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Metamath Proof Explorer

Theorem cdlemc

Description: Lemma C in Crawley p. 113. (Contributed by NM, 26-May-2012)

Ref Expression
Hypotheses cdlemc3.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemc3.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemc3.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemc3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemc3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemc3.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemc3.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemc ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemc3.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemc3.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemc3.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdlemc3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemc3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemc3.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemc3.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
10 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝐹 ‘ 𝑃 ) = 𝑃 )
11 1 2 3 4 5 6 7 cdlemc6 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )
12 8 9 10 11 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )
13 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
15 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
16 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 )
17 1 2 3 4 5 6 7 cdlemc5 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )
18 13 14 15 16 17 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )
19 12 18 pm2.61dane ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )