# Metamath Proof Explorer

## Theorem mapdh6gN

Description: Lemmma for mapdh6N . Part (6) of Baer p. 47 line 39. (Contributed by NM, 1-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses mapdh.q 𝑄 = ( 0g𝐶 )
mapdh.i 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd𝑥 ) = 0 , 𝑄 , ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑥 ) } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑥 ) ) ( 2nd𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st𝑥 ) ) 𝑅 ) } ) ) ) ) )
mapdh.h 𝐻 = ( LHyp ‘ 𝐾 )
mapdh.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
mapdh.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
mapdh.v 𝑉 = ( Base ‘ 𝑈 )
mapdh.s = ( -g𝑈 )
mapdhc.o 0 = ( 0g𝑈 )
mapdh.n 𝑁 = ( LSpan ‘ 𝑈 )
mapdh.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
mapdh.d 𝐷 = ( Base ‘ 𝐶 )
mapdh.r 𝑅 = ( -g𝐶 )
mapdh.j 𝐽 = ( LSpan ‘ 𝐶 )
mapdh.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
mapdhc.f ( 𝜑𝐹𝐷 )
mapdh.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
mapdhcl.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh.p + = ( +g𝑈 )
mapdh.a = ( +g𝐶 )
mapdh6d.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
mapdh6d.yz ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
mapdh6d.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh6d.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh6d.w ( 𝜑𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh6d.wn ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
Assertion mapdh6gN ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) = ( ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )

### Proof

Step Hyp Ref Expression
1 mapdh.q 𝑄 = ( 0g𝐶 )
2 mapdh.i 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd𝑥 ) = 0 , 𝑄 , ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑥 ) } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑥 ) ) ( 2nd𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st𝑥 ) ) 𝑅 ) } ) ) ) ) )
3 mapdh.h 𝐻 = ( LHyp ‘ 𝐾 )
4 mapdh.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
5 mapdh.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6 mapdh.v 𝑉 = ( Base ‘ 𝑈 )
7 mapdh.s = ( -g𝑈 )
8 mapdhc.o 0 = ( 0g𝑈 )
9 mapdh.n 𝑁 = ( LSpan ‘ 𝑈 )
10 mapdh.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
11 mapdh.d 𝐷 = ( Base ‘ 𝐶 )
12 mapdh.r 𝑅 = ( -g𝐶 )
13 mapdh.j 𝐽 = ( LSpan ‘ 𝐶 )
14 mapdh.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
15 mapdhc.f ( 𝜑𝐹𝐷 )
16 mapdh.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17 mapdhcl.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18 mapdh.p + = ( +g𝑈 )
19 mapdh.a = ( +g𝐶 )
20 mapdh6d.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
21 mapdh6d.yz ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
22 mapdh6d.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
23 mapdh6d.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
24 mapdh6d.w ( 𝜑𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
25 mapdh6d.wn ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 mapdh6dN ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + ( 𝑌 + 𝑍 ) ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) )
27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 mapdh6eN ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( ( 𝑤 + 𝑌 ) + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
28 3 5 14 dvhlmod ( 𝜑𝑈 ∈ LMod )
29 24 eldifad ( 𝜑𝑤𝑉 )
30 22 eldifad ( 𝜑𝑌𝑉 )
31 23 eldifad ( 𝜑𝑍𝑉 )
32 6 18 lmodass ( ( 𝑈 ∈ LMod ∧ ( 𝑤𝑉𝑌𝑉𝑍𝑉 ) ) → ( ( 𝑤 + 𝑌 ) + 𝑍 ) = ( 𝑤 + ( 𝑌 + 𝑍 ) ) )
33 28 29 30 31 32 syl13anc ( 𝜑 → ( ( 𝑤 + 𝑌 ) + 𝑍 ) = ( 𝑤 + ( 𝑌 + 𝑍 ) ) )
34 33 oteq3d ( 𝜑 → ⟨ 𝑋 , 𝐹 , ( ( 𝑤 + 𝑌 ) + 𝑍 ) ⟩ = ⟨ 𝑋 , 𝐹 , ( 𝑤 + ( 𝑌 + 𝑍 ) ) ⟩ )
35 34 fveq2d ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( ( 𝑤 + 𝑌 ) + 𝑍 ) ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + ( 𝑌 + 𝑍 ) ) ⟩ ) )
36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 mapdh6fN ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) )
37 36 oveq1d ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) = ( ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
38 27 35 37 3eqtr3d ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + ( 𝑌 + 𝑍 ) ) ⟩ ) = ( ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
39 26 38 eqtr3d ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) = ( ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )