Theorem hvmapidN

Description: The value of the vector to functional map, at the vector, is one. (Contributed by NM, 23-Mar-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hvmapid.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hvmapid.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hvmapid.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hvmapid.z	⊢ 0 = ( 0g ‘ 𝑈 )
	hvmapid.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
	hvmapid.i	⊢ 1 = ( 1r ‘ 𝑆 )
	hvmapid.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
	hvmapid.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hvmapid.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
Assertion	hvmapidN	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ‘ 𝑋 ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		hvmapid.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hvmapid.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hvmapid.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hvmapid.z	⊢ 0 = ( 0g ‘ 𝑈 )
5		hvmapid.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
6		hvmapid.i	⊢ 1 = ( 1r ‘ 𝑆 )
7		hvmapid.m	⊢ 𝑀 = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
8		hvmapid.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		hvmapid.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
10		eqid	⊢ ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
11		eqid	⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 )
12		eqid	⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
13		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
14	1 2 10 3 11 12 4 5 13 7 8 9	hvmapval	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ ( Base ‘ 𝑆 ) ∃ 𝑡 ∈ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ { 𝑋 } ) 𝑣 = ( 𝑡 ( +g ‘ 𝑈 ) ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑋 ) ) ) ) )
15	14	fveq1d	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ‘ 𝑋 ) = ( ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ ( Base ‘ 𝑆 ) ∃ 𝑡 ∈ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ { 𝑋 } ) 𝑣 = ( 𝑡 ( +g ‘ 𝑈 ) ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑋 ) ) ) ) ‘ 𝑋 ) )
16		eqid	⊢ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ ( Base ‘ 𝑆 ) ∃ 𝑡 ∈ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ { 𝑋 } ) 𝑣 = ( 𝑡 ( +g ‘ 𝑈 ) ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑋 ) ) ) ) = ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ ( Base ‘ 𝑆 ) ∃ 𝑡 ∈ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ { 𝑋 } ) 𝑣 = ( 𝑡 ( +g ‘ 𝑈 ) ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑋 ) ) ) )
17	1 10 2 3 11 12 4 5 13 6 8 9 16	dochfl1	⊢ ( 𝜑 → ( ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑗 ∈ ( Base ‘ 𝑆 ) ∃ 𝑡 ∈ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ { 𝑋 } ) 𝑣 = ( 𝑡 ( +g ‘ 𝑈 ) ( 𝑗 ( ·𝑠 ‘ 𝑈 ) 𝑋 ) ) ) ) ‘ 𝑋 ) = 1 )
18	15 17	eqtrd	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ‘ 𝑋 ) = 1 )