dvh3dim3N - Mathbox for Norm Megill

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Theorem dvh3dim3N 41648

Description: There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 41647 everywhere. If this one is needed, make dvh3dim2 41647 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 2734	. . . . 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 41309	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
7	6	adantr 480	. . . . 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 20927	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑍, 𝑇}) ∈ (LSubSp‘𝑈))
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑍, 𝑇}) ∈ (LSubSp‘𝑈))
13		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ (𝑁‘{𝑍, 𝑇}))
14	8, 2, 6, 9, 10	lspprid2 20947	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝑍, 𝑇}))
15	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ (𝑁‘{𝑍, 𝑇}))
16	1, 2, 7, 12, 13, 15	lspprss 20941	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌, 𝑇}) ⊆ (𝑁‘{𝑍, 𝑇}))
17		sspss 4052	. . . 4 ⊢ ((𝑁‘{𝑌, 𝑇}) ⊆ (𝑁‘{𝑍, 𝑇}) ↔ ((𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇}) ∨ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})))
18	16, 17	sylib 218	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → ((𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇}) ∨ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})))
19	3, 4, 5	dvhlvec 41308	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec)
20	19	adantr 480	. . . . . 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 20927	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑇}) ∈ (LSubSp‘𝑈))
23	22	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌, 𝑇}) ∈ (LSubSp‘𝑈))
24	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → 𝑍 ∈ 𝑉)
25	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → 𝑇 ∈ 𝑉)
26		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇}))
27	8, 1, 2, 20, 23, 24, 25, 26	lspprat 21106	. . . . 5 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) ⊊ (𝑁‘{𝑍, 𝑇})) → ∃𝑤 ∈ 𝑉 (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤}))
28	5	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
29		simp2 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → 𝑤 ∈ 𝑉)
30		dvh3dim.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
31	30	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → 𝑋 ∈ 𝑉)
32	9	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → 𝑍 ∈ 𝑉)
33	3, 4, 8, 2, 28, 29, 31, 32	dvh3dim2 41647	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑍})))
34	6	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → 𝑈 ∈ LMod)
35	1	lsssssubg 20907	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
36	34, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
37	8, 1, 2	lspsncl 20926	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
38	6, 30, 37	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
39	38	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
40	36, 39	sseldd 3932	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑈))
41	8, 1, 2	lspsncl 20926	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉) → (𝑁‘{𝑤}) ∈ (LSubSp‘𝑈))
42	34, 29, 41	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ∈ (LSubSp‘𝑈))
43	36, 42	sseldd 3932	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ∈ (SubGrp‘𝑈))
44		prssi 4775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑇 ∈ 𝑉) → {𝑌, 𝑇} ⊆ 𝑉)
45	21, 10, 44	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑌, 𝑇} ⊆ 𝑉)
46		snsspr1 4768	. . . . . . . . . . . . . . . . 17 ⊢ {𝑌} ⊆ {𝑌, 𝑇}
47	46	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑌} ⊆ {𝑌, 𝑇})
48	8, 2	lspss 20933	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ LMod ∧ {𝑌, 𝑇} ⊆ 𝑉 ∧ {𝑌} ⊆ {𝑌, 𝑇}) → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑇}))
49	6, 45, 47, 48	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑇}))
50	49	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑌, 𝑇}))
51		simp3 1138	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤}))
52	50, 51	sseqtrd 3968	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑤}))
53		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
54	53	lsmless2 19588	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 21039	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))
57	56	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))
58		prcom 4687	. . . . . . . . . . . . . 14 ⊢ {𝑤, 𝑋} = {𝑋, 𝑤}
59	58	fveq2i 6835	. . . . . . . . . . . . 13 ⊢ (𝑁‘{𝑤, 𝑋}) = (𝑁‘{𝑋, 𝑤})
60	8, 2, 53, 34, 31, 29	lsmpr 21039	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋, 𝑤}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑤})))
61	59, 60	eqtrid 2781	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑤, 𝑋}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑤})))
