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Theorem isatl 39300
Description: The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
Hypotheses
Ref Expression
isatlat.b 𝐵 = (Base‘𝐾)
isatlat.g 𝐺 = (glb‘𝐾)
isatlat.l = (le‘𝐾)
isatlat.z 0 = (0.‘𝐾)
isatlat.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
isatl (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦,𝐾
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐺(𝑥,𝑦)   (𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem isatl
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6906 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isatlat.b . . . . . 6 𝐵 = (Base‘𝐾)
31, 2eqtr4di 2795 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6906 . . . . . . 7 (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
5 isatlat.g . . . . . . 7 𝐺 = (glb‘𝐾)
64, 5eqtr4di 2795 . . . . . 6 (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
76dmeqd 5916 . . . . 5 (𝑘 = 𝐾 → dom (glb‘𝑘) = dom 𝐺)
83, 7eleq12d 2835 . . . 4 (𝑘 = 𝐾 → ((Base‘𝑘) ∈ dom (glb‘𝑘) ↔ 𝐵 ∈ dom 𝐺))
9 fveq2 6906 . . . . . . . 8 (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
10 isatlat.z . . . . . . . 8 0 = (0.‘𝐾)
119, 10eqtr4di 2795 . . . . . . 7 (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
1211neeq2d 3001 . . . . . 6 (𝑘 = 𝐾 → (𝑥 ≠ (0.‘𝑘) ↔ 𝑥0 ))
13 fveq2 6906 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
14 isatlat.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
1513, 14eqtr4di 2795 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
16 fveq2 6906 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
17 isatlat.l . . . . . . . . 9 = (le‘𝐾)
1816, 17eqtr4di 2795 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1918breqd 5154 . . . . . . 7 (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑥𝑦 𝑥))
2015, 19rexeqbidv 3347 . . . . . 6 (𝑘 = 𝐾 → (∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥 ↔ ∃𝑦𝐴 𝑦 𝑥))
2112, 20imbi12d 344 . . . . 5 (𝑘 = 𝐾 → ((𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
223, 21raleqbidv 3346 . . . 4 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
238, 22anbi12d 632 . . 3 (𝑘 = 𝐾 → (((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥)) ↔ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))))
24 df-atl 39299 . . 3 AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥))}
2523, 24elrab2 3695 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))))
26 3anass 1095 . 2 ((𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)) ↔ (𝐾 ∈ Lat ∧ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))))
2725, 26bitr4i 278 1 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070   class class class wbr 5143  dom cdm 5685  cfv 6561  Basecbs 17247  lecple 17304  glbcglb 18356  0.cp0 18468  Latclat 18476  Atomscatm 39264  AtLatcal 39265
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569  df-atl 39299
This theorem is referenced by:  atllat  39301  atl0dm  39303  atlex  39317
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