mapdh9a - Mathbox for Norm Megill

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Theorem mapdh9a 38475

Description: Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 38476? (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 1126	. . . . . . 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 1126	. . . . . . 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 1126	. . . . . . 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 1126	. . . . . . 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 1237	. . . . . . 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 1239	. . . . . . 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 773	. . . . . . . . 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 1128	. . . . . . . 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 3039	. . . . . . 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 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‘{𝑤}) ≠ (𝑁‘{𝑋}))
28	27	3ad2ant3 1128	. . . . . . . 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 3039	. . . . . . 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 774	. . . . . . . 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 1128	. . . . . . 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 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‘{𝑤}) ≠ (𝑁‘{𝑇}))
33	32	3ad2ant3 1128	. . . . . . 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 1126	. . . . . . 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 38474	. . . . . 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 1112	. . . . 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 3157	. . . 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 3871	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
40	1, 2, 3, 6, 14, 39, 34	dvh3dim 38132	. . . . . . 7 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}))
41		eqid 2795	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
42	1, 2, 14	dvhlmod 37796	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LMod)
43	42	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LMod)
44	3, 41, 6, 42, 39, 34	lspprcl 19440	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈))
45	44	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈))
46		simplr 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ 𝑉)
47		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}))
48	5, 41, 43, 45, 46, 47	lssneln0 19414	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ (𝑉 ∖ { 0 }))
49	1, 2, 14	dvhlvec 37795	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
50	49	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LVec)
51	39	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑋 ∈ 𝑉)
52	34	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑇 ∈ 𝑉)
53	3, 6, 50, 46, 51, 52, 47	lspindpi 19594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))
54	48, 53	jca 512	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))))
55	54	ex 413	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))))
56	55	reximdva 3237	. . . . . . 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 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
59	16	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝐹 ∈ 𝐷)
60	18	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
61	20	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
62		simplr 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑧 ∈ 𝑉)
63		simprrl 777	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}))
64	63	necomd 3039	. . . . . . . . . . 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 38413	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝐼‘⟨𝑋, 𝐹, 𝑧⟩) ∈ 𝐷)
66		eqidd 2796	. . . . . . . . . . . 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 767	. . . . . . . . . . . . 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 38414	. . . . . . . . . . . 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 233	. . . . . . . . . . 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 495	. . . . . . . . . 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 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑇 ∈ 𝑉)
72		simprrr 778	. . . . . . . . . 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 38413	. . . . . . . . 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 413	. . . . . . . 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 551	. . . . . . 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 3237	. . . . . 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 2865	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝑉 ∖ { 0 }) ↔ 𝑤 ∈ (𝑉 ∖ { 0 })))
79		sneq 4482	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
80	79	fveq2d 6542	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑁‘{𝑧}) = (𝑁‘{𝑤}))
81	80	neeq1d 3043	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ↔ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋})))
82	80	neeq1d 3043	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}) ↔ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))
83	81, 82	anbi12d 630	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})) ↔ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))
84	78, 83	anbi12d 630	. . . . . 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 4719	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩ = ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)
86		oteq3 4721	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ⟨𝑋, 𝐹, 𝑧⟩ = ⟨𝑋, 𝐹, 𝑤⟩)
87	86	fveq2d 6542	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝐼‘⟨𝑋, 𝐹, 𝑧⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
88	87	oteq2d 4723	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩ = ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩)
89	85, 88	eqtrd 2831	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩ = ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩)
90	89	fveq2d 6542	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) = (𝐼‘⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩))
91	84, 90	reusv3 5197	. . . . 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 233	. . 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 978	. . . . . . . 8 ⊢ (¬ (𝑧 ∈ (𝑁‘{𝑋}) ∨ 𝑧 ∈ (𝑁‘{𝑇})) ↔ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})))
95		elun 4046	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ↔ (𝑧 ∈ (𝑁‘{𝑋}) ∨ 𝑧 ∈ (𝑁‘{𝑇})))
96	94, 95	xchnxbir 334	. . . . . . 7 ⊢ (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ↔ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})))
97	42	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑈 ∈ LMod)
98	3, 41, 6	lspsncl 19439	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
99	42, 39, 98	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
100	99	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
101		simplr 765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑧 ∈ 𝑉)
102		simprl 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → ¬ 𝑧 ∈ (𝑁‘{𝑋}))
103	5, 41, 97, 100, 101, 102	lssneln0 19414	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑧 ∈ (𝑉 ∖ { 0 }))
104	103	ex 413	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑧 ∈ (𝑉 ∖ { 0 })))
105	42	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑈 ∈ LMod)
106		simplr 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ 𝑉)
107	39	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑋 ∈ 𝑉)
108		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → ¬ 𝑧 ∈ (𝑁‘{𝑋}))
109	3, 6, 105, 106, 107, 108	lspsnne2 19580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}))
110	109	ex 413	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑋}) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})))
111	42	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑈 ∈ LMod)
112		simplr 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑧 ∈ 𝑉)
113	34	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑇 ∈ 𝑉)
114		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑇}))
115	3, 6, 111, 112, 113, 114	lspsnne2 19580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))
116	115	ex 413	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑇}) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))
117	110, 116	anim12d 608	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))))
118	104, 117	jcad 513	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))))
119	96, 118	syl5bi 243	. . . . . 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 3144	. . . 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 3236	. . 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 19459	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋, 𝑇}))
125	41, 6, 42, 44, 124	lspsnel5a 19458	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑇}))
126	3, 6, 42, 39, 34	lspprid2 19460	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝑋, 𝑇}))
127	41, 6, 42, 44, 126	lspsnel5a 19458	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑋, 𝑇}))
128	125, 127	unssd 4083	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ⊆ (𝑁‘{𝑋, 𝑇}))
129	128	ssneld 3891	. . . . 5 ⊢ (𝜑 → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))))
130	129	reximdv 3236	. . . 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 5189	. . 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 258	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 207 ∧ wa 396 ∨ wo 842 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 ∃wrex 3106 ∃!wreu 3107 Vcvv 3437 ∖ cdif 3856 ∪ cun 3857 ifcif 4381 {csn 4472 {cpr 4474 ⟨cotp 4480 ↦ cmpt 5041 ‘cfv 6225 ℩crio 6976 (class class class)co 7016 1^st c1st 7543 2^nd c2nd 7544 Basecbs 16312 0_gc0g 16542 -_gcsg 17863 LModclmod 19324 LSubSpclss 19393 LSpanclspn 19433 LVecclvec 19564 HLchlt 36036 LHypclh 36670 DVecHcdvh 37764 LCDualclcd 38272 mapdcmpd 38310