62	55, 57, 61	3sstr4d 3987	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑤, 𝑋}))
63	62	ssneld 3933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑋}) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})))
64	8, 1, 2	lspsncl 20926	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
65	6, 9, 64	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
66	65	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
67	36, 66	sseldd 3932	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑈))
68		snsspr2 4769	. . . . . . . . . . . . . . . . 17 ⊢ {𝑇} ⊆ {𝑌, 𝑇}
69	68	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑇} ⊆ {𝑌, 𝑇})
70	8, 2	lspss 20933	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∈ LMod ∧ {𝑌, 𝑇} ⊆ 𝑉 ∧ {𝑇} ⊆ {𝑌, 𝑇}) → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑇}))
71	6, 45, 69, 70	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑇}))
72	71	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑌, 𝑇}))
73	72, 51	sseqtrd 3968	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑤}))
74	53	lsmless2 19588	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 21039	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑍, 𝑇}) = ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑇})))
77	76	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍, 𝑇}) = ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑇})))
78		prcom 4687	. . . . . . . . . . . . . 14 ⊢ {𝑤, 𝑍} = {𝑍, 𝑤}
79	78	fveq2i 6835	. . . . . . . . . . . . 13 ⊢ (𝑁‘{𝑤, 𝑍}) = (𝑁‘{𝑍, 𝑤})
80	8, 2, 53, 34, 32, 29	lsmpr 21039	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍, 𝑤}) = ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑤})))
81	79, 80	eqtrid 2781	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑤, 𝑍}) = ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑤})))
82	75, 77, 81	3sstr4d 3987	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍, 𝑇}) ⊆ (𝑁‘{𝑤, 𝑍}))
83	82	ssneld 3933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑍}) → ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
84	63, 83	anim12d 609	. . . . . . . . 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 3149	. . . . . . . 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 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
87	86	rexlimdv3a 3139	. . . . . 6 ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤}) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}))))
88	87	adantr 480	. . . . 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 15	. . . 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 41647	. . . . . 6 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑇})))
91	90	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑇})))
92		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇}))
93		prcom 4687	. . . . . . . . . . . 12 ⊢ {𝑌, 𝑋} = {𝑋, 𝑌}
94	93	fveq2i 6835	. . . . . . . . . . 11 ⊢ (𝑁‘{𝑌, 𝑋}) = (𝑁‘{𝑋, 𝑌})
95	94	eleq2i 2826	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑁‘{𝑌, 𝑋}) ↔ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))
96	95	notbii 320	. . . . . . . . 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 2823	. . . . . . . . 9 ⊢ ((𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇}) → (𝑧 ∈ (𝑁‘{𝑌, 𝑇}) ↔ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
99	98	notbid 318	. . . . . . . 8 ⊢ ((𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇}) → (¬ 𝑧 ∈ (𝑁‘{𝑌, 𝑇}) ↔ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
100	97, 99	anbi12d 632	. . . . . . 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 17	. . . . . 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 3158	. . . . 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 232	. . . 4 ⊢ ((𝜑 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑍, 𝑇})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
104	89, 103	jaodan 959	. . 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 591	. 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 41647	. . . 4 ⊢ (𝜑 → ∃𝑤 ∈ 𝑉 (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇})))
107	106	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → ∃𝑤 ∈ 𝑉 (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇})))
108		simpl1l 1225	. . . . . . . 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 17	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑈 ∈ LMod)
110		simpl2 1193	. . . . . . 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 17	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑌 ∈ 𝑉)
112		eqid 2734	. . . . . . . 8 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
113	8, 112	lmodvacl 20824	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤(+_g‘𝑈)𝑌) ∈ 𝑉)
