Theorem hdmapadd

Description: Part 11 in Baer p. 48 line 35, (a+b)S = aS+bS in their notation (S = sigma). (Contributed by NM, 22-May-2015)

	Ref	Expression
Hypotheses	hdmap11.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hdmap11.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hdmap11.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	hdmap11.p	⊢ + = ( +g ‘ 𝑈 )
	hdmap11.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	hdmap11.a	⊢ ✚ = ( +g ‘ 𝐶 )
	hdmap11.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
	hdmap11.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hdmap11.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	hdmap11.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	hdmapadd	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hdmap11.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hdmap11.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hdmap11.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
4		hdmap11.p	⊢ + = ( +g ‘ 𝑈 )
5		hdmap11.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
6		hdmap11.a	⊢ ✚ = ( +g ‘ 𝐶 )
7		hdmap11.s	⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
8		hdmap11.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		hdmap11.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
10		hdmap11.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
11		eqid	⊢ ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩
12		eqid	⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
13		eqid	⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
14		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
15		eqid	⊢ ( LSpan ‘ 𝐶 ) = ( LSpan ‘ 𝐶 )
16		eqid	⊢ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
17		eqid	⊢ ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HVMap ‘ 𝐾 ) ‘ 𝑊 )
18		eqid	⊢ ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
19	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18	hdmap11lem2	⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝑋 + 𝑌 ) ) = ( ( 𝑆 ‘ 𝑋 ) ✚ ( 𝑆 ‘ 𝑌 ) ) )