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 ⊢ + = ( +g ‘ 𝑈 )
hdmap1l6.s ⊢ − = ( -g ‘ 𝑈 )
hdmap1l6c.o ⊢ 0 = ( 0g ‘ 𝑈 )
hdmap1l6.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
hdmap1l6.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmap1l6.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
hdmap1l6.a ⊢ ✚ = ( +g ‘ 𝐶 )
hdmap1l6.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
hdmap1l6.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
hdmap1l6.l ⊢ 𝐿 = ( LSpan ‘ 𝐶 )
hdmap1l6.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
hdmap1l6.i ⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 )
hdmap1l6.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
hdmap1l6.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
hdmap1l6cl.x ⊢ ( 𝜑 → 𝑋 ∈ ( 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 ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6d.z ⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
hdmap1l6d.w ⊢ ( 𝜑 → 𝑤 ∈ ( 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 ⊢ + = ( +g ‘ 𝑈 )
5 hdmap1l6.s ⊢ − = ( -g ‘ 𝑈 )
6 hdmap1l6c.o ⊢ 0 = ( 0g ‘ 𝑈 )
7 hdmap1l6.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8 hdmap1l6.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
9 hdmap1l6.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
10 hdmap1l6.a ⊢ ✚ = ( +g ‘ 𝐶 )
11 hdmap1l6.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
12 hdmap1l6.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
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 ∧ 𝑊 ∈ 𝐻 ) )
17 hdmap1l6.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐷 )
18 hdmap1l6cl.x ⊢ ( 𝜑 → 𝑋 ∈ ( 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 ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
23 hdmap1l6d.z ⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
24 hdmap1l6d.w ⊢ ( 𝜑 → 𝑤 ∈ ( 𝑉 ∖ { 0 } ) )
25 hdmap1l6d.wn ⊢ ( 𝜑 → ¬ 𝑤 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) )
26 1 2 16 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
27 24 eldifad ⊢ ( 𝜑 → 𝑤 ∈ 𝑉 )
28 22 eldifad ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
29 3 4 lmodvacl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑤 + 𝑌 ) ∈ 𝑉 )
30 26 27 28 29 syl3anc ⊢ ( 𝜑 → ( 𝑤 + 𝑌 ) ∈ 𝑉 )
31 1 2 16 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
32 18 eldifad ⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
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 ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
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 ⟩ ) ) )