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Metamath Proof Explorer

Theorem cdlemc3

Description: Part of proof of Lemma C in Crawley p. 113. (Contributed by NM, 26-May-2012)

Ref Expression
Hypotheses cdlemc3.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemc3.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemc3.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemc3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemc3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemc3.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemc3.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemc3 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) → 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemc3.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemc3.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemc3.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdlemc3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemc3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemc3.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemc3.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simpll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝐾 ∈ HL )
9 simpl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 simpr1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝐹 ∈ 𝑇 )
11 simpr2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝑃 ∈ 𝐴 )
12 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
13 9 10 11 12 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
14 simpr3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝑄 ∈ 𝐴 )
15 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
16 15 5 6 7 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) )
17 10 16 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) )
18 1 4 5 6 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
19 18 3adant3r3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
20 1 4 5 6 7 trlnle ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) → ¬ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑅 ‘ 𝐹 ) )
21 9 10 19 20 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ¬ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑅 ‘ 𝐹 ) )
22 15 1 2 4 hlexch2 ⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑅 ‘ 𝐹 ) ) → ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) → 𝑄 ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) ) )
23 8 13 14 17 21 22 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) → 𝑄 ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) ) )
24 1 2 4 5 6 7 trljat2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
25 24 3adant3r3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
26 25 breq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝑄 ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐹 ) ) ↔ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) )
27 23 26 sylibd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ≤ ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) → 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) )