mapdh9a - Mathbox for Norm Megill

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Theorem mapdh9a 41772

Description: Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 41773? (Contributed by NM, 14-May-2015.)

**Hypotheses**
Ref	Expression
mapdh8a.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdh8a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdh8a.v	⊢ 𝑉 = (Base‘𝑈)
mapdh8a.s	⊢ − = (-_g‘𝑈)
mapdh8a.o	⊢ 0 = (0_g‘𝑈)
mapdh8a.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdh8a.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdh8a.d	⊢ 𝐷 = (Base‘𝐶)
mapdh8a.r	⊢ 𝑅 = (-_g‘𝐶)
mapdh8a.q	⊢ 𝑄 = (0_g‘𝐶)
mapdh8a.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdh8a.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdh8a.i	⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
mapdh8a.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdh8h.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh8h.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdh9a.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdh9a.t	⊢ (𝜑 → 𝑇 ∈ 𝑉)

**Assertion**
Ref	Expression
mapdh9a	⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))

Distinct variable groups: 𝑥,ℎ, − 0 ,ℎ,𝑥 𝐶,ℎ 𝐷,ℎ,𝑥 ℎ,𝐹,𝑥 ℎ,𝐼 ℎ,𝐽,𝑥 ℎ,𝑀,𝑥 ℎ,𝑁,𝑥 𝜑,ℎ 𝑅,ℎ,𝑥 𝑥,𝑄 𝑇,ℎ,𝑥 𝑈,ℎ ℎ,𝑋,𝑥 𝑥,𝐼 ℎ,𝑉 𝑦,𝑧,𝐷 𝑦,𝐹,𝑧 𝑦,𝐼,𝑧 𝑦,𝑁,𝑧 𝑦, 0 ,𝑧 𝑦,𝑇,𝑧 𝑧,𝑈 𝑦,𝑉,𝑧 𝑦,𝑋,𝑧 𝜑,𝑦,𝑧 𝑧,ℎ,𝑥 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥,𝑦,𝑧) 𝑄(𝑦,𝑧,ℎ) 𝑅(𝑦,𝑧) 𝑈(𝑥,𝑦) 𝐻(𝑥,𝑦,𝑧,ℎ) 𝐽(𝑦,𝑧) 𝐾(𝑥,𝑦,𝑧,ℎ) 𝑀(𝑦,𝑧) − (𝑦,𝑧) 𝑉(𝑥) 𝑊(𝑥,𝑦,𝑧,ℎ)

