Description: Part of proof of Lemma K of Crawley p. 118. Value of the sigma_1 (p) function U . (Contributed by NM, 2-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 ) | ||
| cdlemk1.u | ⊢ 𝑈 = ( 𝑒 ∈ 𝑇 ↦ ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( 𝑂 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) ) ) ) | ||
| Assertion | cdlemkuvN | ⊢ ( 𝐺 ∈ 𝑇 → ( 𝑈 ‘ 𝐺 ) = ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑂 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 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 | cdlemk1.u | ⊢ 𝑈 = ( 𝑒 ∈ 𝑇 ↦ ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑒 ) ) ∧ ( ( 𝑂 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝑒 ∘ ◡ 𝐷 ) ) ) ) ) ) | |
| 12 | 1 2 3 5 6 7 8 4 11 | cdlemksv | ⊢ ( 𝐺 ∈ 𝑇 → ( 𝑈 ‘ 𝐺 ) = ( ℩ 𝑗 ∈ 𝑇 ( 𝑗 ‘ 𝑃 ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑂 ‘ 𝑃 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐷 ) ) ) ) ) ) |