Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 37388 and cdleme7 37389. (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 7170 | . . 3 ⊢ (𝐹 ∨ 𝑉) = (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)) |
4 | 3 | oveq2i 7170 | . 2 ⊢ ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊))) |
5 | 1, 4 | eqtr4i 2850 | 1 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ‘cfv 6358 (class class class)co 7159 lecple 16575 joincjn 17557 meetcmee 17558 Atomscatm 36403 LHypclh 37124 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 |
This theorem is referenced by: cdleme7d 37386 cdleme17a 37426 |
Copyright terms: Public domain | W3C validator |