# Metamath Proof Explorer

## Theorem hdmap1l6

Description: Part (6) of Baer p. 47 line 6. Note that we use -. X e. ( N{ Y , Z } ) which is equivalent to Baer's "Fx i^i (Fy + Fz)" by lspdisjb . (Convert mapdh6N to use the function HDMap1 .) (Contributed by NM, 17-May-2015)

Ref Expression
Hypotheses hdmap1-6.h 𝐻 = ( LHyp ‘ 𝐾 )
hdmap1-6.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmap1-6.v 𝑉 = ( Base ‘ 𝑈 )
hdmap1-6.p + = ( +g𝑈 )
hdmap1-6.o 0 = ( 0g𝑈 )
hdmap1-6.n 𝑁 = ( LSpan ‘ 𝑈 )
hdmap1-6.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmap1-6.d 𝐷 = ( Base ‘ 𝐶 )
hdmap1-6.a = ( +g𝐶 )
hdmap1-6.l 𝐿 = ( LSpan ‘ 𝐶 )
hdmap1-6.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
hdmap1-6.i 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
hdmap1-6.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
hdmap1-6.f ( 𝜑𝐹𝐷 )
hdmap1-6.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1-6.y ( 𝜑𝑌𝑉 )
hdmap1-6.z ( 𝜑𝑍𝑉 )
hdmap1-6.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
hdmap1-6.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
Assertion hdmap1l6 ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )

### Proof

Step Hyp Ref Expression
1 hdmap1-6.h 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmap1-6.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hdmap1-6.v 𝑉 = ( Base ‘ 𝑈 )
4 hdmap1-6.p + = ( +g𝑈 )
5 hdmap1-6.o 0 = ( 0g𝑈 )
6 hdmap1-6.n 𝑁 = ( LSpan ‘ 𝑈 )
7 hdmap1-6.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8 hdmap1-6.d 𝐷 = ( Base ‘ 𝐶 )
9 hdmap1-6.a = ( +g𝐶 )
10 hdmap1-6.l 𝐿 = ( LSpan ‘ 𝐶 )
11 hdmap1-6.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
12 hdmap1-6.i 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
13 hdmap1-6.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
14 hdmap1-6.f ( 𝜑𝐹𝐷 )
15 hdmap1-6.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
16 hdmap1-6.y ( 𝜑𝑌𝑉 )
17 hdmap1-6.z ( 𝜑𝑍𝑉 )
18 hdmap1-6.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
19 hdmap1-6.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
20 eqid ( -g𝑈 ) = ( -g𝑈 )
21 eqid ( -g𝐶 ) = ( -g𝐶 )
22 eqid ( 0g𝐶 ) = ( 0g𝐶 )
23 1 2 3 4 20 5 6 7 8 9 21 22 10 11 12 13 14 15 19 16 17 18 hdmap1l6k ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )