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Mathbox for Norm Megill |
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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 36313. (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 2826 | . 2 ⊢ ((𝑃 ∨ 𝑅) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | cdleme3h 36311 | 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 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 class class class wbr 4874 ‘cfv 6124 (class class class)co 6906 lecple 16313 joincjn 17298 meetcmee 17299 Atomscatm 35339 HLchlt 35426 LHypclh 36060 |
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-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
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-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-1st 7429 df-2nd 7430 df-proset 17282 df-poset 17300 df-plt 17312 df-lub 17328 df-glb 17329 df-join 17330 df-meet 17331 df-p0 17393 df-p1 17394 df-lat 17400 df-clat 17462 df-oposet 35252 df-ol 35254 df-oml 35255 df-covers 35342 df-ats 35343 df-atl 35374 df-cvlat 35398 df-hlat 35427 df-lines 35577 df-psubsp 35579 df-pmap 35580 df-padd 35872 df-lhyp 36064 |
This theorem is referenced by: cdleme3 36313 cdleme7d 36322 cdleme7ga 36324 cdleme11j 36343 cdleme11k 36344 cdleme11 36346 cdleme14 36349 cdleme15a 36350 cdleme16b 36355 cdleme16c 36356 cdleme16d 36357 cdleme16e 36358 cdleme16f 36359 cdleme19d 36382 cdleme20f 36390 cdleme20l1 36396 cdleme20l2 36397 cdleme22f2 36423 cdleme22g 36424 cdlemefr32sn2aw 36480 cdleme35a 36524 cdleme36m 36537 cdleme43bN 36566 |
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