . . . . . . 10 |- ( W e. LMod -> S C_ (SubGrp ` W ) )
27	15, 26	syl 17	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> S C_ (SubGrp ` W ) )
28	16	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> T e. S )
29	27, 28	sseldd 3604	. . . . . . . 8 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> T e. (SubGrp ` W ) )
30	17	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> U e. S )
31	27, 30	sseldd 3604	. . . . . . . 8 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> U e. (SubGrp ` W ) )
32		eqid 2622	. . . . . . . . 9 |- ( +g ` W ) = ( +g ` W )
33	32, 18	lsmelval 18064	. . . . . . . 8 |- ( ( T e. (SubGrp ` W ) /\ U e. (SubGrp ` W ) ) -> ( r e. ( T .(+) U ) <-> E. y e. T E. z e. U r = ( y ( +g ` W ) z ) ) )
34	29, 31, 33	syl2anc 693	. . . . . . 7 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> ( r e. ( T .(+) U ) <-> E. y e. T E. z e. U r = ( y ( +g ` W ) z ) ) )
35	25, 34	mpbid 222	. . . . . 6 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> E. y e. T E. z e. U r = ( y ( +g ` W ) z ) )
36		lsmsat.n	. . . . . . . . . . . . . . 15 |- ( ph -> T =/= { .0. } )
37	5, 14	lssne0 18951	. . . . . . . . . . . . . . . 16 |- ( T e. S -> ( T =/= { .0. } <-> E. q e. T q =/= .0. ) )
38	16, 37	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( T =/= { .0. } <-> E. q e. T q =/= .0. ) )
39	36, 38	mpbid 222	. . . . . . . . . . . . . 14 |- ( ph -> E. q e. T q =/= .0. )
40	39	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) -> E. q e. T q =/= .0. )
41	40	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> E. q e. T q =/= .0. )
42	41	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ y = .0. ) -> E. q e. T q =/= .0. )
43	2	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) -> W e. LMod )
44	43	3ad2ant1 1082	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> W e. LMod )
45	44	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> W e. LMod )
46	16	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) -> T e. S )
47	46	3ad2ant1 1082	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> T e. S )
48	47	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> T e. S )
49		simpr2 1068	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> q e. T )
50	3, 14	lssel 18938	. . . . . . . . . . . . . . . . 17 |- ( ( T e. S /\ q e. T ) -> q e. ( Base ` W ) )
51	48, 49, 50	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> q e. ( Base ` W ) )
52		simpr3 1069	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> q =/= .0. )
53	3, 4, 5, 6	lsatlspsn2 34279	. . . . . . . . . . . . . . . 16 |- ( ( W e. LMod /\ q e. ( Base ` W ) /\ q =/= .0. ) -> ( ( LSpan ` W ) ` { q } ) e. A )
54	45, 51, 52, 53	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> ( ( LSpan ` W ) ` { q } ) e. A )
55	14, 4, 45, 48, 49	lspsnel5a 18996	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> ( ( LSpan ` W ) ` { q } ) C_ T )
56		simpl3 1066	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> r = ( y ( +g ` W ) z ) )
57		simpr1 1067	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> y = .0. )
58	57	oveq1d 6665	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> ( y ( +g ` W ) z ) = ( .0. ( +g ` W ) z ) )
59	17	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) -> U e. S )
60	59	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> U e. S )
61		simp2r 1088	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> z e. U )
62	3, 14	lssel 18938	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( U e. S /\ z e. U ) -> z e. ( Base ` W ) )
63	60, 61, 62	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> z e. ( Base ` W ) )
64	63	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> z e. ( Base ` W ) )
