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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme3fa | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. See cdleme3 37367. (Contributed by NM, 6-Oct-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))) |
Ref | Expression |
---|---|
cdleme3fa | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐹 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme1.l | . 2 ⊢ ≤ = (le‘𝐾) | |
2 | cdleme1.j | . 2 ⊢ ∨ = (join‘𝐾) | |
3 | cdleme1.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
4 | cdleme1.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdleme1.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdleme1.u | . 2 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
7 | cdleme1.f | . 2 ⊢ 𝐹 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) | |
8 | eqid 2821 | . 2 ⊢ ((𝑃 ∨ 𝑅) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | cdleme3h 37365 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐹 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 lecple 16566 joincjn 17548 meetcmee 17549 Atomscatm 36393 HLchlt 36480 LHypclh 37114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 |
This theorem is referenced by: cdleme3 37367 cdleme7d 37376 cdleme7ga 37378 cdleme11j 37397 cdleme11k 37398 cdleme11 37400 cdleme14 37403 cdleme15a 37404 cdleme16b 37409 cdleme16c 37410 cdleme16d 37411 cdleme16e 37412 cdleme16f 37413 cdleme19d 37436 cdleme20f 37444 cdleme20l1 37450 cdleme20l2 37451 cdleme22f2 37477 cdleme22g 37478 cdlemefr32sn2aw 37534 cdleme35a 37578 cdleme36m 37591 cdleme43bN 37620 |
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