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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 29219 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 2622 | . . . . 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 34586 | . . . 4 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom (glb‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))) |
7 | 6 | simp3bi 1078 | . . 3 ⊢ (𝐾 ∈ AtLat → ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
8 | neeq1 2856 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ≠ 0 ↔ 𝑋 ≠ 0 )) | |
9 | breq2 4657 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋)) | |
10 | 9 | rexbidv 3052 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋)) |
11 | 8, 10 | imbi12d 334 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ↔ (𝑋 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋))) |
12 | 11 | rspccv 3306 | . . 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 1256 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 Basecbs 15857 lecple 15948 glbcglb 16943 0.cp0 17037 Latclat 17045 Atomscatm 34550 AtLatcal 34551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-dm 5124 df-iota 5851 df-fv 5896 df-atl 34585 |
This theorem is referenced by: atnle 34604 atlatmstc 34606 cvratlem 34707 cvrat4 34729 2llnmat 34810 2lnat 35070 |
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