Theorem lsmsat 39760

Description: Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 40557 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 2763	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		eqid 2763	. . . . 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 39743	. . . 4 ⊢ (𝑊 ∈ LMod → (𝑄 ∈ 𝐴 ↔ ∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟})))
8	2, 7	syl 18	. . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟})))
9	1, 8	mpbid 235	. 2 ⊢ (𝜑 → ∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟}))
10		simp3 1156	. . . . . . . . 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 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑄 ⊆ (𝑇 ⊕ 𝑈))
13	10, 12	eqsstrrd 3973	. . . . . . . 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 1151	. . . . . . . . 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 21185	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
20	2, 16, 17, 19	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝑆)
21	20	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑇 ⊕ 𝑈) ∈ 𝑆)
22		eldifi 4086	. . . . . . . . . 10 ⊢ (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑟 ∈ (Base‘𝑊))
23	22	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (Base‘𝑊))
24	3, 14, 4, 15, 21, 23	ellspsn5b 21097	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑟 ∈ (𝑇 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑇 ⊕ 𝑈)))
25	13, 24	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (𝑇 ⊕ 𝑈))
26	14	lsssssubg 21060	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
27	15, 26	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑆 ⊆ (SubGrp‘𝑊))
28	16	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇 ∈ 𝑆)
29	27, 28	sseldd 3939	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇 ∈ (SubGrp‘𝑊))
30	17	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈 ∈ 𝑆)
31	27, 30	sseldd 3939	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈 ∈ (SubGrp‘𝑊))
32		eqid 2763	. . . . . . . . 9 ⊢ (+g‘𝑊) = (+g‘𝑊)
33	32, 18	lsmelval 19720	. . . . . . . 8 ⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑟 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑟 = (𝑦(+g‘𝑊)𝑧)))
34	29, 31, 33	syl2anc 595	. . . . . . 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 235	. . . . . 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 21053	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ 𝑆 → (𝑇 ≠ { 0 } ↔ ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 ))
38	16, 37	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑇 ≠ { 0 } ↔ ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 ))
39	36, 38	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 )
40	39	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 )
41	40	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 )
42	41	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑦 = 0 ) → ∃𝑞 ∈ 𝑇 𝑞 ≠ 0 )
43	2	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑊 ∈ LMod)
44	43	3ad2ant1 1151	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑊 ∈ LMod)
45	44	adantr 485	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑇 ∈ 𝑆)
47	46	3ad2ant1 1151	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑇 ∈ 𝑆)
48	47	adantr 485	. . . . . . . . . . . . . . . . 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 1214	. . . . . . . . . . . . . . . . 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 21039	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑞 ∈ 𝑇) → 𝑞 ∈ (Base‘𝑊))
51	48, 49, 50	syl2anc 595	. . . . . . . . . . . . . . . 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 1215	. . . . . . . . . . . . . . . 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 39744	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊) ∧ 𝑞 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑞}) ∈ 𝐴)
54	45, 51, 52, 53	syl3anc 1398	. . . . . . . . . . . . . . 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 21098	. . . . . . . . . . . . . . 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 1212	. . . . . . . . . . . . . . . . . . . 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 1213	. . . . . . . . . . . . . . . . . . . . 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 7427	. . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑈 ∈ 𝑆)
60	59	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑈 ∈ 𝑆)
61		simp2r 1219	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑧 ∈ 𝑈)
62	3, 14	lssel 21039	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (Base‘𝑊))
63	60, 61, 62	syl2anc 595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑧 ∈ (Base‘𝑊))
64	63	adantr 485	. . . . . . . . . . . . . . . . . . . . 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 20994	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ∈ LMod ∧ 𝑧 ∈ (Base‘𝑊)) → ( 0 (+g‘𝑊)𝑧) = 𝑧)
66	45, 64, 65	syl2anc 595	. . . . . . . . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . . . . . . . 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 4602	. . . . . . . . . . . . . . . . . 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 6887	. . . . . . . . . . . . . . . . 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 21098	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑧}) ⊆ 𝑈)
71	70	adantr 485	. . . . . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . . . . 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 21082	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊))
