Theorem cdlemn

Description: Lemma N of Crawley p. 121 line 27. (Contributed by NM, 27-Feb-2014)

	Ref	Expression
Hypotheses	cdlemn11.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdlemn11.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemn11.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemn11.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemn11.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemn11.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
	cdlemn11.J	⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
	cdlemn11.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	cdlemn11.s	⊢ ⊕ = ( LSSum ‘ 𝑈 )
Assertion	cdlemn	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ) → ( 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ↔ ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemn11.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdlemn11.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdlemn11.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdlemn11.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdlemn11.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdlemn11.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
7		cdlemn11.J	⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
8		cdlemn11.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
9		cdlemn11.s	⊢ ⊕ = ( LSSum ‘ 𝑈 )
10	1 2 3 4 5 8 9 6 7	cdlemn5	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) → ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) )
11	10	3expia	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ) → ( 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) → ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) )
12	1 2 3 4 5 6 7 8 9	cdlemn11	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ∧ ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) → 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) )
13	12	3expia	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ) → ( ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) → 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ) )
14	11 13	impbid	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) ) → ( 𝑅 ≤ ( 𝑄 ∨ 𝑋 ) ↔ ( 𝐽 ‘ 𝑅 ) ⊆ ( ( 𝐽 ‘ 𝑄 ) ⊕ ( 𝐼 ‘ 𝑋 ) ) ) )