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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 38057 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐴) |
8 | 1, 7 | eqeltrid 2843 | 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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 lecple 16969 joincjn 18029 meetcmee 18030 Atomscatm 37277 HLchlt 37364 LHypclh 37998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-lhyp 38002 |
This theorem is referenced by: lhpat3 38060 4atexlemu 38078 4atexlemv 38079 cdleme0a 38225 cdleme0dN 38230 cdleme0e 38231 cdleme02N 38236 cdleme0ex1N 38237 cdleme0moN 38239 cdleme3b 38243 cdleme3c 38244 cdleme3g 38248 cdleme3h 38249 cdleme3 38251 cdleme7aa 38256 cdleme7c 38259 cdleme7d 38260 cdleme7e 38261 cdleme7ga 38262 cdleme7 38263 cdleme9a 38265 cdleme16aN 38273 cdleme11a 38274 cdleme11c 38275 cdleme12 38285 cdleme16b 38293 cdleme16c 38294 cdleme16d 38295 cdleme20h 38330 cdleme20j 38332 cdleme20l2 38335 cdlemeg46rgv 38542 cdlemeg46req 38543 |
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