65	3, 32, 5	lmod0vlid 18893	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( W e. LMod /\ z e. ( Base ` W ) ) -> ( .0. ( +g ` W ) z ) = z )
66	45, 64, 65	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> ( .0. ( +g ` W ) z ) = z )
67	56, 58, 66	3eqtrd 2660	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> r = z )
68	67	sneqd 4189	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> { r } = { z } )
69	68	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> ( ( LSpan ` W ) ` { r } ) = ( ( LSpan ` W ) ` { z } ) )
70	14, 4, 44, 60, 61	lspsnel5a 18996	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( ( LSpan ` W ) ` { z } ) C_ U )
71	70	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> ( ( LSpan ` W ) ` { z } ) C_ U )
72	69, 71	eqsstrd 3639	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> ( ( LSpan ` W ) ` { r } ) C_ U )
73	3, 4	lspsnsubg 18980	. . . . . . . . . . . . . . . . . 18 |- ( ( W e. LMod /\ q e. ( Base ` W ) ) -> ( ( LSpan ` W ) ` { q } ) e. (SubGrp ` W ) )
74	45, 51, 73	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> ( ( LSpan ` W ) ` { q } ) e. (SubGrp ` W ) )
75	45, 26	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> S C_ (SubGrp ` W ) )
76	60	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> U e. S )
77	75, 76	sseldd 3604	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> U e. (SubGrp ` W ) )
78	18	lsmub2 18072	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( LSpan ` W ) ` { q } ) e. (SubGrp ` W ) /\ U e. (SubGrp ` W ) ) -> U C_ ( ( ( LSpan ` W ) ` { q } ) .(+) U ) )
79	74, 77, 78	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> U C_ ( ( ( LSpan ` W ) ` { q } ) .(+) U ) )
80	72, 79	sstrd 3613	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> ( ( LSpan ` W ) ` { r } ) C_ ( ( ( LSpan ` W ) ` { q } ) .(+) U ) )
81		sseq1 3626	. . . . . . . . . . . . . . . . 17 |- ( p = ( ( LSpan ` W ) ` { q } ) -> ( p C_ T <-> ( ( LSpan ` W ) ` { q } ) C_ T ) )
82		oveq1 6657	. . . . . . . . . . . . . . . . . 18 |- ( p = ( ( LSpan ` W ) ` { q } ) -> ( p .(+) U ) = ( ( ( LSpan ` W ) ` { q } ) .(+) U ) )
83	82	sseq2d 3633	. . . . . . . . . . . . . . . . 17 |- ( p = ( ( LSpan ` W ) ` { q } ) -> ( ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) <-> ( ( LSpan ` W ) ` { r } ) C_ ( ( ( LSpan ` W ) ` { q } ) .(+) U ) ) )
84	81, 83	anbi12d 747	. . . . . . . . . . . . . . . 16 |- ( p = ( ( LSpan ` W ) ` { q } ) -> ( ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) <-> ( ( ( LSpan ` W ) ` { q } ) C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( ( ( LSpan ` W ) ` { q } ) .(+) U ) ) ) )
85	84	rspcev 3309	. . . . . . . . . . . . . . 15 |- ( ( ( ( LSpan ` W ) ` { q } ) e. A /\ ( ( ( LSpan ` W ) ` { q } ) C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( ( ( LSpan ` W ) ` { q } ) .(+) U ) ) ) -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) )
86	54, 55, 80, 85	syl12anc 1324	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ ( y = .0. /\ q e. T /\ q =/= .0. ) ) -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) )
87	86	3exp2 1285	. . . . . . . . . . . . 13 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( y = .0. -> ( q e. T -> ( q =/= .0. -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) ) ) ) )
88	87	imp 445	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ y = .0. ) -> ( q e. T -> ( q =/= .0. -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) ) ) )
89	88	rexlimdv 3030	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ y = .0. ) -> ( E. q e. T q =/= .0. -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) ) )
90	42, 89	mpd 15	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ y = .0. ) -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) )