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-riotaBAD 35639

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-ot 4481 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-tpos 7743 df-undef 7790 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-n0 11746 df-z 11830 df-uz 12094 df-fz 12743 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-sca 16410 df-vsca 16411 df-0g 16544 df-mre 16686 df-mrc 16687 df-acs 16689 df-proset 17367 df-poset 17385 df-plt 17397 df-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-p0 17478 df-p1 17479 df-lat 17485 df-clat 17547 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-subg 18030 df-cntz 18188 df-oppg 18215 df-lsm 18491 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-dvr 19123 df-drng 19194 df-lmod 19326 df-lss 19394 df-lsp 19434 df-lvec 19565 df-lsatoms 35662 df-lshyp 35663 df-lcv 35705 df-lfl 35744 df-lkr 35772 df-ldual 35810 df-oposet 35862 df-ol 35864 df-oml 35865 df-covers 35952 df-ats 35953 df-atl 35984 df-cvlat 36008 df-hlat 36037 df-llines 36184 df-lplanes 36185 df-lvols 36186 df-lines 36187 df-psubsp 36189 df-pmap 36190 df-padd 36482 df-lhyp 36674 df-laut 36675 df-ldil 36790 df-ltrn 36791 df-trl 36845 df-tgrp 37429 df-tendo 37441 df-edring 37443 df-dveca 37689 df-disoa 37715 df-dvech 37765 df-dib 37825 df-dic 37859 df-dih 37915 df-doch 38034 df-djh 38081 df-lcdual 38273 df-mapd 38311

This theorem is referenced by: hdmap1eulem 38508

W3C validator