Theorem hdmapg

Description: Apply the scalar sigma function (involution) G to an inner product reverses the arguments. The inner product of X and Y is represented by ( ( SY )X ) . Line 15 in Baer p. 111, f(x,y) alpha = f(y,x). (Contributed by NM, 14-Jun-2015)

	Ref	Expression
Hypotheses	hdmapg.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmapg.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmapg.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmapg.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
	hdmapg.g	⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
	hdmapg.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmapg.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	hdmapg.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	hdmapg	⊢ ( 𝜑 → ( 𝐺 ‘ ( ( 𝑆 ‘ 𝑌 ) ‘ 𝑋 ) ) = ( ( 𝑆 ‘ 𝑋 ) ‘ 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		hdmapg.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmapg.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmapg.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmapg.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
5		hdmapg.g	⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
6		hdmapg.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		hdmapg.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
8		hdmapg.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
9		eqid	⊢ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩
10		eqid	⊢ ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
11		eqid	⊢ ( +g ‘ 𝑈 ) = ( +g ‘ 𝑈 )
12		eqid	⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
13		eqid	⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
14		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) )
15		eqid	⊢ ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 )
16		eqid	⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
17		eqid	⊢ ( .r ‘ ( Scalar ‘ 𝑈 ) ) = ( .r ‘ ( Scalar ‘ 𝑈 ) )
18		eqid	⊢ ( 0g ‘ ( Scalar ‘ 𝑈 ) ) = ( 0g ‘ ( Scalar ‘ 𝑈 ) )
19		eqid	⊢ ( +g ‘ ( Scalar ‘ 𝑈 ) ) = ( +g ‘ ( Scalar ‘ 𝑈 ) )
20	1 9 10 2 3 11 12 13 14 15 16 6 7 17 18 19 4 5 8	hdmapglem7	⊢ ( 𝜑 → ( 𝐺 ‘ ( ( 𝑆 ‘ 𝑌 ) ‘ 𝑋 ) ) = ( ( 𝑆 ‘ 𝑋 ) ‘ 𝑌 ) )