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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 35375 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
2 | 1 | adantr 474 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ Poset) |
3 | eqid 2825 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | atnle0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
5 | 3, 4 | atl0cl 35377 | . . 3 ⊢ (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾)) |
6 | 5 | adantr 474 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ∈ (Base‘𝐾)) |
7 | atnle0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 3, 7 | atbase 35363 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
9 | 8 | adantl 475 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾)) |
10 | eqid 2825 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
11 | 4, 10, 7 | atcvr0 35362 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
12 | atnle0.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
13 | 3, 12, 10 | cvrnle 35354 | . 2 ⊢ (((𝐾 ∈ Poset ∧ 0 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) ∧ 0 ( ⋖ ‘𝐾)𝑃) → ¬ 𝑃 ≤ 0 ) |
14 | 2, 6, 9, 11, 13 | syl31anc 1496 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ≤ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 ‘cfv 6127 Basecbs 16229 lecple 16319 Posetcpo 17300 0.cp0 17397 ⋖ ccvr 35336 Atomscatm 35337 AtLatcal 35338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-proset 17288 df-poset 17306 df-plt 17318 df-glb 17335 df-p0 17399 df-lat 17406 df-covers 35340 df-ats 35341 df-atl 35372 |
This theorem is referenced by: pmap0 35839 trlnle 36260 cdlemg27b 36770 |
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