hdmap1eq4N

Description: Convert mapdheq4 to use HDMap1 function. (Contributed by NM, 17-May-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hdmap1eq2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap1eq2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eq2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap1eq2.o	⊢ 0 = ( 0_g ‘ 𝑈 )
	hdmap1eq2.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	hdmap1eq2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eq2.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
	hdmap1eq2.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
	hdmap1eq2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eq2.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
	hdmap1eq2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmap1eq2.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
	hdmap1eq2.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
	hdmap1eq4.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap1eq4.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap1eq4.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
	hdmap1eq4.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
	hdmap1eq4.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
	hdmap1eq4.eg	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
	hdmap1eq4.ee	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐵 )
Assertion	hdmap1eq4N	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑍 ⟩ ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		hdmap1eq2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap1eq2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap1eq2.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap1eq2.o	⊢ 0 = ( 0_g ‘ 𝑈 )
5		hdmap1eq2.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
6		hdmap1eq2.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
7		hdmap1eq2.d	⊢ 𝐷 = ( Base ‘ 𝐶 )
8		hdmap1eq2.l	⊢ 𝐿 = ( LSpan ‘ 𝐶 )
9		hdmap1eq2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
10		hdmap1eq2.i	⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
11		hdmap1eq2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12		hdmap1eq2.f	⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
13		hdmap1eq2.mn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
14		hdmap1eq4.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
15		hdmap1eq4.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
16		hdmap1eq4.z	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
17		hdmap1eq4.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
18		hdmap1eq4.xn	⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
19		hdmap1eq4.eg	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
20		hdmap1eq4.ee	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐵 )
21		eqid	⊢ ( -_g ‘ 𝑈 ) = ( -_g ‘ 𝑈 )
22		eqid	⊢ ( -_g ‘ 𝐶 ) = ( -_g ‘ 𝐶 )
23		eqid	⊢ ( 0_g ‘ 𝐶 ) = ( 0_g ‘ 𝐶 )
24	1 2 11	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
25	14	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
26	15	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
27	16	eldifad	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
28	3 5 24 25 26 27 18	lspindpi	⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
29	28	simpld	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
30	1 2 3 4 5 6 7 8 9 10 11 12 13 29 14 26	hdmap1cl	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
31	19 30	eqeltrrd	⊢ ( 𝜑 → 𝐺 ∈ 𝐷 )
32		eqid	⊢ ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , ( 0_g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝑈 ) ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) = ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , ( 0_g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝑈 ) ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝐶 ) ℎ ) } ) ) ) ) )
33	1 2 3 21 4 5 6 7 22 23 8 9 10 11 15 31 27 32	hdmap1valc	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑍 ⟩ ) = ( ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , ( 0_g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝑈 ) ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) ‘ ⟨ 𝑌 , 𝐺 , 𝑍 ⟩ ) )
34	1 2 3 21 4 5 6 7 22 23 8 9 10 11 14 12 26 32	hdmap1valc	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = ( ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , ( 0_g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝑈 ) ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) )
35	34 19	eqtr3d	⊢ ( 𝜑 → ( ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , ( 0_g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝑈 ) ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
36	1 2 3 21 4 5 6 7 22 23 8 9 10 11 14 12 27 32	hdmap1valc	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = ( ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , ( 0_g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝑈 ) ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) )
37	36 20	eqtr3d	⊢ ( 𝜑 → ( ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , ( 0_g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝑈 ) ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐵 )
38	23 32 1 9 2 3 21 4 5 6 7 22 8 11 12 13 14 15 16 18 17 35 37	mapdheq4	⊢ ( 𝜑 → ( ( 𝑥 ∈ V ↦ if ( ( 2^nd ‘ 𝑥 ) = 0 , ( 0_g ‘ 𝐶 ) , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2^nd ‘ 𝑥 ) } ) ) = ( 𝐿 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1^st ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝑈 ) ( 2^nd ‘ 𝑥 ) ) } ) ) = ( 𝐿 ‘ { ( ( 2^nd ‘ ( 1^st ‘ 𝑥 ) ) ( -_g ‘ 𝐶 ) ℎ ) } ) ) ) ) ) ‘ ⟨ 𝑌 , 𝐺 , 𝑍 ⟩ ) = 𝐵 )
39	33 38	eqtrd	⊢ ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑌 , 𝐺 , 𝑍 ⟩ ) = 𝐵 )

Metamath Proof Explorer

Theorem hdmap1eq4N

Proof