lsmsat - Mathbox for Norm Megill

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Theorem lsmsat 38389

Description: Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 39187 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)

**Hypotheses**
Ref	Expression
lsmsat.o	⊢ 0 = (0_g‘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 2726	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2726	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
5		lsmsat.o	. . . . 5 ⊢ 0 = (0_g‘𝑊)
6		lsmsat.a	. . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊)
7	3, 4, 5, 6	islsat 38372	. . . 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 231	. 2 ⊢ (𝜑 → ∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟}))
10		simp3 1135	. . . . . . . . 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 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑄 ⊆ (𝑇 ⊕ 𝑈))
13	10, 12	eqsstrrd 4016	. . . . . . . 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 1130	. . . . . . . . 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 20929	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
20	2, 16, 17, 19	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝑆)
21	20	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
22		eldifi 4121	. . . . . . . . . 10 ⊢ (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑟 ∈ (Base‘𝑊))
23	22	3ad2ant2 1131	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (Base‘𝑊))
24	3, 14, 4, 15, 21, 23	lspsnel5 20840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑟 ∈ (𝑇 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑇 ⊕ 𝑈)))
25	13, 24	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (𝑇 ⊕ 𝑈))
26	14	lsssssubg 20803	. . . . . . . . . 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 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇 ∈ 𝑆)
29	27, 28	sseldd 3978	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇 ∈ (SubGrp‘𝑊))
30	17	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈 ∈ 𝑆)
31	27, 30	sseldd 3978	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈 ∈ (SubGrp‘𝑊))
32		eqid 2726	. . . . . . . . 9 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
33	32, 18	lsmelval 19567	. . . . . . . 8 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑟 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑟 = (𝑦(+_g‘𝑊)𝑧)))
34	29, 31, 33	syl2anc 583	. . . . . . 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 231	. . . . . 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 20796	. . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 )
40	39	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 )
41	40	3ad2ant1 1130	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 )
42	41	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑦 = 0 ) → ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 )
43	2	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑊 ∈ LMod)
44	43	3ad2ant1 1130	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑊 ∈ LMod)
45	44	adantr 480	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑇 ∈ 𝑆)
47	46	3ad2ant1 1130	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑇 ∈ 𝑆)
48	47	adantr 480	. . . . . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . . . . 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 20782	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑞 ∈ 𝑇) → 𝑞 ∈ (Base‘𝑊))
51	48, 49, 50	syl2anc 583	. . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . 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 38373	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊) ∧ 𝑞 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑞}) ∈ 𝐴)
54	45, 51, 52, 53	syl3anc 1368	. . . . . . . . . . . . . . 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 20841	. . . . . . . . . . . . . . 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 1190	. . . . . . . . . . . . . . . . . . . 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 1191	. . . . . . . . . . . . . . . . . . . . 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 7419	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑈 ∈ 𝑆)
60	59	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑈 ∈ 𝑆)
61		simp2r 1197	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑧 ∈ 𝑈)
62	3, 14	lssel 20782	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (Base‘𝑊))
63	60, 61, 62	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑧 ∈ (Base‘𝑊))
64	63	adantr 480	. . . . . . . . . . . . . . . . . . . . 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 20736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ LMod ∧ 𝑧 ∈ (Base‘𝑊)) → ( 0 (+_g‘𝑊)𝑧) = 𝑧)
66	45, 64, 65	syl2anc 583	. . . . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . . 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 4635	. . . . . . . . . . . . . . . . . 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 6888	. . . . . . . . . . . . . . . . 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 20841	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑧}) ⊆ 𝑈)
71	70	adantr 480	. . . . . . . . . . . . . . . . 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 4015	. . . . . . . . . . . . . . . 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 20825	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊))
