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Metamath Proof Explorer

Theorem cdlemf

Description: Lemma F in Crawley p. 116. If u is an atom under w, there exists a translation whose trace is u. (Contributed by NM, 12-Apr-2013)

Ref Expression
Hypotheses cdlemf.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemf.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemf.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemf.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemf.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemf ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → ∃ 𝑓 ∈ 𝑇 ( 𝑅 ‘ 𝑓 ) = 𝑈 )

Proof

Step Hyp Ref Expression
1 cdlemf.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemf.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 cdlemf.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 cdlemf.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 cdlemf.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
7 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
8 1 6 2 3 7 cdlemf2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
9 simp1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) → 𝑝 ∈ 𝐴 )
11 simp3ll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) → ¬ 𝑝 ≤ 𝑊 )
12 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) → 𝑞 ∈ 𝐴 )
13 simp3lr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) → ¬ 𝑞 ≤ 𝑊 )
14 1 2 3 4 cdleme50ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ∧ ( 𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊 ) ) → ∃ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 )
15 9 10 11 12 13 14 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) → ∃ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 )
16 simp3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → ( 𝑓 ‘ 𝑝 ) = 𝑞 )
17 16 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → ( 𝑝 ( join ‘ 𝐾 ) ( 𝑓 ‘ 𝑝 ) ) = ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) )
18 17 oveq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( 𝑓 ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) )
19 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
20 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → 𝑓 ∈ 𝑇 )
21 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → 𝑝 ∈ 𝐴 )
22 simp2ll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → ¬ 𝑝 ≤ 𝑊 )
23 1 6 7 2 3 4 5 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ 𝑇 ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝑓 ) = ( ( 𝑝 ( join ‘ 𝐾 ) ( 𝑓 ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
24 19 20 21 22 23 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → ( 𝑅 ‘ 𝑓 ) = ( ( 𝑝 ( join ‘ 𝐾 ) ( 𝑓 ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
25 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) )
26 18 24 25 3eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) ) → ( 𝑅 ‘ 𝑓 ) = 𝑈 )
27 26 3exp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ) → ( ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) → ( 𝑅 ‘ 𝑓 ) = 𝑈 ) ) )
28 27 3expia ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → ( ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) → ( ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) → ( 𝑅 ‘ 𝑓 ) = 𝑈 ) ) ) )
29 28 3imp ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) → ( ( 𝑓 ∈ 𝑇 ∧ ( 𝑓 ‘ 𝑝 ) = 𝑞 ) → ( 𝑅 ‘ 𝑓 ) = 𝑈 ) )
30 29 expd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) → ( 𝑓 ∈ 𝑇 → ( ( 𝑓 ‘ 𝑝 ) = 𝑞 → ( 𝑅 ‘ 𝑓 ) = 𝑈 ) ) )
31 30 reximdvai ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) → ( ∃ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑝 ) = 𝑞 → ∃ 𝑓 ∈ 𝑇 ( 𝑅 ‘ 𝑓 ) = 𝑈 ) )
32 15 31 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) ∧ ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) → ∃ 𝑓 ∈ 𝑇 ( 𝑅 ‘ 𝑓 ) = 𝑈 )
33 32 3exp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → ( ( 𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴 ) → ( ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) → ∃ 𝑓 ∈ 𝑇 ( 𝑅 ‘ 𝑓 ) = 𝑈 ) ) )
34 33 rexlimdvv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → ( ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ) ∧ 𝑈 = ( ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) → ∃ 𝑓 ∈ 𝑇 ( 𝑅 ‘ 𝑓 ) = 𝑈 ) )
35 8 34 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → ∃ 𝑓 ∈ 𝑇 ( 𝑅 ‘ 𝑓 ) = 𝑈 )