Theorem 4atexlemex4

Description: Lemma for 4atexlem7 . Show that when C = S , D satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012)

	Ref	Expression
Hypotheses	4thatlem.ph	⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
	4thatlem0.l	⊢ ≤ = ( le ‘ 𝐾 )
	4thatlem0.j	⊢ ∨ = ( join ‘ 𝐾 )
	4thatlem0.m	⊢ ∧ = ( meet ‘ 𝐾 )
	4thatlem0.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	4thatlem0.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	4thatlem0.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
	4thatlem0.v	⊢ 𝑉 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
	4thatlem0.c	⊢ 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) )
	4thatlem0.d	⊢ 𝐷 = ( ( 𝑅 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) )
Assertion	4atexlemex4	⊢ ( ( 𝜑 ∧ 𝐶 = 𝑆 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )

Proof

Step	Hyp	Ref	Expression
1		4thatlem.ph	⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( 𝑈 ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
2		4thatlem0.l	⊢ ≤ = ( le ‘ 𝐾 )
3		4thatlem0.j	⊢ ∨ = ( join ‘ 𝐾 )
4		4thatlem0.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		4thatlem0.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		4thatlem0.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7		4thatlem0.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8		4thatlem0.v	⊢ 𝑉 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
9		4thatlem0.c	⊢ 𝐶 = ( ( 𝑄 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) )
10		4thatlem0.d	⊢ 𝐷 = ( ( 𝑅 ∨ 𝑇 ) ∧ ( 𝑃 ∨ 𝑆 ) )
11	1 2 3 5 7	4atexlemswapqr	⊢ ( 𝜑 → ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) )
12	1 2 3 4 5 6 7 8 9 10	4atexlemcnd	⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )
13		pm13.18	⊢ ( ( 𝐶 = 𝑆 ∧ 𝐶 ≠ 𝐷 ) → 𝑆 ≠ 𝐷 )
14	13	necomd	⊢ ( ( 𝐶 = 𝑆 ∧ 𝐶 ≠ 𝐷 ) → 𝐷 ≠ 𝑆 )
15	14	expcom	⊢ ( 𝐶 ≠ 𝐷 → ( 𝐶 = 𝑆 → 𝐷 ≠ 𝑆 ) )
16	12 15	syl	⊢ ( 𝜑 → ( 𝐶 = 𝑆 → 𝐷 ≠ 𝑆 ) )
17		biid	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) )
18		eqid	⊢ ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 )
19	17 2 3 4 5 6 18 8 10	4atexlemex2	⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑄 ) ) ∧ ( 𝑇 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ∨ 𝑇 ) = ( 𝑉 ∨ 𝑇 ) ) ) ∧ ( 𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑅 ) ) ) ∧ 𝐷 ≠ 𝑆 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )
20	11 16 19	syl6an	⊢ ( 𝜑 → ( 𝐶 = 𝑆 → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) )
21	20	imp	⊢ ( ( 𝜑 ∧ 𝐶 = 𝑆 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )