Metamath Proof Explorer


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 ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) ∧ ( 𝑇𝐴 ∧ ( 𝑈 𝑇 ) = ( 𝑉 𝑇 ) ) ) ∧ ( 𝑃𝑄 ∧ ¬ 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 ( ( 𝜑𝐶 = 𝑆 ) → ∃ 𝑧𝐴 ( ¬ 𝑧 𝑊 ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) )

Proof

Step Hyp Ref Expression
1 4thatlem.ph ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ∧ ( 𝑃 𝑅 ) = ( 𝑄 𝑅 ) ) ∧ ( 𝑇𝐴 ∧ ( 𝑈 𝑇 ) = ( 𝑉 𝑇 ) ) ) ∧ ( 𝑃𝑄 ∧ ¬ 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 ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ ( 𝑃 𝑄 ) = ( 𝑅 𝑄 ) ) ∧ ( 𝑇𝐴 ∧ ( ( ( 𝑃 𝑅 ) 𝑊 ) 𝑇 ) = ( 𝑉 𝑇 ) ) ) ∧ ( 𝑃𝑅 ∧ ¬ 𝑆 ( 𝑃 𝑅 ) ) ) )
12 1 2 3 4 5 6 7 8 9 10 4atexlemcnd ( 𝜑𝐶𝐷 )
13 pm13.18 ( ( 𝐶 = 𝑆𝐶𝐷 ) → 𝑆𝐷 )
14 13 necomd ( ( 𝐶 = 𝑆𝐶𝐷 ) → 𝐷𝑆 )
15 14 expcom ( 𝐶𝐷 → ( 𝐶 = 𝑆𝐷𝑆 ) )
16 12 15 syl ( 𝜑 → ( 𝐶 = 𝑆𝐷𝑆 ) )
17 biid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ ( 𝑃 𝑄 ) = ( 𝑅 𝑄 ) ) ∧ ( 𝑇𝐴 ∧ ( ( ( 𝑃 𝑅 ) 𝑊 ) 𝑇 ) = ( 𝑉 𝑇 ) ) ) ∧ ( 𝑃𝑅 ∧ ¬ 𝑆 ( 𝑃 𝑅 ) ) ) ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ ( 𝑃 𝑄 ) = ( 𝑅 𝑄 ) ) ∧ ( 𝑇𝐴 ∧ ( ( ( 𝑃 𝑅 ) 𝑊 ) 𝑇 ) = ( 𝑉 𝑇 ) ) ) ∧ ( 𝑃𝑅 ∧ ¬ 𝑆 ( 𝑃 𝑅 ) ) ) )
18 eqid ( ( 𝑃 𝑅 ) 𝑊 ) = ( ( 𝑃 𝑅 ) 𝑊 )
19 17 2 3 4 5 6 18 8 10 4atexlemex2 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( 𝑆𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ∧ ( 𝑃 𝑄 ) = ( 𝑅 𝑄 ) ) ∧ ( 𝑇𝐴 ∧ ( ( ( 𝑃 𝑅 ) 𝑊 ) 𝑇 ) = ( 𝑉 𝑇 ) ) ) ∧ ( 𝑃𝑅 ∧ ¬ 𝑆 ( 𝑃 𝑅 ) ) ) ∧ 𝐷𝑆 ) → ∃ 𝑧𝐴 ( ¬ 𝑧 𝑊 ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) )
20 11 16 19 syl6an ( 𝜑 → ( 𝐶 = 𝑆 → ∃ 𝑧𝐴 ( ¬ 𝑧 𝑊 ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) ) )
21 20 imp ( ( 𝜑𝐶 = 𝑆 ) → ∃ 𝑧𝐴 ( ¬ 𝑧 𝑊 ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) )