Theorem dihmeetlem4N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dihmeetlem4.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihmeetlem4.l	⊢ ≤ = ( le ‘ 𝐾 )
	dihmeetlem4.m	⊢ ∧ = ( meet ‘ 𝐾 )
	dihmeetlem4.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	dihmeetlem4.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihmeetlem4.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dihmeetlem4.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dihmeetlem4.z	⊢ 0 = ( 0g ‘ 𝑈 )
Assertion	dihmeetlem4N	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑄 ) ∩ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = { 0 } )

Proof

Step	Hyp	Ref	Expression
1		dihmeetlem4.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihmeetlem4.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dihmeetlem4.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		dihmeetlem4.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		dihmeetlem4.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		dihmeetlem4.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
7		dihmeetlem4.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
8		dihmeetlem4.z	⊢ 0 = ( 0g ‘ 𝑈 )
9		eqid	⊢ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) = ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 )
10		eqid	⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
11		eqid	⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
12		eqid	⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
13		eqid	⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
14		eqid	⊢ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) )
15	1 2 3 4 5 6 7 8 9 10 11 12 13 14	dihmeetlem4preN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑄 ) ∩ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = { 0 } )