Metamath Proof Explorer

Theorem hdmaplna2

Description: Additive property of second (inner product) argument. (Contributed by NM, 10-Jun-2015)

Ref Expression
Hypotheses hdmaplna2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
hdmaplna2.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmaplna2.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
hdmaplna2.p ⊢ + = ( +g ‘ 𝑈 )
hdmaplna2.r ⊢ 𝑅 = ( Scalar ‘ 𝑈 )
hdmaplna2.q ⊢ ⨣ = ( +g ‘ 𝑅 )
hdmaplna2.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
hdmaplna2.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
hdmaplna2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
hdmaplna2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
hdmaplna2.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
Assertion hdmaplna2 ( 𝜑 → ( ( 𝑆 ‘ ( 𝑌 + 𝑍 ) ) ‘ 𝑋 ) = ( ( ( 𝑆 ‘ 𝑌 ) ‘ 𝑋 ) ⨣ ( ( 𝑆 ‘ 𝑍 ) ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 hdmaplna2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmaplna2.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hdmaplna2.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
4 hdmaplna2.p ⊢ + = ( +g ‘ 𝑈 )
5 hdmaplna2.r ⊢ 𝑅 = ( Scalar ‘ 𝑈 )
6 hdmaplna2.q ⊢ ⨣ = ( +g ‘ 𝑅 )
7 hdmaplna2.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
8 hdmaplna2.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 hdmaplna2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
10 hdmaplna2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
11 hdmaplna2.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
12 eqid ⊢ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
13 eqid ⊢ ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
14 1 2 3 4 12 13 7 8 10 11 hdmapadd ⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝑌 + 𝑍 ) ) = ( ( 𝑆 ‘ 𝑌 ) ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( 𝑆 ‘ 𝑍 ) ) )
15 14 fveq1d ⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝑌 + 𝑍 ) ) ‘ 𝑋 ) = ( ( ( 𝑆 ‘ 𝑌 ) ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( 𝑆 ‘ 𝑍 ) ) ‘ 𝑋 ) )
16 eqid ⊢ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
17 1 2 3 12 16 7 8 10 hdmapcl ⊢ ( 𝜑 → ( 𝑆 ‘ 𝑌 ) ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) )
18 1 2 3 12 16 7 8 11 hdmapcl ⊢ ( 𝜑 → ( 𝑆 ‘ 𝑍 ) ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) )
19 1 2 3 5 6 12 16 13 8 17 18 9 lcdvaddval ⊢ ( 𝜑 → ( ( ( 𝑆 ‘ 𝑌 ) ( +g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( 𝑆 ‘ 𝑍 ) ) ‘ 𝑋 ) = ( ( ( 𝑆 ‘ 𝑌 ) ‘ 𝑋 ) ⨣ ( ( 𝑆 ‘ 𝑍 ) ‘ 𝑋 ) ) )
20 15 19 eqtrd ⊢ ( 𝜑 → ( ( 𝑆 ‘ ( 𝑌 + 𝑍 ) ) ‘ 𝑋 ) = ( ( ( 𝑆 ‘ 𝑌 ) ‘ 𝑋 ) ⨣ ( ( 𝑆 ‘ 𝑍 ) ‘ 𝑋 ) ) )