Theorem hdmaplem1

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)

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

Proof

Step	Hyp	Ref	Expression
1		hdmaplem1.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		hdmaplem1.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
3		hdmaplem1.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
4		hdmaplem1.z	⊢ ( 𝜑 → 𝑍 ∈ 𝑉 )
5		hdmaplem1.j	⊢ ( 𝜑 → ¬ 𝑍 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑌 } ) ) )
6		hdmaplem1.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
7		elun1	⊢ ( 𝑍 ∈ ( 𝑁 ‘ { 𝑋 } ) → 𝑍 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑌 } ) ) )
8	5 7	nsyl	⊢ ( 𝜑 → ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑋 } ) )
9	1 2 3 4 6 8	lspsnne2	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) )