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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme7a | Structured version Visualization version GIF version | ||
| Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 40250 and cdleme7 40251. (Contributed by NM, 7-Jun-2012.) |
| Ref | Expression |
|---|---|
| cdleme4.l | ⊢ ≤ = (le‘𝐾) |
| cdleme4.j | ⊢ ∨ = (join‘𝐾) |
| cdleme4.m | ⊢ ∧ = (meet‘𝐾) |
| cdleme4.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| cdleme4.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| cdleme4.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
| cdleme4.f | ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))) |
| cdleme4.g | ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) |
| cdleme7.v | ⊢ 𝑉 = ((𝑅 ∨ 𝑆) ∧ 𝑊) |
| Ref | Expression |
|---|---|
| cdleme7a | ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdleme4.g | . 2 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) | |
| 2 | cdleme7.v | . . . 4 ⊢ 𝑉 = ((𝑅 ∨ 𝑆) ∧ 𝑊) | |
| 3 | 2 | oveq2i 7442 | . . 3 ⊢ (𝐹 ∨ 𝑉) = (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)) |
| 4 | 3 | oveq2i 7442 | . 2 ⊢ ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) |
| 5 | 1, 4 | eqtr4i 2768 | 1 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ‘cfv 6561 (class class class)co 7431 lecple 17304 joincjn 18357 meetcmee 18358 Atomscatm 39264 LHypclh 39986 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: cdleme7d 40248 cdleme17a 40288 |
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