74	45, 51, 73	syl2anc 595	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . . 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 19729	. . . . . . . . . . . . . . . . 17 ⊢ ((((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑈 ⊆ (((LSpan‘𝑊)‘{𝑞}) ⊕ 𝑈))
79	74, 77, 78	syl2anc 595	. . . . . . . . . . . . . . . 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 3948	. . . . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝 ⊆ 𝑇 ↔ ((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇))
82		oveq1 7419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝 ⊕ 𝑈) = (((LSpan‘𝑊)‘{𝑞}) ⊕ 𝑈))
83	82	sseq2d 3970	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) ⊕ 𝑈)))
84	81, 83	anbi12d 643	. . . . . . . . . . . . . . . 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 3582	. . . . . . . . . . . . . . 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 849	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 3164	. . . . . . . . . . 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 16	. . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑦 ≠ 0 ) → 𝑊 ∈ LMod)
92		simp2l 1218	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑦 ∈ 𝑇)
93	3, 14	lssel 21039	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ (Base‘𝑊))
94	47, 92, 93	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑦 ∈ (Base‘𝑊))
95	94	adantr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑦 ≠ 0 ) → 𝑦 ∈ (Base‘𝑊))
96		simpr 489	. . . . . . . . . . . 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 39744	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑦 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝐴)
98	91, 95, 96, 97	syl3anc 1398	. . . . . . . . . . 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 21098	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇)
100	99	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑦 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇)
101		simp3 1156	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑟 = (𝑦(+g‘𝑊)𝑧))
102	101	sneqd 4602	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → {𝑟} = {(𝑦(+g‘𝑊)𝑧)})
103	102	fveq2d 6887	. . . . . . . . . . . . . . 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 21198	. . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . . 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 21191	. . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑆 ⊆ (SubGrp‘𝑊))
110	3, 14, 4	lspsncl 21079	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
111	44, 94, 110	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
112	109, 111	sseldd 3939	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ (SubGrp‘𝑊))
113	109, 60	sseldd 3939	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ 𝑈) ∧ 𝑟 = (𝑦(+g‘𝑊)𝑧)) → 𝑈 ∈ (SubGrp‘𝑊))
114	18	lsmless2 19732	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 3948	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 3963	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝 ⊆ 𝑇 ↔ ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇))
119		oveq1 7419	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝 ⊕ 𝑈) = (((LSpan‘𝑊)‘{𝑦}) ⊕ 𝑈))
120	119	sseq2d 3970	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) ⊕ 𝑈)))
121	118, 120	anbi12d 643	. . . . . . . . . . . 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 3582	. . . . . . . . . . 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 849	. . . . . . . . . 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 3045	. . . . . . . . 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 1137	. . . . . . . 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 3221	. . . . . . 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 1150	. . . . . 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 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈)))
129		sseq1 3963	. . . . . . . 8 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (𝑄 ⊆ (𝑝 ⊕ 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈)))
130	129	anbi2d 641	. . . . . . 7 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ((𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)) ↔ (𝑝 ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈))))
131	130	rexbidv 3189	. . . . . 6 ⊢ (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)) ↔ ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 ⊕ 𝑈))))
132	131	3ad2ant3 1153	. . . . 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 260	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)))
134	133	3exp 1137	. . 3 ⊢ (𝜑 → (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)))))
135	134	rexlimdv 3164	. 2 ⊢ (𝜑 → (∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈))))
136	9, 135	mpd 16	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 ∧ 𝑄 ⊆ (𝑝 ⊕ 𝑈)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089   ∖ cdif 3903   ⊆ wss 3906  {csn 4590  {cpr 4592  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  +_gcplusg 17311  0_gc0g 17493  SubGrpcsubg 19187  LSSumclsm 19705  LModclmod 20962  LSubSpclss 21033  LSpanclspn 21073  LSAtomsclsa 39726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-0g 17495  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-submnd 18843  df-grp 19004  df-minusg 19005  df-sbg 19006  df-subg 19190  df-cntz 19388  df-lsm 19707  df-cmn 19853  df-abl 19854  df-mgp 20218  df-ur 20265  df-ring 20318  df-lmod 20964  df-lss 21034  df-lsp 21074  df-lsatoms 39728

This theorem is referenced by:  dochexmidlem4  42215