mapdh9a - Mathbox for Norm Megill

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Theorem mapdh9a 41294

Description: Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 41295? (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 1130	. . . . . . 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 1130	. . . . . . 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 1130	. . . . . . 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 1130	. . . . . . 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 1241	. . . . . . 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 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}))))) → 𝑤 ∈ (𝑉 ∖ { 0 }))
24		simplrl 775	. . . . . . . . 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 1132	. . . . . . . 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 2993	. . . . . . 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 779	. . . . . . . . 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 1132	. . . . . . . 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 2993	. . . . . . 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 776	. . . . . . . 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 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}))))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))
32		simprrr 780	. . . . . . . 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 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}))))) → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))
34		mapdh9a.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
35	34	3ad2ant1 1130	. . . . . . 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 41293	. . . . . 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 1116	. . . . 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 3196	. . . 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 3961	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
40	1, 2, 3, 6, 14, 39, 34	dvh3dim 40951	. . . . . . 7 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}))
41		eqid 2728	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
42	1, 2, 14	dvhlmod 40615	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LMod)
43	42	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LMod)
44	3, 41, 6, 42, 39, 34	lspprcl 20869	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈))
45	44	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈))
46		simplr 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ 𝑉)
47		simpr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}))
48	5, 41, 43, 45, 46, 47	lssneln0 20844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ (𝑉 ∖ { 0 }))
49	1, 2, 14	dvhlvec 40614	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
50	49	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LVec)
51	39	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑋 ∈ 𝑉)
52	34	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑇 ∈ 𝑉)
53	3, 6, 50, 46, 51, 52, 47	lspindpi 21027	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))
54	48, 53	jca 510	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))))
55	54	ex 411	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))))
56	55	reximdva 3165	. . . . . . 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 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
59	16	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝐹 ∈ 𝐷)
60	18	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
61	20	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
62		simplr 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑧 ∈ 𝑉)
63		simprrl 779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}))
64	63	necomd 2993	. . . . . . . . . . 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 41232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝐼‘⟨𝑋, 𝐹, 𝑧⟩) ∈ 𝐷)
66		eqidd 2729	. . . . . . . . . . . 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 769	. . . . . . . . . . . . 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 41233	. . . . . . . . . . . 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 231	. . . . . . . . . . 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 493	. . . . . . . . . 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 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑇 ∈ 𝑉)
72		simprrr 780	. . . . . . . . . 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 41232	. . . . . . . . 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 411	. . . . . . . 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 549	. . . . . . 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 3165	. . . . . 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 2812	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝑉 ∖ { 0 }) ↔ 𝑤 ∈ (𝑉 ∖ { 0 })))
79		sneq 4642	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
80	79	fveq2d 6906	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑁‘{𝑧}) = (𝑁‘{𝑤}))
81	80	neeq1d 2997	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ↔ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋})))
82	80	neeq1d 2997	. . . . . . . 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 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 6906	. . . . . . . . 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 2768	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩ = ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩)
90	89	fveq2d 6906	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) = (𝐼‘⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩))
91	84, 90	reusv3 5409	. . . . 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 231	. . 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 981	. . . . . . . 8 ⊢ (¬ (𝑧 ∈ (𝑁‘{𝑋}) ∨ 𝑧 ∈ (𝑁‘{𝑇})) ↔ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})))
95		elun 4149	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ↔ (𝑧 ∈ (𝑁‘{𝑋}) ∨ 𝑧 ∈ (𝑁‘{𝑇})))
96	94, 95	xchnxbir 332	. . . . . . 7 ⊢ (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ↔ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})))
97	42	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑈 ∈ LMod)
98	3, 41, 6	lspsncl 20868	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
99	42, 39, 98	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
100	99	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
101		simplr 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑧 ∈ 𝑉)
102		simprl 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → ¬ 𝑧 ∈ (𝑁‘{𝑋}))
103	5, 41, 97, 100, 101, 102	lssneln0 20844	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑧 ∈ (𝑉 ∖ { 0 }))
104	103	ex 411	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑧 ∈ (𝑉 ∖ { 0 })))
105	42	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑈 ∈ LMod)
106		simplr 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ 𝑉)
107	39	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑋 ∈ 𝑉)
108		simpr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → ¬ 𝑧 ∈ (𝑁‘{𝑋}))
109	3, 6, 105, 106, 107, 108	lspsnne2 21013	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}))
110	109	ex 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑋}) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})))
111	42	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑈 ∈ LMod)
112		simplr 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑧 ∈ 𝑉)
113	34	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑇 ∈ 𝑉)
114		simpr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑇}))
115	3, 6, 111, 112, 113, 114	lspsnne2 21013	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))
116	115	ex 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑇}) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))
117	110, 116	anim12d 607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))))
118	104, 117	jcad 511	. . . . . . 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 241	. . . . . 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 3164	. . . 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 3167	. . 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 20888	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋, 𝑇}))
125	41, 6, 42, 44, 124	lspsnel5a 20887	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑇}))
126	3, 6, 42, 39, 34	lspprid2 20889	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝑋, 𝑇}))
127	41, 6, 42, 44, 126	lspsnel5a 20887	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑋, 𝑇}))
128	125, 127	unssd 4188	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ⊆ (𝑁‘{𝑋, 𝑇}))
129	128	ssneld 3984	. . . . 5 ⊢ (𝜑 → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))))
130	129	reximdv 3167	. . . 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 5401	. . 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 256	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 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∃!wreu 3372 Vcvv 3473 ∖ cdif 3946 ∪ cun 3947 ifcif 4532 {csn 4632 {cpr 4634 ⟨cotp 4640 ↦ cmpt 5235 ‘cfv 6553 ℩crio 7381 (class class class)co 7426 1^st c1st 7997 2^nd c2nd 7998 Basecbs 17187 0_gc0g 17428 -_gcsg 18899 LModclmod 20750 LSubSpclss 20822 LSpanclspn 20862 LVecclvec 20994 HLchlt 38854 LHypclh 39489 DVecHcdvh 40583 LCDualclcd 41091 mapdcmpd 41129

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-riotaBAD 38457

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-undef 8285 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-0g 17430 df-mre 17573 df-mrc 17574 df-acs 17576 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-p1 18425 df-lat 18431 df-clat 18498 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cntz 19275 df-oppg 19304 df-lsm 19598 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-dvr 20347 df-drng 20633 df-lmod 20752 df-lss 20823 df-lsp 20863 df-lvec 20995 df-lsatoms 38480 df-lshyp 38481 df-lcv 38523 df-lfl 38562 df-lkr 38590 df-ldual 38628 df-oposet 38680 df-ol 38682 df-oml 38683 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 df-hlat 38855 df-llines 39003 df-lplanes 39004 df-lvols 39005 df-lines 39006 df-psubsp 39008 df-pmap 39009 df-padd 39301 df-lhyp 39493 df-laut 39494 df-ldil 39609 df-ltrn 39610 df-trl 39664 df-tgrp 40248 df-tendo 40260 df-edring 40262 df-dveca 40508 df-disoa 40534 df-dvech 40584 df-dib 40644 df-dic 40678 df-dih 40734 df-doch 40853 df-djh 40900 df-lcdual 41092 df-mapd 41130

This theorem is referenced by: hdmap1eulem 41327

W3C validator