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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 30314 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 2738 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | atnem0.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | 1, 2 | atncmp 36971 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃(le‘𝐾)𝑄 ↔ 𝑃 ≠ 𝑄)) |
4 | eqid 2738 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | 4, 2 | atbase 36948 | . . 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 36976 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ (Base‘𝐾)) → (¬ 𝑃(le‘𝐾)𝑄 ↔ (𝑃 ∧ 𝑄) = 0 )) |
9 | 5, 8 | syl3an3 1166 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (¬ 𝑃(le‘𝐾)𝑄 ↔ (𝑃 ∧ 𝑄) = 0 )) |
10 | 3, 9 | bitr3d 284 | 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 209 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 class class class wbr 5030 ‘cfv 6339 (class class class)co 7172 Basecbs 16588 lecple 16677 meetcmee 17673 0.cp0 17765 Atomscatm 36922 AtLatcal 36923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-proset 17656 df-poset 17674 df-plt 17686 df-lub 17702 df-glb 17703 df-join 17704 df-meet 17705 df-p0 17767 df-lat 17774 df-covers 36925 df-ats 36926 df-atl 36957 |
This theorem is referenced by: cvlatcvr1 37000 atcvrj1 37090 dalem24 37356 lhp2at0 37691 trlval3 37846 cdleme0e 37876 cdleme7c 37904 |
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