lsmsat - Mathbox for Norm Megill

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Theorem lsmsat 38512

Description: Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 39310 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 2728	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2728	. . . . 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 38495	. . . 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 4021	. . . . . . . 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 20975	. . . . . . . . . . 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 4127	. . . . . . . . . 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 20886	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑟 ∈ (𝑇 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑇 ⊕ 𝑈)))
25	13, 24	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (𝑇 ⊕ 𝑈))
26	14	lsssssubg 20849	. . . . . . . . . 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 3983	. . . . . . . 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 3983	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈 ∈ (SubGrp‘𝑊))
32		eqid 2728	. . . . . . . . 9 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
33	32, 18	lsmelval 19611	. . . . . . . 8 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑟 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑟 = (𝑦(+_g‘𝑊)𝑧)))
34	29, 31, 33	syl2anc 582	. . . . . . 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 20842	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . 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 479	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑦 = 0 ) → ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 )
43	2	adantr 479	. . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . 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 20828	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑞 ∈ 𝑇) → 𝑞 ∈ (Base‘𝑊))
51	48, 49, 50	syl2anc 582	. . . . . . . . . . . . . . . 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 38496	. . . . . . . . . . . . . . . 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 20887	. . . . . . . . . . . . . . 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 7441	. . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . 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 20828	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (Base‘𝑊))
63	60, 61, 62	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑧 ∈ (Base‘𝑊))
64	63	adantr 479	. . . . . . . . . . . . . . . . . . . . 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 20782	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ LMod ∧ 𝑧 ∈ (Base‘𝑊)) → ( 0 (+_g‘𝑊)𝑧) = 𝑧)
66	45, 64, 65	syl2anc 582	. . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . 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 4644	. . . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . 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 20887	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑧}) ⊆ 𝑈)
71	70	adantr 479	. . . . . . . . . . . . . . . . 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 4020	. . . . . . . . . . . . . . . 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 20871	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊))
74	45, 51, 73	syl2anc 582	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 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 3983	. . . . . . . . . . . . . . . . 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 19620	. . . . . . . . . . . . . . . . 17 ⊢ ((((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑈 ⊆ (((LSpan‘𝑊)‘{𝑞}) ⊕ 𝑈))
79	74, 77, 78	syl2anc 582	. . . . . . . . . . . . . . . 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 3992	. . . . . . . . . . . . . . 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 4007	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝 ⊆ 𝑇 ↔ ((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇))
82		oveq1 7433	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝 ⊕ 𝑈) = (((LSpan‘𝑊)‘{𝑞}) ⊕ 𝑈))
83	82	sseq2d 4014	. . . . . . . . . . . . . . . . 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 3611	. . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . 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 405	. . . . . . . . . . . 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 3150	. . . . . . . . . . 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 479	. . . . . . . . . . . 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 20828	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ (Base‘𝑊))
94	47, 92, 93	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑦 ∈ (Base‘𝑊))
95	94	adantr 479	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) ∧ 𝑦 ≠ 0 ) → 𝑦 ∈ (Base‘𝑊))
96		simpr 483	. . . . . . . . . . . 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 38496	. . . . . . . . . . . 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 20887	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇)
100	99	adantr 479	. . . . . . . . . . 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 4644	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → {𝑟} = {(𝑦(+_g‘𝑊)𝑧)})
103	102	fveq2d 6906	. . . . . . . . . . . . . . 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 20988	. . . . . . . . . . . . . . . 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 4020	. . . . . . . . . . . . . 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 20981	. . . . . . . . . . . . . 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 4022	. . . . . . . . . . . . 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 20868	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
111	44, 94, 110	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
112	109, 111	sseldd 3983	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ (SubGrp‘𝑊))
113	109, 60	sseldd 3983	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+_g‘𝑊)𝑧)) → 𝑈 ∈ (SubGrp‘𝑊))
114	18	lsmless2 19623	. . . . . . . . . . . . . 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 3992	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 4007	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝 ⊆ 𝑇 ↔ ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇))
119		oveq1 7433	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝 ⊕ 𝑈) = (((LSpan‘𝑊)‘{𝑦}) ⊕ 𝑈))
120	119	sseq2d 4014	. . . . . . . . . . . . 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 3611	. . . . . . . . . . 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 835	. . . . . . . . . 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 3026	. . . . . . . . 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 3208	. . . . . . 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 4007	. . . . . . . 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 3176	. . . . . 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 256	. . . 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 3150	. 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∃wrex 3067 ∖ cdif 3946 ⊆ wss 3949 {csn 4632 {cpr 4634 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +_gcplusg 17240 0_gc0g 17428 SubGrpcsubg 19082 LSSumclsm 19596 LModclmod 20750 LSubSpclss 20822 LSpanclspn 20862 LSAtomsclsa 38478

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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cntz 19275 df-lsm 19598 df-cmn 19744 df-abl 19745 df-mgp 20082 df-ur 20129 df-ring 20182 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lsatoms 38480

This theorem is referenced by: dochexmidlem4 40968

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