Description: TODO: FIX COMMENT. TODO: replace with cdlemg4 . (Contributed by NM, 27-Apr-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cdlemg4.l | ⊢ ≤ = ( le ‘ 𝐾 ) | |
| cdlemg4.a | ⊢ 𝐴 = ( Atoms ‘ 𝐾 ) | ||
| cdlemg4.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | ||
| cdlemg4.t | ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) | ||
| cdlemg4.r | ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) | ||
| cdlemg4.j | ⊢ ∨ = ( join ‘ 𝐾 ) | ||
| cdlemg4b.v | ⊢ 𝑉 = ( 𝑅 ‘ 𝐺 ) | ||
| Assertion | cdlemg6a | ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑟 ) ) = 𝑟 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdlemg4.l | ⊢ ≤ = ( le ‘ 𝐾 ) | |
| 2 | cdlemg4.a | ⊢ 𝐴 = ( Atoms ‘ 𝐾 ) | |
| 3 | cdlemg4.h | ⊢ 𝐻 = ( LHyp ‘ 𝐾 ) | |
| 4 | cdlemg4.t | ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) | |
| 5 | cdlemg4.r | ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) | |
| 6 | cdlemg4.j | ⊢ ∨ = ( join ‘ 𝐾 ) | |
| 7 | cdlemg4b.v | ⊢ 𝑉 = ( 𝑅 ‘ 𝐺 ) | |
| 8 | 1 2 3 4 5 6 7 | cdlemg4 | ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑟 ) ) = 𝑟 ) |