dvh3dim3N - Mathbox for Norm Megill

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Theorem dvh3dim3N 42243

Description: There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 42242 everywhere. If this one is needed, make dvh3dim2 42242 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
dvh3dim.h	⊢ 𝐻 = (LHyp‘𝐾)
dvh3dim.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvh3dim.v	⊢ 𝑉 = (Base‘𝑈)
dvh3dim.n	⊢ 𝑁 = (LSpan‘𝑈)
dvh3dim.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dvh3dim.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
dvh3dim.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
dvh3dim2.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
dvh3dim3.t	⊢ (𝜑 → 𝑇 ∈ 𝑉)

**Assertion**
Ref	Expression
dvh3dim3N	⊢ (𝜑 → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))

Distinct variable groups: 𝑧,𝑁 𝑧,𝑈 𝑧,𝑉 𝑧,𝑋 𝑧,𝑌 𝑧,𝑍 𝜑,𝑧 𝑧,𝑇 Allowed substitution hints: 𝐻(𝑧) 𝐾(𝑧) 𝑊(𝑧)

Proof of Theorem dvh3dim3N Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
2		dvh3dim.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
3		dvh3dim.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
4		dvh3dim.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		dvh3dim.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	3, 4, 5	dvhlmod 41904	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
7	6	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑈 ∈ LMod)
8		dvh3dim.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
9		dvh3dim2.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
10		dvh3dim3.t	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
11	8, 1, 2, 6, 9, 10	lspprcl 21099	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑍, 𝑇}) ∈ (LSubSp‘𝑈))
12	11	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑍, 𝑇}) ∈ (LSubSp‘𝑈))
13		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑍, 𝑇}))
14	8, 2, 6, 9, 10	lspprid2 21119	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝑍, 𝑇}))
15	14	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ (𝑁‘{𝑍, 𝑇}))
16	1, 2, 7, 12, 13, 15	lspprss 21113	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌, 𝑇}) ⊆ (𝑁‘{𝑍, 𝑇}))
17		sspss 4056	. . . 4 ⊢ ((𝑁‘{𝑌, 𝑇}) ⊆ (𝑁‘{𝑍, 𝑇}) ↔ ((𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇}) ∨ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})))
18	16, 17	sylib 221	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → ((𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇}) ∨ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})))
19	3, 4, 5	dvhlvec 41903	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
20	19	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → 𝑈 ∈ LVec)
21		dvh3dim.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
22	8, 1, 2, 6, 21, 10	lspprcl 21099	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑇}) ∈ (LSubSp‘𝑈))
23	22	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌, 𝑇}) ∈ (LSubSp‘𝑈))
24	9	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → 𝑍 ∈ 𝑉)
25	10	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ 𝑉)
26		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇}))
27	8, 1, 2, 20, 23, 24, 25, 26	lspprat 21277	. . . . 5 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → ∃𝑤 ∈ 𝑉 (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤}))
28	5	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
29		simp2 1155	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → 𝑤 ∈ 𝑉)
30		dvh3dim.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
31	30	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → 𝑋 ∈ 𝑉)
32	9	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → 𝑍 ∈ 𝑉)
33	3, 4, 8, 2, 28, 29, 31, 32	dvh3dim2 42242	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑍})))
34	6	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → 𝑈 ∈ LMod)
35	1	lsssssubg 21079	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
36	34, 35	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
37	8, 1, 2	lspsncl 21098	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
38	6, 30, 37	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
39	38	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
40	36, 39	sseldd 3938	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑈))
41	8, 1, 2	lspsncl 21098	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉) → (𝑁‘{𝑤}) ∈ (LSubSp‘𝑈))
42	34, 29, 41	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ∈ (LSubSp‘𝑈))
43	36, 42	sseldd 3938	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ∈ (SubGrp‘𝑈))
44		prssi 4787	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑇 ∈ 𝑉) → {𝑌, 𝑇} ⊆ 𝑉)
45	21, 10, 44	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑌, 𝑇} ⊆ 𝑉)
46		snsspr1 4780	. . . . . . . . . . . . . . . . 17 ⊢ {𝑌} ⊆ {𝑌, 𝑇}
47	46	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑌} ⊆ {𝑌, 𝑇})
48	8, 2	lspss 21105	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ LMod ∧ {𝑌, 𝑇} ⊆ 𝑉 ∧ {𝑌} ⊆ {𝑌, 𝑇}) → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑇}))
49	6, 45, 47, 48	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑇}))
50	49	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑇}))
51		simp3 1156	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤}))
52	50, 51	sseqtrd 3973	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑤}))
53		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
54	53	lsmless2 19726	. . . . . . . . . . . . 13 ⊢ (((𝑁‘{𝑋}) ∈ (SubGrp‘𝑈) ∧ (𝑁‘{𝑤}) ∈ (SubGrp‘𝑈) ∧ (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑤})) → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑤})))
55	40, 43, 52, 54	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑤})))
56	8, 2, 53, 6, 30, 21	lsmpr 21210	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))
57	56	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))
58		prcom 4698	. . . . . . . . . . . . . 14 ⊢ {𝑤, 𝑋} = {𝑋, 𝑤}
59	58	fveq2i 6884	. . . . . . . . . . . . 13 ⊢ (𝑁‘{𝑤, 𝑋}) = (𝑁‘{𝑋, 𝑤})
60	8, 2, 53, 34, 31, 29	lsmpr 21210	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋, 𝑤}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑤})))
61	59, 60	eqtrid 2810	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑤, 𝑋}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑤})))
62	55, 57, 61	3sstr4d 3992	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑤, 𝑋}))
63	62	ssneld 3939	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑋}) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})))
64	8, 1, 2	lspsncl 21098	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
65	6, 9, 64	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
66	65	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
67	36, 66	sseldd 3938	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑈))
68		snsspr2 4781	. . . . . . . . . . . . . . . . 17 ⊢ {𝑇} ⊆ {𝑌, 𝑇}
69	68	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑇} ⊆ {𝑌, 𝑇})
70	8, 2	lspss 21105	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ LMod ∧ {𝑌, 𝑇} ⊆ 𝑉 ∧ {𝑇} ⊆ {𝑌, 𝑇}) → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑇}))
71	6, 45, 69, 70	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑇}))
72	71	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑇}))
73	72, 51	sseqtrd 3973	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑤}))
74	53	lsmless2 19726	. . . . . . . . . . . . 13 ⊢ (((𝑁‘{𝑍}) ∈ (SubGrp‘𝑈) ∧ (𝑁‘{𝑤}) ∈ (SubGrp‘𝑈) ∧ (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑤})) → ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑇})) ⊆ ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑤})))
75	67, 43, 73, 74	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑇})) ⊆ ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑤})))
76	8, 2, 53, 6, 9, 10	lsmpr 21210	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑍, 𝑇}) = ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑇})))
77	76	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍, 𝑇}) = ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑇})))
78		prcom 4698	. . . . . . . . . . . . . 14 ⊢ {𝑤, 𝑍} = {𝑍, 𝑤}
79	78	fveq2i 6884	. . . . . . . . . . . . 13 ⊢ (𝑁‘{𝑤, 𝑍}) = (𝑁‘{𝑍, 𝑤})
80	8, 2, 53, 34, 32, 29	lsmpr 21210	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍, 𝑤}) = ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑤})))
