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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 40230 and cdleme7 40231. (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 7441 | . . 3 ⊢ (𝐹 ∨ 𝑉) = (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)) |
4 | 3 | oveq2i 7441 | . 2 ⊢ ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) |
5 | 1, 4 | eqtr4i 2765 | 1 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ‘cfv 6562 (class class class)co 7430 lecple 17304 joincjn 18368 meetcmee 18369 Atomscatm 39244 LHypclh 39966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-ov 7433 |
This theorem is referenced by: cdleme7d 40228 cdleme17a 40268 |
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