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Mathbox for Norm Megill |
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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 32383 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 2734 | . . . . 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 39204 | . . . 4 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))) |
7 | 6 | simp3bi 1147 | . . 3 ⊢ (𝐾 ∈ AtLat → ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
8 | neeq1 3005 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ 0 ↔ 𝑋 ≠ 0 )) | |
9 | breq2 5173 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋)) | |
10 | 9 | rexbidv 3181 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)) |
11 | 8, 10 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ↔ (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋))) |
12 | 11 | rspccv 3628 | . . 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 1111 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 ∀wral 3063 ∃wrex 3072 class class class wbr 5169 dom cdm 5699 ‘cfv 6572 Basecbs 17253 lecple 17313 glbcglb 18375 0.cp0 18488 Latclat 18496 Atomscatm 39168 AtLatcal 39169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-dm 5709 df-iota 6524 df-fv 6580 df-atl 39203 |
This theorem is referenced by: atnle 39222 atlatmstc 39224 cvratlem 39327 cvrat4 39349 2llnmat 39430 2lnat 39690 |
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