91	44	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ y =/= .0. ) -> W e. LMod )
92		simp2l 1087	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> y e. T )
93	3, 14	lssel 18938	. . . . . . . . . . . . . 14 |- ( ( T e. S /\ y e. T ) -> y e. ( Base ` W ) )
94	47, 92, 93	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> y e. ( Base ` W ) )
95	94	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ y =/= .0. ) -> y e. ( Base ` W ) )
96		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ y =/= .0. ) -> y =/= .0. )
97	3, 4, 5, 6	lsatlspsn2 34279	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ y e. ( Base ` W ) /\ y =/= .0. ) -> ( ( LSpan ` W ) ` { y } ) e. A )
98	91, 95, 96, 97	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ y =/= .0. ) -> ( ( LSpan ` W ) ` { y } ) e. A )
99	14, 4, 44, 47, 92	lspsnel5a 18996	. . . . . . . . . . . 12 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( ( LSpan ` W ) ` { y } ) C_ T )
100	99	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ y =/= .0. ) -> ( ( LSpan ` W ) ` { y } ) C_ T )
101		simp3 1063	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> r = ( y ( +g ` W ) z ) )
102	101	sneqd 4189	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> { r } = { ( y ( +g ` W ) z ) } )
103	102	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( ( LSpan ` W ) ` { r } ) = ( ( LSpan ` W ) ` { ( y ( +g ` W ) z ) } ) )
104	3, 32, 4	lspvadd 19096	. . . . . . . . . . . . . . . 16 |- ( ( W e. LMod /\ y e. ( Base ` W ) /\ z e. ( Base ` W ) ) -> ( ( LSpan ` W ) ` { ( y ( +g ` W ) z ) } ) C_ ( ( LSpan ` W ) ` { y , z } ) )
105	44, 94, 63, 104	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( ( LSpan ` W ) ` { ( y ( +g ` W ) z ) } ) C_ ( ( LSpan ` W ) ` { y , z } ) )
106	103, 105	eqsstrd 3639	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( ( LSpan ` W ) ` { r } ) C_ ( ( LSpan ` W ) ` { y , z } ) )
107	3, 4, 18, 44, 94, 63	lsmpr 19089	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( ( LSpan ` W ) ` { y , z } ) = ( ( ( LSpan ` W ) ` { y } ) .(+) ( ( LSpan ` W ) ` { z } ) ) )
108	106, 107	sseqtrd 3641	. . . . . . . . . . . . 13 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( ( LSpan ` W ) ` { r } ) C_ ( ( ( LSpan ` W ) ` { y } ) .(+) ( ( LSpan ` W ) ` { z } ) ) )
109	44, 26	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> S C_ (SubGrp ` W ) )
110	3, 14, 4	lspsncl 18977	. . . . . . . . . . . . . . . 16 |- ( ( W e. LMod /\ y e. ( Base ` W ) ) -> ( ( LSpan ` W ) ` { y } ) e. S )
111	44, 94, 110	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( ( LSpan ` W ) ` { y } ) e. S )
112	109, 111	sseldd 3604	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( ( LSpan ` W ) ` { y } ) e. (SubGrp ` W ) )
113	109, 60	sseldd 3604	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> U e. (SubGrp ` W ) )
114	18	lsmless2 18075	. . . . . . . . . . . . . 14 |- ( ( ( ( LSpan ` W ) ` { y } ) e. (SubGrp ` W ) /\ U e. (SubGrp ` W ) /\ ( ( LSpan ` W ) ` { z } ) C_ U ) -> ( ( ( LSpan ` W ) ` { y } ) .(+) ( ( LSpan ` W ) ` { z } ) ) C_ ( ( ( LSpan ` W ) ` { y } ) .(+) U ) )
115	112, 113, 70, 114	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( ( ( LSpan ` W ) ` { y } ) .(+) ( ( LSpan ` W ) ` { z } ) ) C_ ( ( ( LSpan ` W ) ` { y } ) .(+) U ) )
116	108, 115	sstrd 3613	. . . . . . . . . . . 12 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> ( ( LSpan ` W ) ` { r } ) C_ ( ( ( LSpan ` W ) ` { y } ) .(+) U ) )
117	116	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ y =/= .0. ) -> ( ( LSpan ` W ) ` { r } ) C_ ( ( ( LSpan ` W ) ` { y } ) .(+) U ) )
