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	⊢ (𝜑 → 𝑊 ∈ LMod)
lsmsat.t	⊢ (𝜑 → 𝑇 ∈ 𝑆)
lsmsat.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)
lsmsat.q	⊢ (𝜑 → 𝑄 ∈ 𝐴)
lsmsat.n	⊢ (𝜑 → 𝑇 ≠ { 0 })
lsmsat.l	⊢ (𝜑 → 𝑄 ⊆ (𝑇 ⊕ 𝑈))

Assertion

Ref	Expression
lsmsat	⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)))

Distinct variable groups:   𝐴,𝑝   ⊕ ,𝑝   𝑄,𝑝   𝑇,𝑝   𝑈,𝑝   𝑊,𝑝

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

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574  {csn 4177  {cpr 4179  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  0_gc0g 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