Metamath Proof Explorer

Theorem cdleme35h

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one outside of P .\/ Q line. TODO: FIX COMMENT. (Contributed by NM, 11-Mar-2013)

Ref Expression
Hypotheses cdleme35.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme35.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme35.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme35.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme35.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme35.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme35.f ⊢ 𝐹 = ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ) )
cdleme35.g ⊢ 𝐺 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
Assertion cdleme35h ( ( ( ( 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 ) ∧ 𝐹 = 𝐺 ) ) → 𝑅 = 𝑆 )

Proof

Step Hyp Ref Expression
1 cdleme35.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme35.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme35.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme35.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme35.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme35.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme35.f ⊢ 𝐹 = ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ) )
8 cdleme35.g ⊢ 𝐺 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
9 oveq1 ⊢ ( 𝐹 = 𝐺 → ( 𝐹 ∨ 𝑈 ) = ( 𝐺 ∨ 𝑈 ) )
10 oveq2 ⊢ ( 𝐹 = 𝐺 → ( 𝑄 ∨ 𝐹 ) = ( 𝑄 ∨ 𝐺 ) )
11 10 oveq1d ⊢ ( 𝐹 = 𝐺 → ( ( 𝑄 ∨ 𝐹 ) ∧ 𝑊 ) = ( ( 𝑄 ∨ 𝐺 ) ∧ 𝑊 ) )
12 11 oveq2d ⊢ ( 𝐹 = 𝐺 → ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐹 ) ∧ 𝑊 ) ) = ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐺 ) ∧ 𝑊 ) ) )
13 9 12 oveq12d ⊢ ( 𝐹 = 𝐺 → ( ( 𝐹 ∨ 𝑈 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐹 ) ∧ 𝑊 ) ) ) = ( ( 𝐺 ∨ 𝑈 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐺 ) ∧ 𝑊 ) ) ) )
14 13 3ad2ant3 ⊢ ( ( ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) → ( ( 𝐹 ∨ 𝑈 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐹 ) ∧ 𝑊 ) ) ) = ( ( 𝐺 ∨ 𝑈 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐺 ) ∧ 𝑊 ) ) ) )
15 14 3ad2ant3 ⊢ ( ( ( ( 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 ) ∧ 𝐹 = 𝐺 ) ) → ( ( 𝐹 ∨ 𝑈 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐹 ) ∧ 𝑊 ) ) ) = ( ( 𝐺 ∨ 𝑈 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐺 ) ∧ 𝑊 ) ) ) )
16 simp1 ⊢ ( ( ( ( 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 ) ∧ 𝐹 = 𝐺 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
17 simp21 ⊢ ( ( ( ( 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 ) ∧ 𝐹 = 𝐺 ) ) → 𝑃 ≠ 𝑄 )
18 simp22 ⊢ ( ( ( ( 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 ) ∧ 𝐹 = 𝐺 ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) )
19 simp31 ⊢ ( ( ( ( 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 ) ∧ 𝐹 = 𝐺 ) ) → ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
20 1 2 3 4 5 6 7 cdleme35g ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝐹 ∨ 𝑈 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐹 ) ∧ 𝑊 ) ) ) = 𝑅 )
21 16 17 18 19 20 syl121anc ⊢ ( ( ( ( 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 ) ∧ 𝐹 = 𝐺 ) ) → ( ( 𝐹 ∨ 𝑈 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐹 ) ∧ 𝑊 ) ) ) = 𝑅 )
22 simp23 ⊢ ( ( ( ( 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 ) ∧ 𝐹 = 𝐺 ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) )
23 simp32 ⊢ ( ( ( ( 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 ) ∧ 𝐹 = 𝐺 ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
24 1 2 3 4 5 6 8 cdleme35g ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝐺 ∨ 𝑈 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐺 ) ∧ 𝑊 ) ) ) = 𝑆 )
25 16 17 22 23 24 syl121anc ⊢ ( ( ( ( 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 ) ∧ 𝐹 = 𝐺 ) ) → ( ( 𝐺 ∨ 𝑈 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝐺 ) ∧ 𝑊 ) ) ) = 𝑆 )
26 15 21 25 3eqtr3d ⊢ ( ( ( ( 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 ) ∧ 𝐹 = 𝐺 ) ) → 𝑅 = 𝑆 )