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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 1057 | . . . 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 33608 | . . . . 5 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ∈ 𝐵) |
6 | simpl2 1058 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ∈ 𝐵) | |
7 | 1, 5, 6 | 3jca 1235 | . . 3 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → (𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) |
8 | simpl3 1059 | . . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐴) | |
9 | atlen0.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | 2, 9 | atbase 33594 | . . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐵) |
12 | eqid 2610 | . . . . . . 7 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
13 | 3, 12, 9 | atcvr0 33593 | . . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
14 | 1, 8, 13 | syl2anc 691 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ( ⋖ ‘𝐾)𝑃) |
15 | eqid 2610 | . . . . . 6 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
16 | 2, 15, 12 | cvrlt 33575 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) ∧ 0 ( ⋖ ‘𝐾)𝑃) → 0 (lt‘𝐾)𝑃) |
17 | 1, 5, 11, 14, 16 | syl31anc 1321 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑃) |
18 | simpr 476 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ≤ 𝑋) | |
19 | atlpos 33606 | . . . . . 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 16794 | . . . . 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 1320 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → (( 0 (lt‘𝐾)𝑃 ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋)) |
24 | 17, 18, 23 | mp2and 711 | . . 3 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋) |
25 | 15 | pltne 16785 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑋 → 0 ≠ 𝑋)) |
26 | 7, 24, 25 | sylc 63 | . 2 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ≠ 𝑋) |
27 | 26 | necomd 2837 | 1 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 Posetcpo 16763 ltcplt 16764 0.cp0 16860 ⋖ ccvr 33567 Atomscatm 33568 AtLatcal 33569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-preset 16751 df-poset 16769 df-plt 16781 df-glb 16798 df-p0 16862 df-lat 16869 df-covers 33571 df-ats 33572 df-atl 33603 |
This theorem is referenced by: ps-2b 33786 2atm 33831 2llnm4 33874 dalem21 33998 dalem54 34030 trlval3 34492 cdlemc5 34500 |
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