dvh3dim3N - Mathbox for Norm Megill

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Theorem dvh3dim3N 41731

Description: There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 41730 everywhere. If this one is needed, make dvh3dim2 41730 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 2736	. . . . 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 41392	. . . . . 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 20931	. . . . . 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 20951	. . . . . 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 20945	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘{𝑍, 𝑇})) → (𝑁‘{𝑌, 𝑇}) ⊆ (𝑁‘{𝑍, 𝑇}))
17		sspss 4054	. . . 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 41391	. . . . . . 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 20931	. . . . . . 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 21110	. . . . 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 41730	. . . . . . . 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 20911	. . . . . . . . . . . . . . 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 20930	. . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑈))
41	8, 1, 2	lspsncl 20930	. . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑤}) ∈ (SubGrp‘𝑈))
44		prssi 4777	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝑇 ∈ 𝑉) → {𝑌, 𝑇} ⊆ 𝑉)
45	21, 10, 44	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑌, 𝑇} ⊆ 𝑉)
46		snsspr1 4770	. . . . . . . . . . . . . . . . 17 ⊢ {𝑌} ⊆ {𝑌, 𝑇}
47	46	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑌} ⊆ {𝑌, 𝑇})
48	8, 2	lspss 20937	. . . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑤}))
53		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
54	53	lsmless2 19592	. . . . . . . . . . . . 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 21043	. . . . . . . . . . . . 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 4689	. . . . . . . . . . . . . 14 ⊢ {𝑤, 𝑋} = {𝑋, 𝑤}
59	58	fveq2i 6837	. . . . . . . . . . . . 13 ⊢ (𝑁‘{𝑤, 𝑋}) = (𝑁‘{𝑋, 𝑤})
60	8, 2, 53, 34, 31, 29	lsmpr 21043	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋, 𝑤}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑤})))
61	59, 60	eqtrid 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑤, 𝑋}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑤})))
62	55, 57, 61	3sstr4d 3989	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑤, 𝑋}))
63	62	ssneld 3935	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (¬ 𝑧 ∈ (𝑁‘{𝑤, 𝑋}) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})))
64	8, 1, 2	lspsncl 20930	. . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍}) ∈ (SubGrp‘𝑈))
68		snsspr2 4771	. . . . . . . . . . . . . . . . 17 ⊢ {𝑇} ⊆ {𝑌, 𝑇}
69	68	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑇} ⊆ {𝑌, 𝑇})
70	8, 2	lspss 20937	. . . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑤}))
74	53	lsmless2 19592	. . . . . . . . . . . . 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 21043	. . . . . . . . . . . . 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 4689	. . . . . . . . . . . . . 14 ⊢ {𝑤, 𝑍} = {𝑍, 𝑤}
79	78	fveq2i 6837	. . . . . . . . . . . . 13 ⊢ (𝑁‘{𝑤, 𝑍}) = (𝑁‘{𝑍, 𝑤})
80	8, 2, 53, 34, 32, 29	lsmpr 21043	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍, 𝑤}) = ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑤})))
81	79, 80	eqtrid 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑤, 𝑍}) = ((𝑁‘{𝑍})(LSSum‘𝑈)(𝑁‘{𝑤})))
82	75, 77, 81	3sstr4d 3989	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ (𝑁‘{𝑌, 𝑇}) = (𝑁‘{𝑤})) → (𝑁‘{𝑍, 𝑇}) ⊆ (𝑁‘{𝑤, 𝑍}))
83	82	ssneld 3935	. . . . . . . . . 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 3151	. . . . . . . 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 3141	. . . . . 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 41730	. . . . . 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 4689	. . . . . . . . . . . 12 ⊢ {𝑌, 𝑋} = {𝑋, 𝑌}
94	93	fveq2i 6837	. . . . . . . . . . 11 ⊢ (𝑁‘{𝑌, 𝑋}) = (𝑁‘{𝑋, 𝑌})
95	94	eleq2i 2828	. . . . . . . . . 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 2825	. . . . . . . . 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 3160	. . . . 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 41730	. . . 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 2736	. . . . . . . 8 ⊢ (+_g‘𝑈) = (+_g‘𝑈)
113	8, 112	lmodvacl 20828	. . . . . . 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 20931	. . . . . . . 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 20951	. . . . . . . 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 2828	. . . . . . . 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 20899	. . . . . 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 20898	. . . . . 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 2824	. . . . . . . . 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 2824	. . . . . . . . 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 3576	. . . . . 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 2824	. . . . . . . . 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 2824	. . . . . . . . 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 3576	. . . . . 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 3141	. . 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 3060 ⊆ wss 3901 ⊊ wpss 3902 {csn 4580 {cpr 4582 ‘cfv 6492 (class class class)co 7358 Basecbs 17138 +_gcplusg 17179 SubGrpcsubg 19052 LSSumclsm 19565 LModclmod 20813 LSubSpclss 20884 LSpanclspn 20924 LVecclvec 21056 HLchlt 39632 LHypclh 40266 DVecHcdvh 41360

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-riotaBAD 39235

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 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 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8767 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-n0 12404 df-z 12491 df-uz 12754 df-fz 13426 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-0g 17363 df-proset 18219 df-poset 18238 df-plt 18253 df-lub 18269 df-glb 18270 df-join 18271 df-meet 18272 df-p0 18348 df-p1 18349 df-lat 18357 df-clat 18424 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cntz 19248 df-lsm 19567 df-cmn 19713 df-abl 19714 df-mgp 20078 df-rng 20090 df-ur 20119 df-ring 20172 df-oppr 20275 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-drng 20666 df-lmod 20815 df-lss 20885 df-lsp 20925 df-lvec 21057 df-lsatoms 39258 df-oposet 39458 df-ol 39460 df-oml 39461 df-covers 39548 df-ats 39549 df-atl 39580 df-cvlat 39604 df-hlat 39633 df-llines 39780 df-lplanes 39781 df-lvols 39782 df-lines 39783 df-psubsp 39785 df-pmap 39786 df-padd 40078 df-lhyp 40270 df-laut 40271 df-ldil 40386 df-ltrn 40387 df-trl 40441 df-tgrp 41025 df-tendo 41037 df-edring 41039 df-dveca 41285 df-disoa 41311 df-dvech 41361 df-dib 41421 df-dic 41455 df-dih 41511 df-doch 41630 df-djh 41677

This theorem is referenced by: (None)

W3C validator