Theorem hvmapvalvalN

Description: Value of value of map (i.e. functional value) from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hvmapval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hvmapval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hvmapval.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	hvmapval.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hvmapval.p	⊢ + = ( +g ‘ 𝑈 )
	hvmapval.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	hvmapval.z	⊢ 0 = ( 0g ‘ 𝑈 )
	hvmapval.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
	hvmapval.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
	hvmapval.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
	hvmapval.k	⊢ ( 𝜑 → ( 𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) )
	hvmapval.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	hvmapval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	hvmapvalvalN	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ‘ 𝑌 ) = ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑌 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		hvmapval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hvmapval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hvmapval.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
4		hvmapval.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		hvmapval.p	⊢ + = ( +g ‘ 𝑈 )
6		hvmapval.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
7		hvmapval.z	⊢ 0 = ( 0g ‘ 𝑈 )
8		hvmapval.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
9		hvmapval.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
10		hvmapval.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
11		hvmapval.k	⊢ ( 𝜑 → ( 𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) )
12		hvmapval.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
13		hvmapval.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
14	1 2 3 4 5 6 7 8 9 10 11 12	hvmapval	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( 𝑦 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑦 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ) )
15	14	fveq1d	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ‘ 𝑌 ) = ( ( 𝑦 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑦 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ) ‘ 𝑌 ) )
16		riotaex	⊢ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑌 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ∈ V
17		eqeq1	⊢ ( 𝑦 = 𝑌 → ( 𝑦 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ↔ 𝑌 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) )
18	17	rexbidv	⊢ ( 𝑦 = 𝑌 → ( ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑦 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ↔ ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑌 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) )
19	18	riotabidv	⊢ ( 𝑦 = 𝑌 → ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑦 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) = ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑌 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) )
20		eqid	⊢ ( 𝑦 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑦 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ) = ( 𝑦 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑦 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) )
21	19 20	fvmptg	⊢ ( ( 𝑌 ∈ 𝑉 ∧ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑌 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ∈ V ) → ( ( 𝑦 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑦 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ) ‘ 𝑌 ) = ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑌 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) )
22	13 16 21	sylancl	⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑦 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ) ‘ 𝑌 ) = ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑌 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) )
23	15 22	eqtrd	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ‘ 𝑌 ) = ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑌 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) )