Theorem hdmaplem3

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.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	hdmaplem3.o	⊢ 0 = ( 0g ‘ 𝑊 )
Assertion	hdmaplem3	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )

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.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
7		hdmaplem3.o	⊢ 0 = ( 0g ‘ 𝑊 )
8		eqid	⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
9	1 8 2	lspsncl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) )
10	3 6 9	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑊 ) )
11		elun2	⊢ ( 𝑍 ∈ ( 𝑁 ‘ { 𝑌 } ) → 𝑍 ∈ ( ( 𝑁 ‘ { 𝑋 } ) ∪ ( 𝑁 ‘ { 𝑌 } ) ) )
12	5 11	nsyl	⊢ ( 𝜑 → ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑌 } ) )
13	7 8 3 10 4 12	lssneln0	⊢ ( 𝜑 → 𝑍 ∈ ( 𝑉 ∖ { 0 } ) )