74	45, 51, 73	syl2anc 583	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 3978	. . . . . . . . . . . . . . . . 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 19576	. . . . . . . . . . . . . . . . 17 ⊢ ((((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑈 ⊆ (((LSpan‘𝑊)‘{𝑞}) ⊕ 𝑈))
79	74, 77, 78	syl2anc 583	. . . . . . . . . . . . . . . 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 3987	. . . . . . . . . . . . . . 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 4002	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝 ⊆ 𝑇 ↔ ((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇))
82		oveq1 7411	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝 ⊕ 𝑈) = (((LSpan‘𝑊)‘{𝑞}) ⊕ 𝑈))
83	82	sseq2d 4009	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) ⊕ 𝑈)))
84	81, 83	anbi12d 630	. . . . . . . . . . . . . . . 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 3606	. . . . . . . . . . . . . . 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 834	. . . . . . . . . . . . . 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 1351	. . . . . . . . . . . . 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 406	. . . . . . . . . . . 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 3147	. . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑦 ≠ 0 ) → 𝑊 ∈ LMod)
92		simp2l 1196	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑦 ∈ 𝑇)
93	3, 14	lssel 20782	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ (Base‘𝑊))
94	47, 92, 93	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑦 ∈ (Base‘𝑊))
95	94	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑦 ≠ 0 ) → 𝑦 ∈ (Base‘𝑊))
96		simpr 484	. . . . . . . . . . . 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 38373	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑦 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝐴)
98	91, 95, 96, 97	syl3anc 1368	. . . . . . . . . . 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 20841	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇)
100	99	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑦 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇)
101		simp3 1135	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑟 = (𝑦(+_g‘𝑊)𝑧))
102	101	sneqd 4635	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → {𝑟} = {(𝑦(+_g‘𝑊)𝑧)})
103	102	fveq2d 6888	. . . . . . . . . . . . . . 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 20942	. . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . 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 4015	. . . . . . . . . . . . . 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 20935	. . . . . . . . . . . . . 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 4017	. . . . . . . . . . . . 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 20822	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
111	44, 94, 110	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
112	109, 111	sseldd 3978	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ (SubGrp‘𝑊))
113	109, 60	sseldd 3978	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑈 ∈ (SubGrp‘𝑊))
114	18	lsmless2 19579	. . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . 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 3987	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 4002	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝 ⊆ 𝑇 ↔ ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇))
119		oveq1 7411	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝 ⊕ 𝑈) = (((LSpan‘𝑊)‘{𝑦}) ⊕ 𝑈))
120	119	sseq2d 4009	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) ⊕ 𝑈)))
121	118, 120	anbi12d 630	. . . . . . . . . . . 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 3606	. . . . . . . . . . 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 834	. . . . . . . . . 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 3023	. . . . . . . . 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 1116	. . . . . . . 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 3204	. . . . . . 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 1129	. . . . . 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 4002	. . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (𝑄 ⊆ (𝑝 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈)))
130	129	anbi2d 628	. . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ((𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)) ↔ (𝑝 ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈))))
131	130	rexbidv 3172	. . . . . 6 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)) ↔ ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈))))
132	131	3ad2ant3 1132	. . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)))
134	133	3exp 1116	. . 3 ⊢ (𝜑 → (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)))))
135	134	rexlimdv 3147	. 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)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∃wrex 3064 ∖ cdif 3940 ⊆ wss 3943 {csn 4623 {cpr 4625 ‘cfv 6536 (class class class)co 7404 Basecbs 17151 +_gcplusg 17204 0_gc0g 17392 SubGrpcsubg 19045 LSSumclsm 19552 LModclmod 20704 LSubSpclss 20776 LSpanclspn 20816 LSAtomsclsa 38355

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19048 df-cntz 19231 df-lsm 19554 df-cmn 19700 df-abl 19701 df-mgp 20038 df-ur 20085 df-ring 20138 df-lmod 20706 df-lss 20777 df-lsp 20817 df-lsatoms 38357

This theorem is referenced by: dochexmidlem4 40845

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