Description: TODO: FIX COMMENT. TODO: Can this be moved up as a stand-alone theorem in ltrn* area? (Contributed by NM, 4-May-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cdlemg8.l | ⊢ ≤ = ( le ‘ 𝐾 ) | |
| cdlemg8.j | ⊢ ∨ = ( join ‘ 𝐾 ) | ||
| cdlemg8.m | ⊢ ∧ = ( meet ‘ 𝐾 ) | ||
| cdlemg8.a | ⊢ 𝐴 = ( Atoms ‘ 𝐾 ) | ||
| cdlemg8.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | ||
| cdlemg8.t | ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) | ||
| Assertion | cdlemg10b | ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdlemg8.l | ⊢ ≤ = ( le ‘ 𝐾 ) | |
| 2 | cdlemg8.j | ⊢ ∨ = ( join ‘ 𝐾 ) | |
| 3 | cdlemg8.m | ⊢ ∧ = ( meet ‘ 𝐾 ) | |
| 4 | cdlemg8.a | ⊢ 𝐴 = ( Atoms ‘ 𝐾 ) | |
| 5 | cdlemg8.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | |
| 6 | cdlemg8.t | ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) | |
| 7 | eqid | ⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) | |
| 8 | 5 6 1 2 4 3 7 | cdlemg2m | ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) |