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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 36269 and cdleme7 36270. (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 6889 | . . 3 ⊢ (𝐹 ∨ 𝑉) = (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)) |
4 | 3 | oveq2i 6889 | . 2 ⊢ ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) |
5 | 1, 4 | eqtr4i 2824 | 1 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ‘cfv 6101 (class class class)co 6878 lecple 16274 joincjn 17259 meetcmee 17260 Atomscatm 35284 LHypclh 36005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-iota 6064 df-fv 6109 df-ov 6881 |
This theorem is referenced by: cdleme7d 36267 cdleme17a 36307 |
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