# Metamath Proof Explorer

## Theorem hdmap1l6e

Description: Lemmma for hdmap1l6 . Part (6) in Baer p. 47 line 38. (Contributed by NM, 1-May-2015)

Ref Expression
Hypotheses hdmap1l6.h 𝐻 = ( LHyp ‘ 𝐾 )
hdmap1l6.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmap1l6.v 𝑉 = ( Base ‘ 𝑈 )
hdmap1l6.p + = ( +g𝑈 )
hdmap1l6.s = ( -g𝑈 )
hdmap1l6c.o 0 = ( 0g𝑈 )
hdmap1l6.n 𝑁 = ( LSpan ‘ 𝑈 )
hdmap1l6.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmap1l6.d 𝐷 = ( Base ‘ 𝐶 )
hdmap1l6.a = ( +g𝐶 )
hdmap1l6.r 𝑅 = ( -g𝐶 )
hdmap1l6.q 𝑄 = ( 0g𝐶 )
hdmap1l6.l 𝐿 = ( LSpan ‘ 𝐶 )
hdmap1l6.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
hdmap1l6.i 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
hdmap1l6.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
hdmap1l6.f ( 𝜑𝐹𝐷 )
hdmap1l6cl.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
hdmap1l6d.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
hdmap1l6d.yz ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
hdmap1l6d.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6d.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6d.w ( 𝜑𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6d.wn ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
Assertion hdmap1l6e ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( ( 𝑤 + 𝑌 ) + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )

### Proof

Step Hyp Ref Expression
1 hdmap1l6.h 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmap1l6.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hdmap1l6.v 𝑉 = ( Base ‘ 𝑈 )
4 hdmap1l6.p + = ( +g𝑈 )
5 hdmap1l6.s = ( -g𝑈 )
6 hdmap1l6c.o 0 = ( 0g𝑈 )
7 hdmap1l6.n 𝑁 = ( LSpan ‘ 𝑈 )
8 hdmap1l6.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
9 hdmap1l6.d 𝐷 = ( Base ‘ 𝐶 )
10 hdmap1l6.a = ( +g𝐶 )
11 hdmap1l6.r 𝑅 = ( -g𝐶 )
12 hdmap1l6.q 𝑄 = ( 0g𝐶 )
13 hdmap1l6.l 𝐿 = ( LSpan ‘ 𝐶 )
14 hdmap1l6.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
15 hdmap1l6.i 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
16 hdmap1l6.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
17 hdmap1l6.f ( 𝜑𝐹𝐷 )
18 hdmap1l6cl.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
19 hdmap1l6.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐿 ‘ { 𝐹 } ) )
20 hdmap1l6d.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
21 hdmap1l6d.yz ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) = ( 𝑁 ‘ { 𝑍 } ) )
22 hdmap1l6d.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
23 hdmap1l6d.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
24 hdmap1l6d.w ( 𝜑𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
25 hdmap1l6d.wn ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
26 1 2 16 dvhlmod ( 𝜑𝑈 ∈ LMod )
27 24 eldifad ( 𝜑𝑤𝑉 )
28 22 eldifad ( 𝜑𝑌𝑉 )
29 3 4 lmodvacl ( ( 𝑈 ∈ LMod ∧ 𝑤𝑉𝑌𝑉 ) → ( 𝑤 + 𝑌 ) ∈ 𝑉 )
30 26 27 28 29 syl3anc ( 𝜑 → ( 𝑤 + 𝑌 ) ∈ 𝑉 )
31 1 2 16 dvhlvec ( 𝜑𝑈 ∈ LVec )
32 18 eldifad ( 𝜑𝑋𝑉 )
33 3 7 31 27 32 28 25 lspindpi ( 𝜑 → ( ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) )
34 33 simprd ( 𝜑 → ( 𝑁 ‘ { 𝑤 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
35 3 4 6 7 26 27 28 34 lmodindp1 ( 𝜑 → ( 𝑤 + 𝑌 ) ≠ 0 )
36 eldifsn ( ( 𝑤 + 𝑌 ) ∈ ( 𝑉 ∖ { 0 } ) ↔ ( ( 𝑤 + 𝑌 ) ∈ 𝑉 ∧ ( 𝑤 + 𝑌 ) ≠ 0 ) )
37 30 35 36 sylanbrc ( 𝜑 → ( 𝑤 + 𝑌 ) ∈ ( 𝑉 ∖ { 0 } ) )
38 23 eldifad ( 𝜑𝑍𝑉 )
39 3 7 31 32 28 38 20 lspindpi ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ) )
40 39 simpld ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
41 3 4 6 7 31 18 22 23 24 21 40 25 mapdindp3 ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) )
42 3 4 6 7 31 18 22 23 24 21 40 25 mapdindp4 ( 𝜑 → ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑋 , ( 𝑤 + 𝑌 ) } ) )
43 3 6 7 31 18 30 38 41 42 lspindp1 ( 𝜑 → ( ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ∧ ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑍 , ( 𝑤 + 𝑌 ) } ) ) )
44 43 simprd ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑍 , ( 𝑤 + 𝑌 ) } ) )
45 prcom { ( 𝑤 + 𝑌 ) , 𝑍 } = { 𝑍 , ( 𝑤 + 𝑌 ) }
46 45 fveq2i ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) , 𝑍 } ) = ( 𝑁 ‘ { 𝑍 , ( 𝑤 + 𝑌 ) } )
47 46 eleq2i ( 𝑋 ∈ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) , 𝑍 } ) ↔ 𝑋 ∈ ( 𝑁 ‘ { 𝑍 , ( 𝑤 + 𝑌 ) } ) )
48 44 47 sylnibr ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) , 𝑍 } ) )
49 3 7 31 38 32 30 42 lspindpi ( 𝜑 → ( ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ) )
50 49 simprd ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) )
51 50 necomd ( 𝜑 → ( 𝑁 ‘ { ( 𝑤 + 𝑌 ) } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
52 eqidd ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) )
53 eqidd ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) )
54 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 37 23 48 51 52 53 hdmap1l6a ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( ( 𝑤 + 𝑌 ) + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑤 + 𝑌 ) ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )