Theorem lsmsat 39507

Description: Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 40304 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 2740	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2740	. . . . 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 39490	. . . 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 233	. 2 ⊢ (𝜑 → ∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟}))
10		simp3 1144	. . . . . . . . 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 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑄 ⊆ (𝑇 ⊕ 𝑈))
13	10, 12	eqsstrrd 3957	. . . . . . . 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 1139	. . . . . . . . 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 21080	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
20	2, 16, 17, 19	syl3anc 1379	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝑆)
21	20	3ad2ant1 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
22		eldifi 4068	. . . . . . . . . 10 ⊢ (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑟 ∈ (Base‘𝑊))
23	22	3ad2ant2 1140	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (Base‘𝑊))
24	3, 14, 4, 15, 21, 23	ellspsn5b 20992	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑟 ∈ (𝑇 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑇 ⊕ 𝑈)))
25	13, 24	mpbird 258	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (𝑇 ⊕ 𝑈))
26	14	lsssssubg 20955	. . . . . . . . . 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 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇 ∈ 𝑆)
29	27, 28	sseldd 3923	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇 ∈ (SubGrp‘𝑊))
30	17	3ad2ant1 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈 ∈ 𝑆)
31	27, 30	sseldd 3923	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈 ∈ (SubGrp‘𝑊))
32		eqid 2740	. . . . . . . . 9 ⊢ (+g‘𝑊) = (+g‘𝑊)
33	32, 18	lsmelval 19622	. . . . . . . 8 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑟 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑟 = (𝑦(+g‘𝑊)𝑧)))
34	29, 31, 33	syl2anc 590	. . . . . . 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 233	. . . . . 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 20948	. . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 )
40	39	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 )
41	40	3ad2ant1 1139	. . . . . . . . . . . 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 1139	. . . . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . . . . 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 20934	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑞 ∈ 𝑇) → 𝑞 ∈ (Base‘𝑊))
51	48, 49, 50	syl2anc 590	. . . . . . . . . . . . . . . 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 1203	. . . . . . . . . . . . . . . 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 39491	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊) ∧ 𝑞 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑞}) ∈ 𝐴)
54	45, 51, 52, 53	syl3anc 1379	. . . . . . . . . . . . . . 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	ellspsn5 20993	. . . . . . . . . . . . . . 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 1200	. . . . . . . . . . . . . . . . . . . 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 1201	. . . . . . . . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑈 ∈ 𝑆)
61		simp2r 1207	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑧 ∈ 𝑈)
62	3, 14	lssel 20934	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (Base‘𝑊))
63	60, 61, 62	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . 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 20889	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ LMod ∧ 𝑧 ∈ (Base‘𝑊)) → ( 0 (+g‘𝑊)𝑧) = 𝑧)
66	45, 64, 65	syl2anc 590	. . . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . 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 4574	. . . . . . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . . . . 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	ellspsn5 20993	. . . . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . 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 20977	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊))
74	45, 51, 73	syl2anc 590	. . . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . . . 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 19631	. . . . . . . . . . . . . . . . 17 ⊢ ((((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑈 ⊆ (((LSpan‘𝑊)‘{𝑞}) ⊕ 𝑈))
79	74, 77, 78	syl2anc 590	. . . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝 ⊆ 𝑇 ↔ ((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇))
82		oveq1 7370	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝 ⊕ 𝑈) = (((LSpan‘𝑊)‘{𝑞}) ⊕ 𝑈))
83	82	sseq2d 3954	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) ⊕ 𝑈)))
84	81, 83	anbi12d 638	. . . . . . . . . . . . . . . 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 3567	. . . . . . . . . . . . . . 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 842	. . . . . . . . . . . . . 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 1361	. . . . . . . . . . . . 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 407	. . . . . . . . . . . 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 3139	. . . . . . . . . . 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 1206	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑦 ∈ 𝑇)
93	3, 14	lssel 20934	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ (Base‘𝑊))
94	47, 92, 93	syl2anc 590	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 39491	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑦 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝐴)
98	91, 95, 96, 97	syl3anc 1379	. . . . . . . . . . 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	ellspsn5 20993	. . . . . . . . . . . 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 1144	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑟 = (𝑦(+g‘𝑊)𝑧))
102	101	sneqd 4574	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → {𝑟} = {(𝑦(+g‘𝑊)𝑧)})
103	102	fveq2d 6838	. . . . . . . . . . . . . . 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 21093	. . . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . 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 21086	. . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . 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 20974	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
111	44, 94, 110	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
112	109, 111	sseldd 3923	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ (SubGrp‘𝑊))
113	109, 60	sseldd 3923	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑈 ∈ (SubGrp‘𝑊))
114	18	lsmless2 19634	. . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . 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 3932	. . . . . . . . . . . 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 3947	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝 ⊆ 𝑇 ↔ ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇))
119		oveq1 7370	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝 ⊕ 𝑈) = (((LSpan‘𝑊)‘{𝑦}) ⊕ 𝑈))
120	119	sseq2d 3954	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) ⊕ 𝑈)))
121	118, 120	anbi12d 638	. . . . . . . . . . . 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 3567	. . . . . . . . . . 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 842	. . . . . . . . . 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 3022	. . . . . . . . 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 1125	. . . . . . . 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 3196	. . . . . . 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 1138	. . . . . 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 3947	. . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (𝑄 ⊆ (𝑝 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈)))
130	129	anbi2d 636	. . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ((𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)) ↔ (𝑝 ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈))))
131	130	rexbidv 3164	. . . . . 6 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)) ↔ ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈))))
132	131	3ad2ant3 1141	. . . . 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 258	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)))
134	133	3exp 1125	. . 3 ⊢ (𝜑 → (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)))))
135	134	rexlimdv 3139	. 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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∃wrex 3064   ∖ cdif 3887   ⊆ wss 3890  {csn 4562  {cpr 4564  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  0_gc0g 17400  SubGrpcsubg 19094  LSSumclsm 19607  LModclmod 20857  LSubSpclss 20928  LSpanclspn 20968  LSAtomsclsa 39473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-submnd 18750  df-grp 18910  df-minusg 18911  df-sbg 18912  df-subg 19097  df-cntz 19290  df-lsm 19609  df-cmn 19755  df-abl 19756  df-mgp 20120  df-ur 20161  df-ring 20214  df-lmod 20859  df-lss 20929  df-lsp 20969  df-lsatoms 39475

This theorem is referenced by:  dochexmidlem4  41962