|   | Mathbox for Norm Megill | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme3d | Structured version Visualization version GIF version | ||
| Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 40239 and cdleme3 40240. (Contributed by NM, 6-Jun-2012.) | 
| Ref | Expression | 
|---|---|
| cdleme1.l | ⊢ ≤ = (le‘𝐾) | 
| cdleme1.j | ⊢ ∨ = (join‘𝐾) | 
| cdleme1.m | ⊢ ∧ = (meet‘𝐾) | 
| cdleme1.a | ⊢ 𝐴 = (Atoms‘𝐾) | 
| cdleme1.h | ⊢ 𝐻 = (LHyp‘𝐾) | 
| cdleme1.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | 
| cdleme1.f | ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) | 
| cdleme3.3 | ⊢ 𝑉 = ((𝑃 ∨ 𝑅) ∧ 𝑊) | 
| Ref | Expression | 
|---|---|
| cdleme3d | ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cdleme1.f | . 2 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) | |
| 2 | cdleme3.3 | . . . 4 ⊢ 𝑉 = ((𝑃 ∨ 𝑅) ∧ 𝑊) | |
| 3 | 2 | oveq2i 7443 | . . 3 ⊢ (𝑄 ∨ 𝑉) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) | 
| 4 | 3 | oveq2i 7443 | . 2 ⊢ ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉)) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) | 
| 5 | 1, 4 | eqtr4i 2767 | 1 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ‘cfv 6560 (class class class)co 7432 lecple 17305 joincjn 18358 meetcmee 18359 Atomscatm 39265 LHypclh 39987 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 | 
| This theorem is referenced by: cdleme3g 40237 cdleme3h 40238 cdleme9 40256 | 
| Copyright terms: Public domain | W3C validator |