Proof of Theorem mapdh9a Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapdh8a.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdh8a.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		mapdh8a.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
4		mapdh8a.s	. . . . . . 7 ⊢ − = (-_g‘𝑈)
5		mapdh8a.o	. . . . . . 7 ⊢ 0 = (0_g‘𝑈)
6		mapdh8a.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
7		mapdh8a.c	. . . . . . 7 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		mapdh8a.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝐶)
9		mapdh8a.r	. . . . . . 7 ⊢ 𝑅 = (-_g‘𝐶)
10		mapdh8a.q	. . . . . . 7 ⊢ 𝑄 = (0_g‘𝐶)
11		mapdh8a.j	. . . . . . 7 ⊢ 𝐽 = (LSpan‘𝐶)
12		mapdh8a.m	. . . . . . 7 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
13		mapdh8a.i	. . . . . . 7 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
14		mapdh8a.k	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15	14	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		mapdh8h.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
17	16	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → 𝐹 ∈ 𝐷)
18		mapdh8h.mn	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
19	18	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
20		mapdh9a.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
22		simp3ll 1243	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → 𝑧 ∈ (𝑉 ∖ { 0 }))
23		simp3rl 1245	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → 𝑤 ∈ (𝑉 ∖ { 0 }))
24		simplrl 777	. . . . . . . . 9 ⊢ (((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}))
25	24	3ad2ant3 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}))
26	25	necomd 2994	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑧}))
27		simprrl 781	. . . . . . . . 9 ⊢ (((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}))
28	27	3ad2ant3 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}))
29	28	necomd 2994	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤}))
30		simplrr 778	. . . . . . . 8 ⊢ (((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))
31	30	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))
32		simprrr 782	. . . . . . . 8 ⊢ (((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))
33	32	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))
34		mapdh9a.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
35	34	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → 𝑇 ∈ 𝑉)
36	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 35	mapdh8 41771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) ∧ ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))) → (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) = (𝐼‘⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩))
37	36	3exp 1118	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → (((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))) → (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) = (𝐼‘⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩))))
38	37	ralrimivv 3198	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))) → (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) = (𝐼‘⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩)))
39	20	eldifad 3975	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
40	1, 2, 3, 6, 14, 39, 34	dvh3dim 41429	. . . . . . 7 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}))
41		eqid 2735	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
42	1, 2, 14	dvhlmod 41093	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LMod)
43	42	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LMod)
44	3, 41, 6, 42, 39, 34	lspprcl 20994	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈))
45	44	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈))
46		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ 𝑉)
47		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}))
48	5, 41, 43, 45, 46, 47	lssneln0 20969	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ (𝑉 ∖ { 0 }))
49	1, 2, 14	dvhlvec 41092	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
50	49	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LVec)
51	39	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑋 ∈ 𝑉)
52	34	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑇 ∈ 𝑉)
53	3, 6, 50, 46, 51, 52, 47	lspindpi 21152	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))
54	48, 53	jca 511	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))))
55	54	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))))
56	55	reximdva 3166	. . . . . . 7 ⊢ (𝜑 → (∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → ∃𝑧 ∈ 𝑉 (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))))
57	40, 56	mpd 15	. . . . . 6 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))))
58	14	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
59	16	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝐹 ∈ 𝐷)
60	18	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
61	20	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
62		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑧 ∈ 𝑉)
63		simprrl 781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}))
64	63	necomd 2994	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑧}))
65	10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 58, 59, 60, 61, 62, 64	mapdhcl 41710	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝐼‘⟨𝑋, 𝐹, 𝑧⟩) ∈ 𝐷)
66		eqidd 2736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝐼‘⟨𝑋, 𝐹, 𝑧⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑧⟩))
67		simprl 771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑧 ∈ (𝑉 ∖ { 0 }))
68	10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 58, 59, 60, 61, 67, 65, 64	mapdheq 41711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → ((𝐼‘⟨𝑋, 𝐹, 𝑧⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑧⟩) ↔ ((𝑀‘(𝑁‘{𝑧})) = (𝐽‘{(𝐼‘⟨𝑋, 𝐹, 𝑧⟩)}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑧)})) = (𝐽‘{(𝐹𝑅(𝐼‘⟨𝑋, 𝐹, 𝑧⟩))}))))
69	66, 68	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → ((𝑀‘(𝑁‘{𝑧})) = (𝐽‘{(𝐼‘⟨𝑋, 𝐹, 𝑧⟩)}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑧)})) = (𝐽‘{(𝐹𝑅(𝐼‘⟨𝑋, 𝐹, 𝑧⟩))})))
70	69	simpld 494	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝑀‘(𝑁‘{𝑧})) = (𝐽‘{(𝐼‘⟨𝑋, 𝐹, 𝑧⟩)}))
71	34	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑇 ∈ 𝑉)
72		simprrr 782	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))
73	10, 13, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 58, 65, 70, 67, 71, 72	mapdhcl 41710	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) ∈ 𝐷)
74	73	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) → (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) ∈ 𝐷))
75	74	ancld 550	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) → ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) ∈ 𝐷)))
76	75	reximdva 3166	. . . . . 6 ⊢ (𝜑 → (∃𝑧 ∈ 𝑉 (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) → ∃𝑧 ∈ 𝑉 ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) ∈ 𝐷)))
