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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme9a | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. 𝐶 represents s1, which we prove is an atom. (Contributed by NM, 10-Jun-2012.) |
Ref | Expression |
---|---|
cdleme8.l | ⊢ ≤ = (le‘𝐾) |
cdleme8.j | ⊢ ∨ = (join‘𝐾) |
cdleme8.m | ⊢ ∧ = (meet‘𝐾) |
cdleme8.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme8.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme8.4 | ⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊) |
Ref | Expression |
---|---|
cdleme9a | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆)) → 𝐶 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme8.l | . 2 ⊢ ≤ = (le‘𝐾) | |
2 | cdleme8.j | . 2 ⊢ ∨ = (join‘𝐾) | |
3 | cdleme8.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
4 | cdleme8.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdleme8.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdleme8.4 | . 2 ⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | lhpat2 37175 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆)) → 𝐶 ∈ 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-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-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-lhyp 37118 |
This theorem is referenced by: cdleme9 37383 cdleme9taN 37386 cdleme11h 37396 cdleme11j 37397 cdleme11k 37398 cdleme17c 37418 |
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