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Metamath Proof Explorer

Theorem cdleme21at

Description: Part of proof of Lemma E in Crawley p. 115. (Contributed by NM, 29-Nov-2012)

Ref Expression
Hypotheses cdleme21.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme21.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme21.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme21.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme21.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme21.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
Assertion cdleme21at ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) → 𝑇 ≠ 𝑧 )

Proof

Step Hyp Ref Expression
1 cdleme21.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme21.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme21.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme21.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme21.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme21.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 1 2 3 4 5 6 cdleme21c ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) → ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑧 ) )
8 7 3adant2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) → ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑧 ) )
9 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) → 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) )
10 oveq2 ⊢ ( 𝑇 = 𝑧 → ( 𝑆 ∨ 𝑇 ) = ( 𝑆 ∨ 𝑧 ) )
11 10 breq2d ⊢ ( 𝑇 = 𝑧 → ( 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ↔ 𝑈 ≤ ( 𝑆 ∨ 𝑧 ) ) )
12 9 11 syl5ibcom ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) → ( 𝑇 = 𝑧 → 𝑈 ≤ ( 𝑆 ∨ 𝑧 ) ) )
13 12 necon3bd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) → ( ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑧 ) → 𝑇 ≠ 𝑧 ) )
14 8 13 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) → 𝑇 ≠ 𝑧 )