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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 1188 | . . . 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 36599 | . . . . 5 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ∈ 𝐵) |
6 | simpl2 1189 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ∈ 𝐵) | |
7 | 1, 5, 6 | 3jca 1125 | . . 3 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → (𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) |
8 | simpl3 1190 | . . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐴) | |
9 | atlen0.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | 2, 9 | atbase 36585 | . . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐵) |
12 | eqid 2798 | . . . . . . 7 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
13 | 3, 12, 9 | atcvr0 36584 | . . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
14 | 1, 8, 13 | syl2anc 587 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ( ⋖ ‘𝐾)𝑃) |
15 | eqid 2798 | . . . . . 6 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
16 | 2, 15, 12 | cvrlt 36566 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) ∧ 0 ( ⋖ ‘𝐾)𝑃) → 0 (lt‘𝐾)𝑃) |
17 | 1, 5, 11, 14, 16 | syl31anc 1370 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑃) |
18 | simpr 488 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ≤ 𝑋) | |
19 | atlpos 36597 | . . . . . 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 17573 | . . . . 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 1369 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → (( 0 (lt‘𝐾)𝑃 ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋)) |
24 | 17, 18, 23 | mp2and 698 | . . 3 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋) |
25 | 15 | pltne 17564 | . . 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 3042 | 1 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 lecple 16564 Posetcpo 17542 ltcplt 17543 0.cp0 17639 ⋖ ccvr 36558 Atomscatm 36559 AtLatcal 36560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-proset 17530 df-poset 17548 df-plt 17560 df-glb 17577 df-p0 17641 df-lat 17648 df-covers 36562 df-ats 36563 df-atl 36594 |
This theorem is referenced by: ps-2b 36778 2atm 36823 2llnm4 36866 dalem21 36990 dalem54 37022 trlval3 37483 cdlemc5 37491 |
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