# Metamath Proof Explorer

## Theorem mapdh6hN

Description: Lemmma for mapdh6N . Part (6) of Baer p. 48 line 2. (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 mapdh6hN ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 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 mapdh6gN ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) = ( ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )
27 3 10 14 lcdlmod ( 𝜑𝐶 ∈ LMod )
28 24 eldifad ( 𝜑𝑤𝑉 )
29 3 5 14 dvhlvec ( 𝜑𝑈 ∈ LVec )
30 17 eldifad ( 𝜑𝑋𝑉 )
31 22 eldifad ( 𝜑𝑌𝑉 )
32 6 9 29 28 30 31 25 lspindpi ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) )
33 32 simpld ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
34 33 necomd ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑤 } ) )
35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 28 34 mapdhcl ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ∈ 𝐷 )
36 23 eldifad ( 𝜑𝑍𝑉 )
37 6 9 29 30 31 36 20 lspindpi ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
38 37 simpld ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 31 38 mapdhcl ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
40 37 simprd ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 36 40 mapdhcl ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 )
42 11 19 lmodass ( ( 𝐶 ∈ LMod ∧ ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ∈ 𝐷 ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 ) ) → ( ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) )
43 27 35 39 41 42 syl13anc ( 𝜑 → ( ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) )
44 26 43 eqtrd ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) )
45 3 5 14 dvhlmod ( 𝜑𝑈 ∈ LMod )
46 6 18 lmodvacl ( ( 𝑈 ∈ LMod ∧ 𝑌𝑉𝑍𝑉 ) → ( 𝑌 + 𝑍 ) ∈ 𝑉 )
47 45 31 36 46 syl3anc ( 𝜑 → ( 𝑌 + 𝑍 ) ∈ 𝑉 )
48 6 18 8 9 29 17 22 23 24 21 38 25 mapdindp1 ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) )
49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 47 48 mapdhcl ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ∈ 𝐷 )
50 11 19 lmodvacl ( ( 𝐶 ∈ LMod ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 ) → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ∈ 𝐷 )
51 27 39 41 50 syl3anc ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ∈ 𝐷 )
52 11 19 lmodlcan ( ( 𝐶 ∈ LMod ∧ ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ∈ 𝐷 ∧ ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ∈ 𝐷 ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ∈ 𝐷 ) ) → ( ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) ↔ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) )
53 27 49 51 35 52 syl13anc ( 𝜑 → ( ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑤 ⟩ ) ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) ↔ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) )
54 44 53 mpbid ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )