Theorem hvmapval

Description: Value of map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015)

	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 } ) )
Assertion	hvmapval	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑣 = ( 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	1 2 3 4 5 6 7 8 9 10 11	hvmapfval	⊢ ( 𝜑 → 𝑀 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑥 ) ) ) ) ) )
14	13	fveq1d	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑥 ) ) ) ) ) ‘ 𝑋 ) )
15	4	fvexi	⊢ 𝑉 ∈ V
16	15	mptex	⊢ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ) ∈ V
17		sneq	⊢ ( 𝑥 = 𝑋 → { 𝑥 } = { 𝑋 } )
18	17	fveq2d	⊢ ( 𝑥 = 𝑋 → ( 𝑂 ‘ { 𝑥 } ) = ( 𝑂 ‘ { 𝑋 } ) )
19		oveq2	⊢ ( 𝑥 = 𝑋 → ( 𝑗 · 𝑥 ) = ( 𝑗 · 𝑋 ) )
20	19	oveq2d	⊢ ( 𝑥 = 𝑋 → ( 𝑡 + ( 𝑗 · 𝑥 ) ) = ( 𝑡 + ( 𝑗 · 𝑋 ) ) )
21	20	eqeq2d	⊢ ( 𝑥 = 𝑋 → ( 𝑣 = ( 𝑡 + ( 𝑗 · 𝑥 ) ) ↔ 𝑣 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) )
22	18 21	rexeqbidv	⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑥 ) ) ↔ ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) )
23	22	riotabidv	⊢ ( 𝑥 = 𝑋 → ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑥 ) ) ) = ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) )
24	23	mpteq2dv	⊢ ( 𝑥 = 𝑋 → ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑥 ) ) ) ) = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ) )
25		eqid	⊢ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑥 ) ) ) ) ) = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑥 ) ) ) ) )
26	24 25	fvmptg	⊢ ( ( 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ∧ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ) ∈ V ) → ( ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑥 ) ) ) ) ) ‘ 𝑋 ) = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ) )
27	12 16 26	sylancl	⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑥 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑥 ) ) ) ) ) ‘ 𝑋 ) = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ) )
28	14 27	eqtrd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ 𝑅 ∃ 𝑡 ∈ ( 𝑂 ‘ { 𝑋 } ) 𝑣 = ( 𝑡 + ( 𝑗 · 𝑋 ) ) ) ) )