114	109, 110, 111, 113	syl3anc 1373	. . . . . 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 20927	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
116	108, 115	syl 17	. . . . . . 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 20947	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑌}))
118	108, 117	syl 17	. . . . . . 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 1229	. . . . . . . 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 2826	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ↔ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}))
121	119, 120	sylnib 328	. . . . . . 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 20895	. . . . . 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 17	. . . . . . 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 484	. . . . . . 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 1226	. . . . . . 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 20894	. . . . . 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 2822	. . . . . . . . 9 ⊢ (𝑧 = (𝑤(+_g‘𝑈)𝑌) → (𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑋, 𝑌})))
128	127	notbid 318	. . . . . . . 8 ⊢ (𝑧 = (𝑤(+_g‘𝑈)𝑌) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ¬ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑋, 𝑌})))
129		eleq1 2822	. . . . . . . . 9 ⊢ (𝑧 = (𝑤(+_g‘𝑈)𝑌) → (𝑧 ∈ (𝑁‘{𝑍, 𝑇}) ↔ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑍, 𝑇})))
130	129	notbid 318	. . . . . . . 8 ⊢ (𝑧 = (𝑤(+_g‘𝑈)𝑌) → (¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}) ↔ ¬ (𝑤(+_g‘𝑈)𝑌) ∈ (𝑁‘{𝑍, 𝑇})))
131	128, 130	anbi12d 632	. . . . . . 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 3574	. . . . . 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 836	. . . . 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 1193	. . . . . 6 ⊢ ((((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) ∧ 𝑤 ∈ 𝑉 ∧ (¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑋}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑌, 𝑇}))) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})) → 𝑤 ∈ 𝑉)
135		simpl3l 1229	. . . . . . 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 328	. . . . . 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 484	. . . . . 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 2822	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})))
139	138	notbid 318	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})))
140		eleq1 2822	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝑁‘{𝑍, 𝑇}) ↔ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})))
141	140	notbid 318	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}) ↔ ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇})))
142	139, 141	anbi12d 632	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})) ↔ (¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑤 ∈ (𝑁‘{𝑍, 𝑇}))))
143	142	rspcev 3574	. . . . . 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 836	. . . . 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 812	. . . 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 3139	. . 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 15	. 2 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))
148	105, 147	pm2.61dan 812	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∃wrex 3058 ⊆ wss 3899 ⊊ wpss 3900 {csn 4578 {cpr 4580 ‘cfv 6490 (class class class)co 7356 Basecbs 17134 +_gcplusg 17175 SubGrpcsubg 19048 LSSumclsm 19561 LModclmod 20809 LSubSpclss 20880 LSpanclspn 20920 LVecclvec 21052 HLchlt 39549 LHypclh 40183 DVecHcdvh 41277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-riotaBAD 39152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-n0 12400 df-z 12487 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-0g 17359 df-proset 18215 df-poset 18234 df-plt 18249 df-lub 18265 df-glb 18266 df-join 18267 df-meet 18268 df-p0 18344 df-p1 18345 df-lat 18353 df-clat 18420 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19051 df-cntz 19244 df-lsm 19563 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20271 df-dvdsr 20291 df-unit 20292 df-invr 20322 df-dvr 20335 df-drng 20662 df-lmod 20811 df-lss 20881 df-lsp 20921 df-lvec 21053 df-lsatoms 39175 df-oposet 39375 df-ol 39377 df-oml 39378 df-covers 39465 df-ats 39466 df-atl 39497 df-cvlat 39521 df-hlat 39550 df-llines 39697 df-lplanes 39698 df-lvols 39699 df-lines 39700 df-psubsp 39702 df-pmap 39703 df-padd 39995 df-lhyp 40187 df-laut 40188 df-ldil 40303 df-ltrn 40304 df-trl 40358 df-tgrp 40942 df-tendo 40954 df-edring 40956 df-dveca 41202 df-disoa 41228 df-dvech 41278 df-dib 41338 df-dic 41372 df-dih 41428 df-doch 41547 df-djh 41594

This theorem is referenced by: (None)

W3C validator