mapdh9a - Mathbox for Norm Megill

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Theorem mapdh9a 42374

Description: Lemma for part (9) in [Baer] p. 48. TODO: why is this 50% larger than mapdh9aOLDN 42375? (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 1145	. . . . . . 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 1145	. . . . . . 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 1145	. . . . . . 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 1145	. . . . . . 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 1257	. . . . . . 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 1259	. . . . . . 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 786	. . . . . . . . 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 1147	. . . . . . . 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 3011	. . . . . . 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 790	. . . . . . . . 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 1147	. . . . . . . 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 3011	. . . . . . 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 787	. . . . . . . 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 1147	. . . . . . 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 791	. . . . . . . 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 1147	. . . . . . 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 1145	. . . . . . 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 42373	. . . . . 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 1131	. . . . 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 3202	. . . 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 3914	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
40	1, 2, 3, 6, 14, 39, 34	dvh3dim 42031	. . . . . . 7 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}))
41		eqid 2761	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
42	1, 2, 14	dvhlmod 41695	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LMod)
43	42	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LMod)
44	3, 41, 6, 42, 39, 34	lspprcl 21033	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈))
45	44	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑁‘{𝑋, 𝑇}) ∈ (LSubSp‘𝑈))
46		simplr 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ 𝑉)
47		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}))
48	5, 41, 43, 45, 46, 47	lssneln0 21008	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑧 ∈ (𝑉 ∖ { 0 }))
49	1, 2, 14	dvhlvec 41694	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
50	49	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑈 ∈ LVec)
51	39	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑋 ∈ 𝑉)
52	34	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → 𝑇 ∈ 𝑉)
53	3, 6, 50, 46, 51, 52, 47	lspindpi 21190	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))
54	48, 53	jca 519	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇})) → (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))))
55	54	ex 416	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))))
56	55	reximdva 3174	. . . . . . 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 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
59	16	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝐹 ∈ 𝐷)
60	18	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
61	20	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
62		simplr 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑧 ∈ 𝑉)
63		simprrl 790	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}))
64	63	necomd 3011	. . . . . . . . . . 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 42312	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → (𝐼‘⟨𝑋, 𝐹, 𝑧⟩) ∈ 𝐷)
66		eqidd 2762	. . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 42313	. . . . . . . . . . . 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 234	. . . . . . . . . . 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 498	. . . . . . . . . 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 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (𝑧 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))) → 𝑇 ∈ 𝑉)
72		simprrr 791	. . . . . . . . . 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 42312	. . . . . . . . 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 416	. . . . . . . 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 558	. . . . . . 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 3174	. . . . . 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 2844	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ (𝑉 ∖ { 0 }) ↔ 𝑤 ∈ (𝑉 ∖ { 0 })))
79		sneq 4589	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
80	79	fveq2d 6866	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝑁‘{𝑧}) = (𝑁‘{𝑤}))
81	80	neeq1d 3015	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ↔ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋})))
82	80	neeq1d 3015	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}) ↔ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇})))
83	81, 82	anbi12d 641	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})) ↔ ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑇}))))
84	78, 83	anbi12d 641	. . . . . 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 4837	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩ = ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩)
86		oteq3 4839	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ⟨𝑋, 𝐹, 𝑧⟩ = ⟨𝑋, 𝐹, 𝑤⟩)
87	86	fveq2d 6866	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (𝐼‘⟨𝑋, 𝐹, 𝑧⟩) = (𝐼‘⟨𝑋, 𝐹, 𝑤⟩))
88	87	oteq2d 4841	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩ = ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩)
89	85, 88	eqtrd 2796	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩ = ⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩)
90	89	fveq2d 6866	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝐼‘⟨𝑧, (𝐼‘⟨𝑋, 𝐹, 𝑧⟩), 𝑇⟩) = (𝐼‘⟨𝑤, (𝐼‘⟨𝑋, 𝐹, 𝑤⟩), 𝑇⟩))
91	84, 90	reusv3 5359	. . . . 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 234	. . 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 996	. . . . . . . 8 ⊢ (¬ (𝑧 ∈ (𝑁‘{𝑋}) ∨ 𝑧 ∈ (𝑁‘{𝑇})) ↔ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})))
95		elun 4104	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ↔ (𝑧 ∈ (𝑁‘{𝑋}) ∨ 𝑧 ∈ (𝑁‘{𝑇})))
96	94, 95	xchnxbir 335	. . . . . . 7 ⊢ (¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ↔ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})))
97	42	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑈 ∈ LMod)
98	3, 41, 6	lspsncl 21032	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
99	42, 39, 98	syl2anc 593	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
100	99	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
101		simplr 778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑧 ∈ 𝑉)
102		simprl 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → ¬ 𝑧 ∈ (𝑁‘{𝑋}))
103	5, 41, 97, 100, 101, 102	lssneln0 21008	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ (¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇}))) → 𝑧 ∈ (𝑉 ∖ { 0 }))
104	103	ex 416	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑧 ∈ (𝑉 ∖ { 0 })))
105	42	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑈 ∈ LMod)
106		simplr 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ 𝑉)
107	39	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑋 ∈ 𝑉)
108		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → ¬ 𝑧 ∈ (𝑁‘{𝑋}))
109	3, 6, 105, 106, 107, 108	lspsnne2 21176	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}))
110	109	ex 416	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑋}) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑋})))
111	42	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑈 ∈ LMod)
112		simplr 778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑧 ∈ 𝑉)
113	34	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑇 ∈ 𝑉)
114		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → ¬ 𝑧 ∈ (𝑁‘{𝑇}))
115	3, 6, 111, 112, 113, 114	lspsnne2 21176	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑉) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))
116	115	ex 416	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → (¬ 𝑧 ∈ (𝑁‘{𝑇}) → (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇})))
117	110, 116	anim12d 618	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑉) → ((¬ 𝑧 ∈ (𝑁‘{𝑋}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑇})) → ((𝑁‘{𝑧}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑧}) ≠ (𝑁‘{𝑇}))))
118	104, 117	jcad 520	. . . . . . 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 244	. . . . . 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 3173	. . . 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 3176	. . 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 21052	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋, 𝑇}))
125	41, 6, 42, 44, 124	ellspsn5 21051	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑋, 𝑇}))
126	3, 6, 42, 39, 34	lspprid2 21053	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝑋, 𝑇}))
127	41, 6, 42, 44, 126	ellspsn5 21051	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝑋, 𝑇}))
128	125, 127	unssd 4142	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇})) ⊆ (𝑁‘{𝑋, 𝑇}))
129	128	ssneld 3936	. . . . 5 ⊢ (𝜑 → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑇}) → ¬ 𝑧 ∈ ((𝑁‘{𝑋}) ∪ (𝑁‘{𝑇}))))
130	129	reximdv 3176	. . . 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 5351	. . 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 259	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 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 ∃!wreu 3364 Vcvv 3453 ∖ cdif 3899 ∪ cun 3900 ifcif 4477 {csn 4579 {cpr 4581 ⟨cotp 4587 ↦ cmpt 5178 ‘cfv 6516 ℩crio 7347 (class class class)co 7391 1^st c1st 7963 2^nd c2nd 7964 Basecbs 17236 0_gc0g 17459 -_gcsg 18968 LModclmod 20915 LSubSpclss 20986 LSpanclspn 21026 LVecclvec 21157 HLchlt 39935 LHypclh 40569 DVecHcdvh 41663 LCDualclcd 42171 mapdcmpd 42209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-riotaBAD 39538

