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Metamath Proof Explorer

Theorem cdlemk13

Description: Part of proof of Lemma K of Crawley p. 118. Line 13 on p. 119. O , D are k_1, f_1. (Contributed by NM, 1-Jul-2013)

Ref Expression
Hypotheses cdlemk1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemk1.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemk1.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemk1.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemk1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemk1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemk1.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemk1.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemk1.s ⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
cdlemk1.o ⊢ 𝑂 = ( 𝑆 ‘ 𝐷 )
Assertion cdlemk13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ) ∧ ( 𝑁 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐷 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐷 ) ≠ ( 𝑅 ‘ 𝐹 ) ) ) → ( 𝑂 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐷 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐷 ∘ ◡ 𝐹 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemk1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemk1.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemk1.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemk1.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlemk1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemk1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemk1.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemk1.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
9 cdlemk1.s ⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
10 cdlemk1.o ⊢ 𝑂 = ( 𝑆 ‘ 𝐷 )
11 10 fveq1i ⊢ ( 𝑂 ‘ 𝑃 ) = ( ( 𝑆 ‘ 𝐷 ) ‘ 𝑃 )
12 1 2 3 5 6 7 8 4 9 cdlemksv2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ) ∧ ( 𝑁 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐷 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐷 ) ≠ ( 𝑅 ‘ 𝐹 ) ) ) → ( ( 𝑆 ‘ 𝐷 ) ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐷 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐷 ∘ ◡ 𝐹 ) ) ) ) )
13 11 12 eqtrid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ) ∧ ( 𝑁 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐷 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐷 ) ≠ ( 𝑅 ‘ 𝐹 ) ) ) → ( 𝑂 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐷 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐷 ∘ ◡ 𝐹 ) ) ) ) )