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Mirrors > Home > MPE Home > Th. List > Mathboxes > atnem0 | Structured version Visualization version GIF version |
Description: The meet of distinct atoms is zero. (atnemeq0 30153 analog.) (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
atnem0.m | ⊢ ∧ = (meet‘𝐾) |
atnem0.z | ⊢ 0 = (0.‘𝐾) |
atnem0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atnem0 | ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ (𝑃 ∧ 𝑄) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | atnem0.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | 1, 2 | atncmp 36447 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃(le‘𝐾)𝑄 ↔ 𝑃 ≠ 𝑄)) |
4 | eqid 2821 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | 4, 2 | atbase 36424 | . . 3 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
6 | atnem0.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
7 | atnem0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
8 | 4, 1, 6, 7, 2 | atnle 36452 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ (Base‘𝐾)) → (¬ 𝑃(le‘𝐾)𝑄 ↔ (𝑃 ∧ 𝑄) = 0 )) |
9 | 5, 8 | syl3an3 1161 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃(le‘𝐾)𝑄 ↔ (𝑃 ∧ 𝑄) = 0 )) |
10 | 3, 9 | bitr3d 283 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 ↔ (𝑃 ∧ 𝑄) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 lecple 16571 meetcmee 17554 0.cp0 17646 Atomscatm 36398 AtLatcal 36399 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-lat 17655 df-covers 36401 df-ats 36402 df-atl 36433 |
This theorem is referenced by: cvlatcvr1 36476 atcvrj1 36566 dalem24 36832 lhp2at0 37167 trlval3 37322 cdleme0e 37352 cdleme7c 37380 |
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