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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atnle0 | Structured version Visualization version GIF version |
Description: An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.) |
Ref | Expression |
---|---|
atnle0.l | ⊢ ≤ = (le‘𝐾) |
atnle0.z | ⊢ 0 = (0.‘𝐾) |
atnle0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atnle0 | ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ≤ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atlpos 39283 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ Poset) |
3 | eqid 2735 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | atnle0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
5 | 3, 4 | atl0cl 39285 | . . 3 ⊢ (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾)) |
6 | 5 | adantr 480 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ∈ (Base‘𝐾)) |
7 | atnle0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 3, 7 | atbase 39271 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
9 | 8 | adantl 481 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾)) |
10 | eqid 2735 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
11 | 4, 10, 7 | atcvr0 39270 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
12 | atnle0.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
13 | 3, 12, 10 | cvrnle 39262 | . 2 ⊢ (((𝐾 ∈ Poset ∧ 0 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) ∧ 0 ( ⋖ ‘𝐾)𝑃) → ¬ 𝑃 ≤ 0 ) |
14 | 2, 6, 9, 11, 13 | syl31anc 1372 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ≤ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 lecple 17305 Posetcpo 18365 0.cp0 18481 ⋖ ccvr 39244 Atomscatm 39245 AtLatcal 39246 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-proset 18352 df-poset 18371 df-plt 18388 df-glb 18405 df-p0 18483 df-lat 18490 df-covers 39248 df-ats 39249 df-atl 39280 |
This theorem is referenced by: pmap0 39748 trlnle 40169 cdlemg27b 40679 |
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