hdmap1eulemOLDN

Description: Lemma for hdmap1euOLDN . TODO: combine with hdmap1euOLDN or at least share some hypotheses. (Contributed by NM, 15-May-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hdmap1eulem.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap1eulem.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eulem.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap1eulem.s	⊢ − = ( -_g ‘ 𝑈 )
	hdmap1eulem.o	⊢ 0 = ( 0_g ‘ 𝑈 )
	hdmap1eulem.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	hdmap1eulem.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eulem.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	hdmap1eulem.r	⊢ 𝑅 = ( -_g ‘ 𝐶 )
	hdmap1eulem.q	⊢ 𝑄 = ( 0_g ‘ 𝐶 )
	hdmap1eulem.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	hdmap1eulem.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eulem.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eulem.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmap1eulem.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
	hdmap1eulem.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap1eulem.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	hdmap1eulem.y	⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
	hdmap1eulem.l	⊢ 𝐿 = ( 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 ) ) 𝑅 ℎ ) } ) ) ) ) )
Assertion	hdmap1eulemOLDN	⊢ ( 𝜑 → ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )

Proof

Step	Hyp	Ref	Expression
1		hdmap1eulem.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap1eulem.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap1eulem.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap1eulem.s	⊢ − = ( -_g ‘ 𝑈 )
5		hdmap1eulem.o	⊢ 0 = ( 0_g ‘ 𝑈 )
6		hdmap1eulem.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		hdmap1eulem.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		hdmap1eulem.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
9		hdmap1eulem.r	⊢ 𝑅 = ( -_g ‘ 𝐶 )
10		hdmap1eulem.q	⊢ 𝑄 = ( 0_g ‘ 𝐶 )
11		hdmap1eulem.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12		hdmap1eulem.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
13		hdmap1eulem.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
14		hdmap1eulem.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15		hdmap1eulem.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
16		hdmap1eulem.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
17		hdmap1eulem.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
18		hdmap1eulem.y	⊢ ( 𝜑 → 𝑇 ∈ 𝑉 )
19		hdmap1eulem.l	⊢ 𝐿 = ( 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 ) ) 𝑅 ℎ ) } ) ) ) ) )
20	1 2 3 4 5 6 7 8 9 10 11 12 19 14 17 15 16 18	mapdh9aOLDN	⊢ ( 𝜑 → ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )
21	14	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
22	16	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
23	17	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝐹 ∈ 𝐷 )
24		simplr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑧 ∈ 𝑉 )
25	1 2 3 4 5 6 7 8 9 10 11 12 13 21 22 23 24 19	hdmap1valc	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) = ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) )
26	25	oteq2d	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ = ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ )
27	26	fveq2d	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) )
28		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
29	1 2 14	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
30	29	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑈 ∈ LMod )
31	16	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
32	3 28 6 29 31 18	lspprcl	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∈ ( LSubSp ‘ 𝑈 ) )
33	32	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ∈ ( LSubSp ‘ 𝑈 ) )
34		simpr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) )
35	5 28 30 33 24 34	lssneln0	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑧 ∈ ( 𝑉 ∖ { 0 } ) )
36	15	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
37	1 2 14	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
38	37	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑈 ∈ LVec )
39	31	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑋 ∈ 𝑉 )
40	18	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → 𝑇 ∈ 𝑉 )
41	3 6 38 24 39 40 34	lspindpi	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑇 } ) ) )
42	41	simpld	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑧 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
43	42	necomd	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑧 } ) )
44	10 19 1 12 2 3 4 5 6 7 8 9 11 21 23 36 22 24 43	mapdhcl	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) ∈ 𝐷 )
45	1 2 3 4 5 6 7 8 9 10 11 12 13 21 35 44 40 19	hdmap1valc	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) )
46	27 45	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) )
47	46	eqeq2d	⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) ∧ ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) ) → ( 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ↔ 𝑦 = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )
48	47	pm5.74da	⊢ ( ( 𝜑 ∧ 𝑧 ∈ 𝑉 ) → ( ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
49	48	ralbidva	⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
50	49	reubidv	⊢ ( 𝜑 → ( ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ↔ ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐿 ‘ ⟨ 𝑧 , ( 𝐿 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) ) )
51	20 50	mpbird	⊢ ( 𝜑 → ∃! 𝑦 ∈ 𝐷 ∀ 𝑧 ∈ 𝑉 ( ¬ 𝑧 ∈ ( 𝑁 ‘ { 𝑋 , 𝑇 } ) → 𝑦 = ( 𝐼 ‘ ⟨ 𝑧 , ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑧 ⟩ ) , 𝑇 ⟩ ) ) )

Metamath Proof Explorer

Theorem hdmap1eulemOLDN

Proof