![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg3a | Structured version Visualization version GIF version |
Description: Part of proof of Lemma G in [Crawley] p. 116, line 19. Show p ∨ q = p ∨ u. TODO: reformat cdleme0cp 37775 to match this, then replace with cdleme0cp 37775. (Contributed by NM, 19-Apr-2013.) |
Ref | Expression |
---|---|
cdlemg3.l | ⊢ ≤ = (le‘𝐾) |
cdlemg3.j | ⊢ ∨ = (join‘𝐾) |
cdlemg3.m | ⊢ ∧ = (meet‘𝐾) |
cdlemg3.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg3.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg3.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
Ref | Expression |
---|---|
cdlemg3a | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg3.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | cdlemg3.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | cdlemg3.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
4 | cdlemg3.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdlemg3.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdlemg3.u | . . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | cdleme8 37811 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑈) = (𝑃 ∨ 𝑄)) |
8 | 7 | eqcomd 2765 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 class class class wbr 5025 ‘cfv 6328 (class class class)co 7143 lecple 16615 joincjn 17605 meetcmee 17606 Atomscatm 36824 HLchlt 36911 LHypclh 37545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-iin 4879 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-1st 7686 df-2nd 7687 df-proset 17589 df-poset 17607 df-plt 17619 df-lub 17635 df-glb 17636 df-join 17637 df-meet 17638 df-p0 17700 df-p1 17701 df-lat 17707 df-clat 17769 df-oposet 36737 df-ol 36739 df-oml 36740 df-covers 36827 df-ats 36828 df-atl 36859 df-cvlat 36883 df-hlat 36912 df-psubsp 37064 df-pmap 37065 df-padd 37357 df-lhyp 37549 |
This theorem is referenced by: cdlemg9a 38193 cdlemg9b 38194 cdlemg12b 38205 |
Copyright terms: Public domain | W3C validator |