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Theorem isatl 38038
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 6879 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isatlat.b . . . . . 6 𝐵 = (Base‘𝐾)
31, 2eqtr4di 2790 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6879 . . . . . . 7 (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
5 isatlat.g . . . . . . 7 𝐺 = (glb‘𝐾)
64, 5eqtr4di 2790 . . . . . 6 (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
76dmeqd 5898 . . . . 5 (𝑘 = 𝐾 → dom (glb‘𝑘) = dom 𝐺)
83, 7eleq12d 2827 . . . 4 (𝑘 = 𝐾 → ((Base‘𝑘) ∈ dom (glb‘𝑘) ↔ 𝐵 ∈ dom 𝐺))
9 fveq2 6879 . . . . . . . 8 (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
10 isatlat.z . . . . . . . 8 0 = (0.‘𝐾)
119, 10eqtr4di 2790 . . . . . . 7 (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
1211neeq2d 3001 . . . . . 6 (𝑘 = 𝐾 → (𝑥 ≠ (0.‘𝑘) ↔ 𝑥0 ))
13 fveq2 6879 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
14 isatlat.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
1513, 14eqtr4di 2790 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
16 fveq2 6879 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
17 isatlat.l . . . . . . . . 9 = (le‘𝐾)
1816, 17eqtr4di 2790 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1918breqd 5153 . . . . . . 7 (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑥𝑦 𝑥))
2015, 19rexeqbidv 3343 . . . . . 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 3342 . . . 4 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
238, 22anbi12d 631 . . 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 38037 . . 3 AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥))}
2523, 24elrab2 3683 . 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 277 1 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wral 3061  wrex 3070   class class class wbr 5142  dom cdm 5670  cfv 6533  Basecbs 17128  lecple 17188  glbcglb 18247  0.cp0 18360  Latclat 18368  Atomscatm 38002  AtLatcal 38003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-dm 5680  df-iota 6485  df-fv 6541  df-atl 38037
This theorem is referenced by:  atllat  38039  atl0dm  38041  atlex  38055
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