118		sseq1 3626	. . . . . . . . . . . . 13 |- ( p = ( ( LSpan ` W ) ` { y } ) -> ( p C_ T <-> ( ( LSpan ` W ) ` { y } ) C_ T ) )
119		oveq1 6657	. . . . . . . . . . . . . 14 |- ( p = ( ( LSpan ` W ) ` { y } ) -> ( p .(+) U ) = ( ( ( LSpan ` W ) ` { y } ) .(+) U ) )
120	119	sseq2d 3633	. . . . . . . . . . . . 13 |- ( p = ( ( LSpan ` W ) ` { y } ) -> ( ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) <-> ( ( LSpan ` W ) ` { r } ) C_ ( ( ( LSpan ` W ) ` { y } ) .(+) U ) ) )
121	118, 120	anbi12d 747	. . . . . . . . . . . 12 |- ( p = ( ( LSpan ` W ) ` { y } ) -> ( ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) <-> ( ( ( LSpan ` W ) ` { y } ) C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( ( ( LSpan ` W ) ` { y } ) .(+) U ) ) ) )
122	121	rspcev 3309	. . . . . . . . . . 11 |- ( ( ( ( LSpan ` W ) ` { y } ) e. A /\ ( ( ( LSpan ` W ) ` { y } ) C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( ( ( LSpan ` W ) ` { y } ) .(+) U ) ) ) -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) )
123	98, 100, 117, 122	syl12anc 1324	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) /\ y =/= .0. ) -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) )
124	90, 123	pm2.61dane 2881	. . . . . . . . 9 |- ( ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) /\ ( y e. T /\ z e. U ) /\ r = ( y ( +g ` W ) z ) ) -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) )
125	124	3exp 1264	. . . . . . . 8 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) -> ( ( y e. T /\ z e. U ) -> ( r = ( y ( +g ` W ) z ) -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) ) ) )
126	125	rexlimdvv 3037	. . . . . . 7 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) ) -> ( E. y e. T E. z e. U r = ( y ( +g ` W ) z ) -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) ) )
127	126	3adant3 1081	. . . . . 6 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> ( E. y e. T E. z e. U r = ( y ( +g ` W ) z ) -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) ) )
128	35, 127	mpd 15	. . . . 5 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) )
129		sseq1 3626	. . . . . . . 8 |- ( Q = ( ( LSpan ` W ) ` { r } ) -> ( Q C_ ( p .(+) U ) <-> ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) )
130	129	anbi2d 740	. . . . . . 7 |- ( Q = ( ( LSpan ` W ) ` { r } ) -> ( ( p C_ T /\ Q C_ ( p .(+) U ) ) <-> ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) ) )
131	130	rexbidv 3052	. . . . . 6 |- ( Q = ( ( LSpan ` W ) ` { r } ) -> ( E. p e. A ( p C_ T /\ Q C_ ( p .(+) U ) ) <-> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) ) )
132	131	3ad2ant3 1084	. . . . 5 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> ( E. p e. A ( p C_ T /\ Q C_ ( p .(+) U ) ) <-> E. p e. A ( p C_ T /\ ( ( LSpan ` W ) ` { r } ) C_ ( p .(+) U ) ) ) )
133	128, 132	mpbird 247	. . . 4 |- ( ( ph /\ r e. ( ( Base ` W ) \ { .0. } ) /\ Q = ( ( LSpan ` W ) ` { r } ) ) -> E. p e. A ( p C_ T /\ Q C_ ( p .(+) U ) ) )
134	133	3exp 1264	. . 3 |- ( ph -> ( r e. ( ( Base ` W ) \ { .0. } ) -> ( Q = ( ( LSpan ` W ) ` { r } ) -> E. p e. A ( p C_ T /\ Q C_ ( p .(+) U ) ) ) ) )
135	134	rexlimdv 3030	. 2 |- ( ph -> ( E. r e. ( ( Base ` W ) \ { .0. } ) Q = ( ( LSpan ` W ) ` { r } ) -> E. p e. A ( p C_ T /\ Q C_ ( p .(+) U ) ) ) )
136	9, 135	mpd 15	1 |- ( ph -> E. p e. A ( p C_ T /\ Q C_ ( p .(+) U ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100  SubGrpcsubg 17588   LSSumclsm 18049   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971  LSAtomsclsa 34261

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-int 4476  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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lsatoms 34263

This theorem is referenced by:  dochexmidlem4  36752