Theorem lcfrlem18

Description: Lemma for lcfr . (Contributed by NM, 24-Feb-2015)

	Ref	Expression
Hypotheses	lcfrlem17.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcfrlem17.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem17.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem17.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	lcfrlem17.p	⊢ + = ( +g ‘ 𝑈 )
	lcfrlem17.z	⊢ 0 = ( 0g ‘ 𝑈 )
	lcfrlem17.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	lcfrlem17.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
	lcfrlem17.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lcfrlem17.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	lcfrlem17.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	lcfrlem17.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
Assertion	lcfrlem18	⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 , 𝑌 } ) = ( ( ⊥ ‘ { 𝑋 } ) ∩ ( ⊥ ‘ { 𝑌 } ) ) )

Proof

Step	Hyp	Ref	Expression
1		lcfrlem17.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcfrlem17.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		lcfrlem17.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		lcfrlem17.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		lcfrlem17.p	⊢ + = ( +g ‘ 𝑈 )
6		lcfrlem17.z	⊢ 0 = ( 0g ‘ 𝑈 )
7		lcfrlem17.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8		lcfrlem17.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
9		lcfrlem17.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10		lcfrlem17.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
11		lcfrlem17.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
12		lcfrlem17.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
13		df-pr	⊢ { 𝑋 , 𝑌 } = ( { 𝑋 } ∪ { 𝑌 } )
14	13	fveq2i	⊢ ( ⊥ ‘ { 𝑋 , 𝑌 } ) = ( ⊥ ‘ ( { 𝑋 } ∪ { 𝑌 } ) )
15	10	eldifad	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
16	15	snssd	⊢ ( 𝜑 → { 𝑋 } ⊆ 𝑉 )
17	11	eldifad	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
18	17	snssd	⊢ ( 𝜑 → { 𝑌 } ⊆ 𝑉 )
19	1 3 4 2	dochdmj1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ { 𝑋 } ⊆ 𝑉 ∧ { 𝑌 } ⊆ 𝑉 ) → ( ⊥ ‘ ( { 𝑋 } ∪ { 𝑌 } ) ) = ( ( ⊥ ‘ { 𝑋 } ) ∩ ( ⊥ ‘ { 𝑌 } ) ) )
20	9 16 18 19	syl3anc	⊢ ( 𝜑 → ( ⊥ ‘ ( { 𝑋 } ∪ { 𝑌 } ) ) = ( ( ⊥ ‘ { 𝑋 } ) ∩ ( ⊥ ‘ { 𝑌 } ) ) )
21	14 20	eqtrid	⊢ ( 𝜑 → ( ⊥ ‘ { 𝑋 , 𝑌 } ) = ( ( ⊥ ‘ { 𝑋 } ) ∩ ( ⊥ ‘ { 𝑌 } ) ) )