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Metamath Proof Explorer

Theorem cdlemksat

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 27-Jun-2013)

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

Proof

Step Hyp Ref Expression
1 cdlemk.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemk.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemk.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemk.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemk.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemk.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemk.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemk.m ⊢ ∧ = ( meet ‘ 𝐾 )
9 cdlemk.s ⊢ 𝑆 = ( 𝑓 ∈ 𝑇 ↦ ( ℩ 𝑖 ∈ 𝑇 ( 𝑖 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑓 ) ) ∧ ( ( 𝑁 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑓 ∘ ◡ 𝐹 ) ) ) ) ) )
10 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑁 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ 𝐹 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 1 2 3 4 5 6 7 8 9 cdlemksel ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑁 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ 𝐹 ) ) ) → ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 )
12 simp22l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑁 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ 𝐹 ) ) ) → 𝑃 ∈ 𝐴 )
13 2 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝑆 ‘ 𝐺 ) ‘ 𝑃 ) ∈ 𝐴 )
14 10 11 12 13 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑁 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝑁 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ 𝐹 ) ) ) → ( ( 𝑆 ‘ 𝐺 ) ‘ 𝑃 ) ∈ 𝐴 )