This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cdleme15a

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, showing, in their notation, ((s \/ p) /\ (f(s) \/ q)) \/ ((t \/ p) /\ (f(t) \/ q))=((p \/ s_1) /\ (q \/ s_1)) \/ ((p \/ t_1) /\ (q \/ t_1)). We represent f(s), f(t), s_1, and t_1 with F , G , C , and X respectively. The order of our operations is slightly different. (Contributed by NM, 9-Oct-2012)

Ref Expression
Hypotheses cdleme12.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme12.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme12.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme12.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme12.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme12.g ⊢ 𝐺 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
cdleme15.c ⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
cdleme15.x ⊢ 𝑋 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 )
Assertion cdleme15a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( ( 𝑇 ∨ 𝑃 ) ∧ ( 𝐺 ∨ 𝑄 ) ) ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝐹 ) ) ) = ( ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑄 ∨ 𝑋 ) ) ∨ ( ( 𝑃 ∨ 𝐶 ) ∧ ( 𝑄 ∨ 𝐶 ) ) ) )

Proof

Step Hyp Ref Expression
1 cdleme12.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme12.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme12.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme12.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme12.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
8 cdleme12.g ⊢ 𝐺 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
9 cdleme15.c ⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
10 cdleme15.x ⊢ 𝑋 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 )
11 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐾 ∈ HL )
12 simp11r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑊 ∈ 𝐻 )
13 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑃 ∈ 𝐴 )
14 simp12r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ¬ 𝑃 ≤ 𝑊 )
15 simp22l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑇 ∈ 𝐴 )
16 1 2 3 4 5 10 cdleme8 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑋 ) = ( 𝑃 ∨ 𝑇 ) )
17 11 12 13 14 15 16 syl221anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑃 ∨ 𝑋 ) = ( 𝑃 ∨ 𝑇 ) )
18 2 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑇 ) = ( 𝑇 ∨ 𝑃 ) )
19 11 13 15 18 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑃 ∨ 𝑇 ) = ( 𝑇 ∨ 𝑃 ) )
20 17 19 eqtr2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑇 ∨ 𝑃 ) = ( 𝑃 ∨ 𝑋 ) )
21 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
22 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
23 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
24 simp22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) )
25 simp23l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑃 ≠ 𝑄 )
26 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) )
27 1 2 3 4 5 6 8 cdleme3fa ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐺 ∈ 𝐴 )
28 21 22 23 24 25 26 27 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐺 ∈ 𝐴 )
29 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑄 ∈ 𝐴 )
30 2 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐺 ∨ 𝑄 ) = ( 𝑄 ∨ 𝐺 ) )
31 11 28 29 30 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝐺 ∨ 𝑄 ) = ( 𝑄 ∨ 𝐺 ) )
32 1 2 3 4 5 6 10 6 8 cdleme11g ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑇 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ 𝐺 ) = ( 𝑄 ∨ 𝑋 ) )
33 21 13 23 15 25 32 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑄 ∨ 𝐺 ) = ( 𝑄 ∨ 𝑋 ) )
34 31 33 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝐺 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑋 ) )
35 20 34 oveq12d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝑇 ∨ 𝑃 ) ∧ ( 𝐺 ∨ 𝑄 ) ) = ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑄 ∨ 𝑋 ) ) )
36 simp21l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑆 ∈ 𝐴 )
37 1 2 3 4 5 9 cdleme8 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) → ( 𝑃 ∨ 𝐶 ) = ( 𝑃 ∨ 𝑆 ) )
38 11 12 13 14 36 37 syl221anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑃 ∨ 𝐶 ) = ( 𝑃 ∨ 𝑆 ) )
39 38 eqcomd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑃 ∨ 𝑆 ) = ( 𝑃 ∨ 𝐶 ) )
40 1 2 3 4 5 6 9 6 7 cdleme11g ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑄 ∨ 𝐹 ) = ( 𝑄 ∨ 𝐶 ) )
41 21 13 23 36 25 40 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑄 ∨ 𝐹 ) = ( 𝑄 ∨ 𝐶 ) )
42 39 41 oveq12d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝐹 ) ) = ( ( 𝑃 ∨ 𝐶 ) ∧ ( 𝑄 ∨ 𝐶 ) ) )
43 35 42 oveq12d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( ( 𝑇 ∨ 𝑃 ) ∧ ( 𝐺 ∨ 𝑄 ) ) ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ ( 𝑄 ∨ 𝐹 ) ) ) = ( ( ( 𝑃 ∨ 𝑋 ) ∧ ( 𝑄 ∨ 𝑋 ) ) ∨ ( ( 𝑃 ∨ 𝐶 ) ∧ ( 𝑄 ∨ 𝐶 ) ) ) )