| 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 32262 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 2729 | . . . . 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 39265 | . . . 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 2987 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ 0 ↔ 𝑋 ≠ 0 )) | |
| 9 | breq2 5106 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋)) | |
| 10 | 9 | rexbidv 3157 | . . . . 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 3582 | . . 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 1110 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 class class class wbr 5102 dom cdm 5631 ‘cfv 6499 Basecbs 17155 lecple 17203 glbcglb 18247 0.cp0 18358 Latclat 18366 Atomscatm 39229 AtLatcal 39230 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2701 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-dif 3914 df-un 3916 df-ss 3928 df-nul 4293 df-if 4485 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-dm 5641 df-iota 6452 df-fv 6507 df-atl 39264 |
| This theorem is referenced by: atnle 39283 atlatmstc 39285 cvratlem 39388 cvrat4 39410 2llnmat 39491 2lnat 39751 |
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