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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 37341 | 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 lecple 16564 joincjn 17546 meetcmee 17547 Atomscatm 36559 HLchlt 36646 LHypclh 37280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-lhyp 37284 |
This theorem is referenced by: cdleme21c 37623 cdleme21ct 37625 cdleme22aa 37635 cdleme22e 37640 cdleme22eALTN 37641 cdleme35a 37744 cdleme35b 37746 cdleme35c 37747 cdleme35f 37750 cdleme36a 37756 cdleme42k 37780 cdlemg9a 37928 cdlemg12a 37939 cdlemg18b 37975 cdlemg18c 37976 |
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