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Mirrors > Home > MPE Home > Th. List > Mathboxes > atlex | Structured version Visualization version GIF version |
Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 28603 analog.) (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
atlex.b | ⊢ 𝐵 = (Base‘𝐾) |
atlex.l | ⊢ ≤ = (le‘𝐾) |
atlex.z | ⊢ 0 = (0.‘𝐾) |
atlex.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atlex | ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atlex.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2610 | . . . . 5 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | atlex.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
4 | atlex.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
5 | atlex.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isatl 33604 | . . . 4 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))) |
7 | 6 | simp3bi 1071 | . . 3 ⊢ (𝐾 ∈ AtLat → ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
8 | neeq1 2844 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ 0 ↔ 𝑋 ≠ 0 )) | |
9 | breq2 4587 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋)) | |
10 | 9 | rexbidv 3034 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)) |
11 | 8, 10 | imbi12d 333 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ↔ (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋))) |
12 | 11 | rspccv 3279 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → (𝑋 ∈ 𝐵 → (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋))) |
13 | 7, 12 | syl 17 | . 2 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝐵 → (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋))) |
14 | 13 | 3imp 1249 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 Basecbs 15695 lecple 15775 glbcglb 16766 0.cp0 16860 Latclat 16868 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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-dm 5048 df-iota 5768 df-fv 5812 df-atl 33603 |
This theorem is referenced by: atnle 33622 atlatmstc 33624 cvratlem 33725 cvrat4 33747 2llnmat 33828 2lnat 34088 |
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