Metamath Proof Explorer

Theorem cdlemg17f

Description: TODO: fix comment. (Contributed by NM, 8-May-2013)

Ref Expression
Hypotheses cdlemg12.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg12.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg12.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemg12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg12.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg12b.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg17f ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg12.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg12.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemg12.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdlemg12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemg12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemg12.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemg12b.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 1 2 3 4 5 6 7 cdlemg17e ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐺 ) ) )
9 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 simp22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝐺 ∈ 𝑇 )
11 simp21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝐹 ∈ 𝑇 )
12 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
13 1 4 5 6 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
14 9 11 12 13 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
15 1 2 3 4 5 6 7 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ) ∧ 𝑊 ) )
16 9 10 14 15 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ) ∧ 𝑊 ) )
17 16 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐺 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) )
18 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝑃 ∈ 𝐴 )
19 1 4 5 6 ltrncoat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝐹 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ∈ 𝐴 )
20 9 10 11 18 19 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ∈ 𝐴 )
21 eqid ⊢ ( ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ) ∧ 𝑊 ) = ( ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ) ∧ 𝑊 )
22 1 2 3 4 5 21 cdleme0cp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ∧ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ∈ 𝐴 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ) )
23 9 14 20 22 syl12anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ) )
24 8 17 23 3eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐺 ‘ ( 𝐹 ‘ 𝑃 ) ) ) )