Metamath Proof Explorer

Theorem cdlemg17iqN

Description: cdlemg17i with P and Q swapped. (Contributed by NM, 13-May-2013) (New usage is discouraged.)

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 cdlemg17iqN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑄 ) ) = ( 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 simp11 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝐾 ∈ HL )
9 simp12 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑊 ∈ 𝐻 )
10 8 9 jca ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
12 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
13 simp13l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝐹 ∈ 𝑇 )
14 simp13r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝐺 ∈ 𝑇 )
15 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑃 ≠ 𝑄 )
16 simp33 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 )
17 simp31 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) )
18 simp32 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) )
19 1 2 3 4 5 6 7 cdlemg17pq ⊢ ( ( ( ( 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 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃 ) ∧ ( ( 𝐺 ‘ 𝑄 ) ≠ 𝑄 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑄 ∨ 𝑃 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑄 ∨ 𝑟 ) = ( 𝑃 ∨ 𝑟 ) ) ) ) )
20 10 11 12 13 14 15 16 17 18 19 syl333anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃 ) ∧ ( ( 𝐺 ‘ 𝑄 ) ≠ 𝑄 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑄 ∨ 𝑃 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑄 ∨ 𝑟 ) = ( 𝑃 ∨ 𝑟 ) ) ) ) )
21 1 2 3 4 5 6 7 cdlemg17i ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃 ) ∧ ( ( 𝐺 ‘ 𝑄 ) ≠ 𝑄 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑄 ∨ 𝑃 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑄 ∨ 𝑟 ) = ( 𝑃 ∨ 𝑟 ) ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑄 ) ) = ( 𝐹 ‘ 𝑃 ) )
22 20 21 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐺 ‘ ( 𝐹 ‘ 𝑄 ) ) = ( 𝐹 ‘ 𝑃 ) )