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-om 7842 df-1st 7965 df-2nd 7966 df-tpos 8200 df-undef 8247 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-n0 12476 df-z 12563 df-uz 12834 df-fz 13507 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-sca 17293 df-vsca 17294 df-0g 17461 df-mre 17605 df-mrc 17606 df-acs 17608 df-proset 18317 df-poset 18336 df-plt 18351 df-lub 18367 df-glb 18368 df-join 18369 df-meet 18370 df-p0 18446 df-p1 18447 df-lat 18455 df-clat 18522 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18969 df-minusg 18970 df-sbg 18971 df-subg 19156 df-cntz 19348 df-oppg 19377 df-lsm 19667 df-cmn 19813 df-abl 19814 df-mgp 20178 df-rng 20190 df-ur 20219 df-ring 20272 df-oppr 20373 df-dvdsr 20393 df-unit 20394 df-invr 20424 df-dvr 20437 df-nzr 20550 df-rlreg 20731 df-domn 20732 df-drng 20768 df-lmod 20917 df-lss 20987 df-lsp 21027 df-lvec 21158 df-lsatoms 39561 df-lshyp 39562 df-lcv 39604 df-lfl 39643 df-lkr 39671 df-ldual 39709 df-oposet 39761 df-ol 39763 df-oml 39764 df-covers 39851 df-ats 39852 df-atl 39883 df-cvlat 39907 df-hlat 39936 df-llines 40083 df-lplanes 40084 df-lvols 40085 df-lines 40086 df-psubsp 40088 df-pmap 40089 df-padd 40381 df-lhyp 40573 df-laut 40574 df-ldil 40689 df-ltrn 40690 df-trl 40744 df-tgrp 41328 df-tendo 41340 df-edring 41342 df-dveca 41588 df-disoa 41614 df-dvech 41664 df-dib 41724 df-dic 41758 df-dih 41814 df-doch 41933 df-djh 41980 df-lcdual 42172 df-mapd 42210

This theorem is referenced by: hdmap1eulem 42407

W3C validator