| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpat2 | Structured version Visualization version GIF version | ||
| Description: Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.) |
| Ref | Expression |
|---|---|
| lhpat.l | ⊢ ≤ = (le‘𝐾) |
| lhpat.j | ⊢ ∨ = (join‘𝐾) |
| lhpat.m | ⊢ ∧ = (meet‘𝐾) |
| lhpat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| lhpat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
| lhpat2.r | ⊢ 𝑅 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
| Ref | Expression |
|---|---|
| lhpat2 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑅 ∈ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lhpat2.r | . 2 ⊢ 𝑅 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
| 2 | lhpat.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
| 3 | lhpat.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
| 4 | lhpat.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
| 5 | lhpat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 6 | lhpat.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 7 | 2, 3, 4, 5, 6 | lhpat 40045 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴) |
| 8 | 1, 7 | eqeltrid 2845 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑅 ∈ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 lecple 17304 joincjn 18357 meetcmee 18358 Atomscatm 39264 HLchlt 39351 LHypclh 39986 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-lhyp 39990 |
| This theorem is referenced by: lhpat3 40048 4atexlemu 40066 4atexlemv 40067 cdleme0a 40213 cdleme0dN 40218 cdleme0e 40219 cdleme02N 40224 cdleme0ex1N 40225 cdleme0moN 40227 cdleme3b 40231 cdleme3c 40232 cdleme3g 40236 cdleme3h 40237 cdleme3 40239 cdleme7aa 40244 cdleme7c 40247 cdleme7d 40248 cdleme7e 40249 cdleme7ga 40250 cdleme7 40251 cdleme9a 40253 cdleme16aN 40261 cdleme11a 40262 cdleme11c 40263 cdleme12 40273 cdleme16b 40281 cdleme16c 40282 cdleme16d 40283 cdleme20h 40318 cdleme20j 40320 cdleme20l2 40323 cdlemeg46rgv 40530 cdlemeg46req 40531 |
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