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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 30131 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 2821 | . . . . 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 36429 | . . . 4 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))) |
7 | 6 | simp3bi 1143 | . . 3 ⊢ (𝐾 ∈ AtLat → ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
8 | neeq1 3078 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ 0 ↔ 𝑋 ≠ 0 )) | |
9 | breq2 5062 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋)) | |
10 | 9 | rexbidv 3297 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)) |
11 | 8, 10 | imbi12d 347 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ↔ (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋))) |
12 | 11 | rspccv 3619 | . . 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 1107 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 class class class wbr 5058 dom cdm 5549 ‘cfv 6349 Basecbs 16477 lecple 16566 glbcglb 17547 0.cp0 17641 Latclat 17649 Atomscatm 36393 AtLatcal 36394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-dm 5559 df-iota 6308 df-fv 6357 df-atl 36428 |
This theorem is referenced by: atnle 36447 atlatmstc 36449 cvratlem 36551 cvrat4 36573 2llnmat 36654 2lnat 36914 |
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