![]() |
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 36311 and cdleme3 36312. (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 6916 | . . 3 ⊢ (𝑄 ∨ 𝑉) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)) |
4 | 3 | oveq2i 6916 | . 2 ⊢ ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉)) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) |
5 | 1, 4 | eqtr4i 2852 | 1 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ‘cfv 6123 (class class class)co 6905 lecple 16312 joincjn 17297 meetcmee 17298 Atomscatm 35338 LHypclh 36059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-iota 6086 df-fv 6131 df-ov 6908 |
This theorem is referenced by: cdleme3g 36309 cdleme3h 36310 cdleme9 36328 |
Copyright terms: Public domain | W3C validator |