Metamath Proof Explorer


Theorem hdmaplem2N

Description: Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmaplem1.v 𝑉 = ( Base ‘ 𝑊 )
hdmaplem1.n 𝑁 = ( LSpan ‘ 𝑊 )
hdmaplem1.w ( 𝜑𝑊 ∈ LMod )
hdmaplem1.z ( 𝜑𝑍𝑉 )
hdmaplem1.j ( 𝜑 → ¬ 𝑍 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑌 } ) ) )
hdmaplem1.y ( 𝜑𝑌𝑉 )
Assertion hdmaplem2N ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )

Proof

Step Hyp Ref Expression
1 hdmaplem1.v 𝑉 = ( Base ‘ 𝑊 )
2 hdmaplem1.n 𝑁 = ( LSpan ‘ 𝑊 )
3 hdmaplem1.w ( 𝜑𝑊 ∈ LMod )
4 hdmaplem1.z ( 𝜑𝑍𝑉 )
5 hdmaplem1.j ( 𝜑 → ¬ 𝑍 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑌 } ) ) )
6 hdmaplem1.y ( 𝜑𝑌𝑉 )
7 elun2 ( 𝑍 ∈ ( 𝑁 ‘ { 𝑌 } ) → 𝑍 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑌 } ) ) )
8 5 7 nsyl ( 𝜑 → ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑌 } ) )
9 1 2 3 4 6 8 lspsnne2 ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )