![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 38635 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
2 | 1 | adantr 480 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ Poset) |
3 | eqid 2731 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | atnle0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
5 | 3, 4 | atl0cl 38637 | . . 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 38623 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
9 | 8 | adantl 481 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾)) |
10 | eqid 2731 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
11 | 4, 10, 7 | atcvr0 38622 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
12 | atnle0.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
13 | 3, 12, 10 | cvrnle 38614 | . 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 1540 ∈ wcel 2105 class class class wbr 5148 ‘cfv 6543 Basecbs 17151 lecple 17211 Posetcpo 18270 0.cp0 18386 ⋖ ccvr 38596 Atomscatm 38597 AtLatcal 38598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-proset 18258 df-poset 18276 df-plt 18293 df-glb 18310 df-p0 18388 df-lat 18395 df-covers 38600 df-ats 38601 df-atl 38632 |
This theorem is referenced by: pmap0 39100 trlnle 39521 cdlemg27b 40031 |
Copyright terms: Public domain | W3C validator |