Metamath Proof Explorer

Theorem cdleme39n

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one on P .\/ Q line. TODO: FIX COMMENT. E , Y , G , Z serve as f(t), f(u), f_t( R ), f_t( S ). Put hypotheses of cdleme38n in convention of cdleme32sn1awN . TODO see if this hypothesis conversion would be better if done earlier. (Contributed by NM, 15-Mar-2013)

Ref Expression
Hypotheses cdleme39.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme39.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme39.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme39.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme39.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme39.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme39.e ⊢ 𝐸 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme39.g ⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme39.y ⊢ 𝑌 = ( ( 𝑢 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑢 ) ∧ 𝑊 ) ) )
cdleme39.z ⊢ 𝑍 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑌 ∨ ( ( 𝑆 ∨ 𝑢 ) ∧ 𝑊 ) ) )
Assertion cdleme39n ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝐺 ≠ 𝑍 )

Proof

Step Hyp Ref Expression
1 cdleme39.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme39.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme39.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme39.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme39.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme39.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme39.e ⊢ 𝐸 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
8 cdleme39.g ⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) )
9 cdleme39.y ⊢ 𝑌 = ( ( 𝑢 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑢 ) ∧ 𝑊 ) ) )
10 cdleme39.z ⊢ 𝑍 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑌 ∨ ( ( 𝑆 ∨ 𝑢 ) ∧ 𝑊 ) ) )
11 eqid ⊢ ( ( 𝑡 ∨ 𝐸 ) ∧ 𝑊 ) = ( ( 𝑡 ∨ 𝐸 ) ∧ 𝑊 )
12 eqid ⊢ ( ( 𝑢 ∨ 𝑌 ) ∧ 𝑊 ) = ( ( 𝑢 ∨ 𝑌 ) ∧ 𝑊 )
13 eqid ⊢ ( ( 𝑅 ∨ ( ( 𝑡 ∨ 𝐸 ) ∧ 𝑊 ) ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑅 ) ∧ 𝑊 ) ) ) = ( ( 𝑅 ∨ ( ( 𝑡 ∨ 𝐸 ) ∧ 𝑊 ) ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑅 ) ∧ 𝑊 ) ) )
14 eqid ⊢ ( ( 𝑆 ∨ ( ( 𝑢 ∨ 𝑌 ) ∧ 𝑊 ) ) ∧ ( 𝑌 ∨ ( ( 𝑢 ∨ 𝑆 ) ∧ 𝑊 ) ) ) = ( ( 𝑆 ∨ ( ( 𝑢 ∨ 𝑌 ) ∧ 𝑊 ) ) ∧ ( 𝑌 ∨ ( ( 𝑢 ∨ 𝑆 ) ∧ 𝑊 ) ) )
15 1 2 3 4 5 6 7 9 11 12 13 14 cdleme38n ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝑅 ∨ ( ( 𝑡 ∨ 𝐸 ) ∧ 𝑊 ) ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑅 ) ∧ 𝑊 ) ) ) ≠ ( ( 𝑆 ∨ ( ( 𝑢 ∨ 𝑌 ) ∧ 𝑊 ) ) ∧ ( 𝑌 ∨ ( ( 𝑢 ∨ 𝑆 ) ∧ 𝑊 ) ) ) )
16 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
17 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑃 ∈ 𝐴 )
18 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑄 ∈ 𝐴 )
19 simp22l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑅 ∈ 𝐴 )
20 simp22r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑅 ≤ 𝑊 )
21 simp311 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
22 simp32l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) )
23 1 2 3 4 5 6 7 8 11 cdleme39a ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ) → 𝐺 = ( ( 𝑅 ∨ ( ( 𝑡 ∨ 𝐸 ) ∧ 𝑊 ) ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑅 ) ∧ 𝑊 ) ) ) )
24 16 17 18 19 20 21 22 23 syl322anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝐺 = ( ( 𝑅 ∨ ( ( 𝑡 ∨ 𝐸 ) ∧ 𝑊 ) ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑅 ) ∧ 𝑊 ) ) ) )
25 simp23l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑆 ∈ 𝐴 )
26 simp23r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑆 ≤ 𝑊 )
27 simp312 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
28 simp33l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) )
29 1 2 3 4 5 6 9 10 12 cdleme39a ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ) ) → 𝑍 = ( ( 𝑆 ∨ ( ( 𝑢 ∨ 𝑌 ) ∧ 𝑊 ) ) ∧ ( 𝑌 ∨ ( ( 𝑢 ∨ 𝑆 ) ∧ 𝑊 ) ) ) )
30 16 17 18 25 26 27 28 29 syl322anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑍 = ( ( 𝑆 ∨ ( ( 𝑢 ∨ 𝑌 ) ∧ 𝑊 ) ) ∧ ( 𝑌 ∨ ( ( 𝑢 ∨ 𝑆 ) ∧ 𝑊 ) ) ) )
31 15 24 30 3netr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝐺 ≠ 𝑍 )