# Metamath Proof Explorer

## Theorem hdmap1l6a

Description: Lemma for hdmap1l6 . Part (6) in Baer p. 47, case 1. (Contributed by NM, 23-Apr-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 } ) )
hdmap1l6e.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6e.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6e.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
hdmap1l6.yz ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
hdmap1l6.fg ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
hdmap1l6.fe ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
Assertion hdmap1l6a ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 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 hdmap1l6e.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
21 hdmap1l6e.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
22 hdmap1l6e.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
23 hdmap1l6.yz ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
24 hdmap1l6.fg ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
25 hdmap1l6.fe ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
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 hdmap1l6lem2 ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) = ( 𝐿 ‘ { ( 𝐺 𝐸 ) } ) )
27 24 25 oveq12d ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) = ( 𝐺 𝐸 ) )
28 27 sneqd ( 𝜑 → { ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) } = { ( 𝐺 𝐸 ) } )
29 28 fveq2d ( 𝜑 → ( 𝐿 ‘ { ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) } ) = ( 𝐿 ‘ { ( 𝐺 𝐸 ) } ) )
30 26 29 eqtr4d ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) = ( 𝐿 ‘ { ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) } ) )
31 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 hdmap1l6lem1 ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ) = ( 𝐿 ‘ { ( 𝐹 𝑅 ( 𝐺 𝐸 ) ) } ) )
32 27 oveq2d ( 𝜑 → ( 𝐹 𝑅 ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) = ( 𝐹 𝑅 ( 𝐺 𝐸 ) ) )
33 32 sneqd ( 𝜑 → { ( 𝐹 𝑅 ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) } = { ( 𝐹 𝑅 ( 𝐺 𝐸 ) ) } )
34 33 fveq2d ( 𝜑 → ( 𝐿 ‘ { ( 𝐹 𝑅 ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) } ) = ( 𝐿 ‘ { ( 𝐹 𝑅 ( 𝐺 𝐸 ) ) } ) )
35 31 34 eqtr4d ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ) = ( 𝐿 ‘ { ( 𝐹 𝑅 ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) } ) )
36 1 2 16 dvhlmod ( 𝜑𝑈 ∈ LMod )
37 20 eldifad ( 𝜑𝑌𝑉 )
38 21 eldifad ( 𝜑𝑍𝑉 )
39 3 4 lmodvacl ( ( 𝑈 ∈ LMod ∧ 𝑌𝑉𝑍𝑉 ) → ( 𝑌 + 𝑍 ) ∈ 𝑉 )
40 36 37 38 39 syl3anc ( 𝜑 → ( 𝑌 + 𝑍 ) ∈ 𝑉 )
41 3 4 6 7 36 37 38 23 lmodindp1 ( 𝜑 → ( 𝑌 + 𝑍 ) ≠ 0 )
42 eldifsn ( ( 𝑌 + 𝑍 ) ∈ ( 𝑉 ∖ { 0 } ) ↔ ( ( 𝑌 + 𝑍 ) ∈ 𝑉 ∧ ( 𝑌 + 𝑍 ) ≠ 0 ) )
43 40 41 42 sylanbrc ( 𝜑 → ( 𝑌 + 𝑍 ) ∈ ( 𝑉 ∖ { 0 } ) )
44 1 8 16 lcdlmod ( 𝜑𝐶 ∈ LMod )
45 1 2 16 dvhlvec ( 𝜑𝑈 ∈ LVec )
46 18 eldifad ( 𝜑𝑋𝑉 )
47 3 6 7 45 37 21 46 23 22 lspindp2 ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) )
48 47 simpld ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
49 1 2 3 6 7 8 9 13 14 15 16 17 19 48 18 37 hdmap1cl ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
50 3 6 7 45 20 38 46 23 22 lspindp1 ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ∧ ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) )
51 50 simpld ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
52 1 2 3 6 7 8 9 13 14 15 16 17 19 51 18 38 hdmap1cl ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 )
53 9 10 lmodvacl ( ( 𝐶 ∈ LMod ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 ∧ ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 ) → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ∈ 𝐷 )
54 44 49 52 53 syl3anc ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ∈ 𝐷 )
55 eqid ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
56 3 55 7 36 37 38 lspprcl ( 𝜑 → ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) )
57 3 4 7 36 37 38 lspprvacl ( 𝜑 → ( 𝑌 + 𝑍 ) ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
58 55 7 36 56 57 lspsnel5a ( 𝜑 → ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
59 3 55 7 36 56 46 lspsnel5 ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) )
60 22 59 mtbid ( 𝜑 → ¬ ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
61 nssne2 ( ( ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ∧ ¬ ( 𝑁 ‘ { 𝑋 } ) ⊆ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) → ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
62 58 60 61 syl2anc ( 𝜑 → ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )
63 62 necomd ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) )
64 1 2 3 5 6 7 8 9 11 13 14 15 16 18 17 43 54 63 19 hdmap1eq ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) = ( 𝐿 ‘ { ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ) = ( 𝐿 ‘ { ( 𝐹 𝑅 ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) ) } ) ) ) )
65 30 35 64 mpbir2and ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , ( 𝑌 + 𝑍 ) ⟩ ) = ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ) )