81	79, 80	eqtrid 2810	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑤, 𝑍}) = ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑤})))
82	75, 77, 81	3sstr4d 3992	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍, 𝑇}) ⊆ (𝑁‘{𝑤, 𝑍}))
83	82	ssneld 3939	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑍}) → ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
84	63, 83	anim12d 620	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → ((¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑍})) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}))))
85	84	reximdv 3180	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑍})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}))))
86	33, 85	mpd 16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
87	86	rexlimdv3a 3170	. . . . . 6 ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤}) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}))))
88	87	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → (∃𝑤 ∈ 𝑉 (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤}) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}))))
89	27, 88	mpd 16	. . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
90	3, 4, 8, 2, 5, 21, 30, 10	dvh3dim2 42242	. . . . . 6 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑇})))
91	90	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑇})))
92		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇}))
93		prcom 4698	. . . . . . . . . . . 12 ⊢ {𝑌, 𝑋} = {𝑋, 𝑌}
94	93	fveq2i 6884	. . . . . . . . . . 11 ⊢ (𝑁‘{𝑌, 𝑋}) = (𝑁‘{𝑋, 𝑌})
95	94	eleq2i 2855	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑁‘{𝑌, 𝑋}) ↔ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
96	95	notbii 323	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑋}) ↔ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
97	96	a1i 11	. . . . . . . 8 ⊢ ((𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇}) → (¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑋}) ↔ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})))
98		eleq2 2852	. . . . . . . . 9 ⊢ ((𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇}) → (𝑧 ∈ (𝑁‘{𝑌, 𝑇}) ↔ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
99	98	notbid 321	. . . . . . . 8 ⊢ ((𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇}) → (¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑇}) ↔ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
100	97, 99	anbi12d 643	. . . . . . 7 ⊢ ((𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇}) → ((¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑇})) ↔ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}))))
101	92, 100	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})) → ((¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑇})) ↔ (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}))))
102	101	rexbidv 3189	. . . . 5 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})) → (∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑇})) ↔ ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}))))
103	91, 102	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
104	89, 103	jaodan 972	. . 3 ⊢ ((𝜑 ∧ ((𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇}) ∨ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇}))) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
105	18, 104	syldan 602	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
106	3, 4, 8, 2, 5, 21, 30, 10	dvh3dim2 42242	. . . 4 ⊢ (𝜑 → ∃𝑤 ∈ 𝑉 (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇})))
107	106	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → ∃𝑤 ∈ 𝑉 (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇})))
108		simpl1l 1243	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → 𝜑)
109	108, 6	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑈 ∈ LMod)
110		simpl2 1211	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑤 ∈ 𝑉)
111	108, 21	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ 𝑉)
112		eqid 2763	. . . . . . . 8 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
113	8, 112	lmodvacl 20996	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤(+_g‘𝑈)𝑌) ∈ 𝑉)
114	109, 110, 111, 113	syl3anc 1398	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑤(+_g‘𝑈)𝑌) ∈ 𝑉)
115	8, 1, 2, 6, 30, 21	lspprcl 21099	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
116	108, 115	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
117	8, 2, 6, 30, 21	lspprid2 21119	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑌}))
118	108, 117	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑋, 𝑌}))
119		simpl3l 1247	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}))
120	94	eleq2i 2855	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ↔ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
121	119, 120	sylnib 331	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
122	8, 112, 1, 109, 116, 118, 110, 121	lssvancl2 21067	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → ¬ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑋, 𝑌}))
123	108, 11	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑍, 𝑇}) ∈ (LSubSp‘𝑈))
124		simpr 489	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑤 ∈ (𝑁‘{𝑍, 𝑇}))
125		simpl1r 1244	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇}))
126	8, 112, 1, 109, 123, 124, 111, 125	lssvancl1 21066	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → ¬ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑍, 𝑇}))
127		eleq1 2851	. . . . . . . . 9 ⊢ (𝑧 = (𝑤(+_g‘𝑈)𝑌) → (𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑋, 𝑌})))
128	127	notbid 321	. . . . . . . 8 ⊢ (𝑧 = (𝑤(+_g‘𝑈)𝑌) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ¬ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑋, 𝑌})))
129		eleq1 2851	. . . . . . . . 9 ⊢ (𝑧 = (𝑤(+_g‘𝑈)𝑌) → (𝑧 ∈ (𝑁‘{𝑍, 𝑇}) ↔ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑍, 𝑇})))
130	129	notbid 321	. . . . . . . 8 ⊢ (𝑧 = (𝑤(+_g‘𝑈)𝑌) → (¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}) ↔ ¬ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑍, 𝑇})))
131	128, 130	anbi12d 643	. . . . . . 7 ⊢ (𝑧 = (𝑤(+_g‘𝑈)𝑌) → ((¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})) ↔ (¬ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑍, 𝑇}))))
132	131	rspcev 3581	. . . . . 6 ⊢ (((𝑤(+_g‘𝑈)𝑌) ∈ 𝑉 ∧ (¬ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑍, 𝑇}))) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
133	114, 122, 126, 132	syl12anc 849	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
134		simpl2 1211	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑤 ∈ 𝑉)
135		simpl3l 1247	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}))
136	135, 120	sylnib 331	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
137		simpr 489	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇}))
138		eleq1 2851	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})))
139	138	notbid 321	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})))
140		eleq1 2851	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝑁‘{𝑍, 𝑇}) ↔ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})))
141	140	notbid 321	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}) ↔ ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})))
142	139, 141	anbi12d 643	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})) ↔ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇}))))
143	142	rspcev 3581	. . . . . 6 ⊢ ((𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇}))) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
144	134, 136, 137, 143	syl12anc 849	. . . . 5 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
145	133, 144	pm2.61dan 824	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
146	145	rexlimdv3a 3170	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → (∃𝑤 ∈ 𝑉 (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}))))
147	107, 146	mpd 16	. 2 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
148	105, 147	pm2.61dan 824	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∃wrex 3089 ⊆ wss 3905 ⊊ wpss 3906 {csn 4589 {cpr 4591 ‘cfv 6536 (class class class)co 7410 Basecbs 17264 +_gcplusg 17305 SubGrpcsubg 19181 LSSumclsm 19699 LModclmod 20981 LSubSpclss 21052 LSpanclspn 21092 LVecclvec 21223 HLchlt 40144 LHypclh 40778 DVecHcdvh 41872

