Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 3-Jul-2013) (New usage is discouraged.)
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 | cdlemk5auN | ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ 𝐷 ) ∧ ( 𝐷 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ) → ( ( ( 𝐷 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐷 ) ) ∧ ( ( 𝐷 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) ) ≤ ( ( 𝑋 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑋 ∘ ◡ 𝐷 ) ) ) ) |
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 | 1 2 3 5 6 7 8 4 | cdlemk5a | ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ 𝐷 ) ∧ ( 𝐷 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ) → ( ( ( 𝐷 ‘ 𝑃 ) ∨ ( 𝑅 ‘ 𝐷 ) ) ∧ ( ( 𝐷 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) ) ≤ ( ( 𝑋 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑋 ∘ ◡ 𝐷 ) ) ) ) |