Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 1189 | . . . 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 37244 | . . . . 5 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ∈ 𝐵) |
6 | simpl2 1190 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ∈ 𝐵) | |
7 | 1, 5, 6 | 3jca 1126 | . . 3 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → (𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) |
8 | simpl3 1191 | . . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐴) | |
9 | atlen0.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | 2, 9 | atbase 37230 | . . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐵) |
12 | eqid 2738 | . . . . . . 7 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
13 | 3, 12, 9 | atcvr0 37229 | . . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
14 | 1, 8, 13 | syl2anc 583 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ( ⋖ ‘𝐾)𝑃) |
15 | eqid 2738 | . . . . . 6 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
16 | 2, 15, 12 | cvrlt 37211 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) ∧ 0 ( ⋖ ‘𝐾)𝑃) → 0 (lt‘𝐾)𝑃) |
17 | 1, 5, 11, 14, 16 | syl31anc 1371 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑃) |
18 | simpr 484 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ≤ 𝑋) | |
19 | atlpos 37242 | . . . . . 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 17976 | . . . . 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 1370 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → (( 0 (lt‘𝐾)𝑃 ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋)) |
24 | 17, 18, 23 | mp2and 695 | . . 3 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋) |
25 | 15 | pltne 17967 | . . 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 2998 | 1 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 ‘cfv 6418 Basecbs 16840 lecple 16895 Posetcpo 17940 ltcplt 17941 0.cp0 18056 ⋖ ccvr 37203 Atomscatm 37204 AtLatcal 37205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-proset 17928 df-poset 17946 df-plt 17963 df-glb 17980 df-p0 18058 df-lat 18065 df-covers 37207 df-ats 37208 df-atl 37239 |
This theorem is referenced by: ps-2b 37423 2atm 37468 2llnm4 37511 dalem21 37635 dalem54 37667 trlval3 38128 cdlemc5 38136 |
Copyright terms: Public domain | W3C validator |