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 34588 | . . 3 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ Poset) |
3 | eqid 2622 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | atnle0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
5 | 3, 4 | atl0cl 34590 | . . 3 ⊢ (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾)) |
6 | 5 | adantr 481 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ∈ (Base‘𝐾)) |
7 | atnle0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 3, 7 | atbase 34576 | . . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
9 | 8 | adantl 482 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾)) |
10 | eqid 2622 | . . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
11 | 4, 10, 7 | atcvr0 34575 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
12 | atnle0.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
13 | 3, 12, 10 | cvrnle 34567 | . 2 ⊢ (((𝐾 ∈ Poset ∧ 0 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) ∧ 0 ( ⋖ ‘𝐾)𝑃) → ¬ 𝑃 ≤ 0 ) |
14 | 2, 6, 9, 11, 13 | syl31anc 1329 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 ≤ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 Basecbs 15857 lecple 15948 Posetcpo 16940 0.cp0 17037 ⋖ ccvr 34549 Atomscatm 34550 AtLatcal 34551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-preset 16928 df-poset 16946 df-plt 16958 df-glb 16975 df-p0 17039 df-lat 17046 df-covers 34553 df-ats 34554 df-atl 34585 |
This theorem is referenced by: pmap0 35051 trlnle 35473 cdlemg27b 35984 |
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