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Mathbox for Norm Megill |
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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 38912 and cdleme3 38913. (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 7404 | . . 3 ⊢ (𝑄 ∨ 𝑉) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) |
4 | 3 | oveq2i 7404 | . 2 ⊢ ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉)) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) |
5 | 1, 4 | eqtr4i 2762 | 1 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ‘cfv 6532 (class class class)co 7393 lecple 17186 joincjn 18246 meetcmee 18247 Atomscatm 37938 LHypclh 38660 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-iota 6484 df-fv 6540 df-ov 7396 |
This theorem is referenced by: cdleme3g 38910 cdleme3h 38911 cdleme9 38929 |
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