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Metamath Proof Explorer

Theorem cdlemg43

Description: Part of proof of Lemma G of Crawley p. 116, third line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013)

Ref Expression
Hypotheses cdlemg42.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg42.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg42.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg42.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg42.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg42.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemg42.m ⊢ ∧ = ( meet ‘ 𝐾 )
Assertion cdlemg43 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐺 ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg42.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg42.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemg42.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 cdlemg42.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 cdlemg42.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 cdlemg42.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemg42.m ⊢ ∧ = ( meet ‘ 𝐾 )
8 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝐹 ∈ 𝑇 )
10 simp31 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
11 simp2r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝐺 ∈ 𝑇 )
12 1 3 4 5 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
13 8 11 10 12 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
14 1 2 3 4 5 6 cdlemg42 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ¬ ( 𝐺 ‘ 𝑃 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
15 1 2 7 3 4 5 6 cdlemc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ) ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ) ) )
16 8 9 10 13 14 15 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ) ) )
17 1 2 7 3 4 5 6 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) )
18 8 11 10 17 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) )
19 18 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐺 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ) )
20 19 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐺 ) ) ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ) ) )
21 16 20 eqtr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐺 ) ) ) )