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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatexchb1 | Structured version Visualization version GIF version |
Description: A version of hlexchb1 38987 for atoms. (Contributed by NM, 15-Nov-2011.) |
Ref | Expression |
---|---|
hlatexchb.l | ⊢ ≤ = (le‘𝐾) |
hlatexchb.j | ⊢ ∨ = (join‘𝐾) |
hlatexchb.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
hlatexchb1 | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcvl 38961 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat) | |
2 | hlatexchb.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | hlatexchb.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | hlatexchb.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 2, 3, 4 | cvlatexchb1 38936 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄))) |
6 | 1, 5 | syl3an1 1160 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑄) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 lecple 17243 joincjn 18306 Atomscatm 38865 CvLatclc 38867 HLchlt 38952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-lat 18427 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 |
This theorem is referenced by: 3dimlem2 39062 ps-1 39080 3atlem6 39091 cdlemblem 39396 cdleme11a 39863 |
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