77	57, 76	mpd 15	. . . . 5 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) ∈ 𝐷))
78		eleq1w 2822	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝑉 ∖ { 0 }) ↔ 𝑤 ∈ (𝑉 ∖ { 0 })))
79		sneq 4641	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
80	79	fveq2d 6911	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑁‘{𝑧}) = (𝑁‘{𝑤}))
81	80	neeq1d 2998	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ↔ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋})))
82	80	neeq1d 2998	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}) ↔ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))
83	81, 82	anbi12d 632	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})) ↔ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))
84	78, 83	anbi12d 632	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ↔ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))))
85		oteq1 4887	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩ = ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)
86		oteq3 4889	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ⟨𝑋, 𝐹, 𝑧⟩ = ⟨𝑋, 𝐹, 𝑤⟩)
87	86	fveq2d 6911	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝐼‘⟨𝑋, 𝐹, 𝑧⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
88	87	oteq2d 4891	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩ = ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩)
89	85, 88	eqtrd 2775	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩ = ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩)
90	89	fveq2d 6911	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) = (𝐼‘⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩))
91	84, 90	reusv3 5411	. . . . 5 ⊢ (∃𝑧 ∈ 𝑉 ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) ∈ 𝐷) → (∀𝑧 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))) → (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) = (𝐼‘⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩)) ↔ ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩))))
92	77, 91	syl 17	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) ∧ (𝑤 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))) → (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) = (𝐼‘⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩)) ↔ ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩))))
93	38, 92	mpbid 232	. . 3 ⊢ (𝜑 → ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
94		ioran 985	. . . . . . . 8 ⊢ (¬ (𝑧 ∈ (𝑁‘{𝑋}) ∨ 𝑧 ∈ (𝑁‘{𝑇})) ↔ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})))
95		elun 4163	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ↔ (𝑧 ∈ (𝑁‘{𝑋}) ∨ 𝑧 ∈ (𝑁‘{𝑇})))
96	94, 95	xchnxbir 333	. . . . . . 7 ⊢ (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ↔ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})))
97	42	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑈 ∈ LMod)
98	3, 41, 6	lspsncl 20993	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
99	42, 39, 98	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
100	99	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
101		simplr 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑧 ∈ 𝑉)
102		simprl 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → ¬ 𝑧 ∈ (𝑁‘{𝑋}))
103	5, 41, 97, 100, 101, 102	lssneln0 20969	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑧 ∈ (𝑉 ∖ { 0 }))
104	103	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑧 ∈ (𝑉 ∖ { 0 })))
105	42	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑈 ∈ LMod)
106		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ 𝑉)
107	39	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑋 ∈ 𝑉)
108		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → ¬ 𝑧 ∈ (𝑁‘{𝑋}))
109	3, 6, 105, 106, 107, 108	lspsnne2 21138	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}))
110	109	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑋}) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})))
111	42	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑈 ∈ LMod)
112		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑧 ∈ 𝑉)
113	34	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑇 ∈ 𝑉)
114		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑇}))
115	3, 6, 111, 112, 113, 114	lspsnne2 21138	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))
116	115	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑇}) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))
117	110, 116	anim12d 609	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))))
118	104, 117	jcad 512	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))))
119	96, 118	biimtrid 242	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))))
120	119	imim1d 82	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)) → (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩))))
121	120	ralimdva 3165	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ 𝑉 ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)) → ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩))))
122	121	reximdv 3168	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 ((𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)) → ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩))))
123	93, 122	mpd 15	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))
124	3, 6, 42, 39, 34	lspprid1 21013	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋, 𝑇}))
125	41, 6, 42, 44, 124	ellspsn5 21012	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑇}))
126	3, 6, 42, 39, 34	lspprid2 21014	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝑋, 𝑇}))
127	41, 6, 42, 44, 126	ellspsn5 21012	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑋, 𝑇}))
128	125, 127	unssd 4202	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ⊆ (𝑁‘{𝑋, 𝑇}))
129	128	ssneld 3997	. . . . 5 ⊢ (𝜑 → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))))
130	129	reximdv 3168	. . . 4 ⊢ (𝜑 → (∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))))
131	40, 130	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})))
132		reusv1 5403	. . 3 ⊢ (∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → (∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)) ↔ ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩))))
133	131, 132	syl 17	. 2 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)) ↔ ∃𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩))))
134	123, 133	mpbird 257	1 ⊢ (𝜑 → ∃!𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ∃!wreu 3376 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ifcif 4531 {csn 4631 {cpr 4633 ⟨cotp 4639 ↦ cmpt 5231 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 1^st c1st 8011 2^nd c2nd 8012 Basecbs 17245 0_gc0g 17486 -_gcsg 18966 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 LVecclvec 21119 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 LCDualclcd 41569 mapdcmpd 41607

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lcv 39001 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378 df-lcdual 41570 df-mapd 41608

This theorem is referenced by: hdmap1eulem 41805

W3C validator