Metamath Proof Explorer

Theorem hdmap1l6b0N

Description: Lemmma for hdmap1l6 . (Contributed by NM, 23-Apr-2015) (New usage is discouraged.)

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 } ) )
hdmap1l6b0.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
hdmap1l6b0.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
hdmap1l6b0.ne ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) = { 0 } )
Assertion hdmap1l6b0N ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 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 hdmap1l6b0.y ⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
21 hdmap1l6b0.z ⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
22 hdmap1l6b0.ne ⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) = { 0 } )
23 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
24 1 2 16 dvhlvec ⊢ ( 𝜑 → 𝑈 ∈ LVec )
25 1 2 16 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
26 3 23 7 25 20 21 lspprcl ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) )
27 3 6 7 23 24 26 18 lspdisjb ⊢ ( 𝜑 → ( ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ↔ ( ( 𝑁 ‘ { 𝑋 } ) ∩ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) ) = { 0 } ) )
28 22 27 mpbird ⊢ ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )