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Mirrors > Home > MPE Home > Th. List > Mathboxes > atlen0 | Structured version Visualization version GIF version |
Description: A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.) |
Ref | Expression |
---|---|
atlen0.b | ⊢ 𝐵 = (Base‘𝐾) |
atlen0.l | ⊢ ≤ = (le‘𝐾) |
atlen0.z | ⊢ 0 = (0.‘𝐾) |
atlen0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atlen0 | ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1228 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝐾 ∈ AtLat) | |
2 | atlen0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
3 | atlen0.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
4 | 2, 3 | atl0cl 35111 | . . . . 5 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ∈ 𝐵) |
6 | simpl2 1230 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ∈ 𝐵) | |
7 | 1, 5, 6 | 3jca 1123 | . . 3 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → (𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) |
8 | simpl3 1232 | . . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐴) | |
9 | atlen0.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | 2, 9 | atbase 35097 | . . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐵) |
12 | eqid 2760 | . . . . . . 7 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
13 | 3, 12, 9 | atcvr0 35096 | . . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
14 | 1, 8, 13 | syl2anc 696 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ( ⋖ ‘𝐾)𝑃) |
15 | eqid 2760 | . . . . . 6 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
16 | 2, 15, 12 | cvrlt 35078 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) ∧ 0 ( ⋖ ‘𝐾)𝑃) → 0 (lt‘𝐾)𝑃) |
17 | 1, 5, 11, 14, 16 | syl31anc 1480 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑃) |
18 | simpr 479 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ≤ 𝑋) | |
19 | atlpos 35109 | . . . . . 6 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
20 | 1, 19 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝐾 ∈ Poset) |
21 | atlen0.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
22 | 2, 21, 15 | pltletr 17192 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ ( 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (( 0 (lt‘𝐾)𝑃 ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋)) |
23 | 20, 5, 11, 6, 22 | syl13anc 1479 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → (( 0 (lt‘𝐾)𝑃 ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋)) |
24 | 17, 18, 23 | mp2and 717 | . . 3 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋) |
25 | 15 | pltne 17183 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑋 → 0 ≠ 𝑋)) |
26 | 7, 24, 25 | sylc 65 | . 2 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ≠ 𝑋) |
27 | 26 | necomd 2987 | 1 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 class class class wbr 4804 ‘cfv 6049 Basecbs 16079 lecple 16170 Posetcpo 17161 ltcplt 17162 0.cp0 17258 ⋖ ccvr 35070 Atomscatm 35071 AtLatcal 35072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-preset 17149 df-poset 17167 df-plt 17179 df-glb 17196 df-p0 17260 df-lat 17267 df-covers 35074 df-ats 35075 df-atl 35106 |
This theorem is referenced by: ps-2b 35289 2atm 35334 2llnm4 35377 dalem21 35501 dalem54 35533 trlval3 35995 cdlemc5 36003 |
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