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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme0a | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.) |
Ref | Expression |
---|---|
cdleme0.l | ⊢ ≤ = (le‘𝐾) |
cdleme0.j | ⊢ ∨ = (join‘𝐾) |
cdleme0.m | ⊢ ∧ = (meet‘𝐾) |
cdleme0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme0.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
Ref | Expression |
---|---|
cdleme0a | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑈 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme0.l | . 2 ⊢ ≤ = (le‘𝐾) | |
2 | cdleme0.j | . 2 ⊢ ∨ = (join‘𝐾) | |
3 | cdleme0.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
4 | cdleme0.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdleme0.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdleme0.u | . 2 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | lhpat2 35649 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑈 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 lecple 15995 joincjn 16991 meetcmee 16992 Atomscatm 34868 HLchlt 34955 LHypclh 35588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-lhyp 35592 |
This theorem is referenced by: cdleme21c 35932 cdleme21ct 35934 cdleme22aa 35944 cdleme22e 35949 cdleme22eALTN 35950 cdleme35a 36053 cdleme35b 36055 cdleme35c 36056 cdleme35f 36059 cdleme36a 36065 cdleme42k 36089 cdlemg9a 36237 cdlemg12a 36248 cdlemg18b 36284 cdlemg18c 36285 |
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