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 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-riotaBAD 39747

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 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-undef 8265 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-n0 12500 df-z 12587 df-uz 12858 df-fz 13531 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-sca 17321 df-vsca 17322 df-0g 17489 df-proset 18345 df-poset 18364 df-plt 18379 df-lub 18395 df-glb 18396 df-join 18397 df-meet 18398 df-p0 18474 df-p1 18475 df-lat 18483 df-clat 18550 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-grp 18998 df-minusg 18999 df-sbg 19000 df-subg 19184 df-cntz 19382 df-lsm 19701 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-drng 20829 df-lmod 20983 df-lss 21053 df-lsp 21093 df-lvec 21224 df-lsatoms 39770 df-oposet 39970 df-ol 39972 df-oml 39973 df-covers 40060 df-ats 40061 df-atl 40092 df-cvlat 40116 df-hlat 40145 df-llines 40292 df-lplanes 40293 df-lvols 40294 df-lines 40295 df-psubsp 40297 df-pmap 40298 df-padd 40590 df-lhyp 40782 df-laut 40783 df-ldil 40898 df-ltrn 40899 df-trl 40953 df-tgrp 41537 df-tendo 41549 df-edring 41551 df-dveca 41797 df-disoa 41823 df-dvech 41873 df-dib 41933 df-dic 41967 df-dih 42023 df-doch 42142 df-djh 42189

This theorem is referenced by: (None)

W3C validator