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Theorem isatl 35109
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 6333 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isatlat.b . . . . . 6 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2823 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6333 . . . . . . 7 (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
5 isatlat.g . . . . . . 7 𝐺 = (glb‘𝐾)
64, 5syl6eqr 2823 . . . . . 6 (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
76dmeqd 5465 . . . . 5 (𝑘 = 𝐾 → dom (glb‘𝑘) = dom 𝐺)
83, 7eleq12d 2844 . . . 4 (𝑘 = 𝐾 → ((Base‘𝑘) ∈ dom (glb‘𝑘) ↔ 𝐵 ∈ dom 𝐺))
9 fveq2 6333 . . . . . . . 8 (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
10 isatlat.z . . . . . . . 8 0 = (0.‘𝐾)
119, 10syl6eqr 2823 . . . . . . 7 (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
1211neeq2d 3003 . . . . . 6 (𝑘 = 𝐾 → (𝑥 ≠ (0.‘𝑘) ↔ 𝑥0 ))
13 fveq2 6333 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
14 isatlat.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
1513, 14syl6eqr 2823 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
16 fveq2 6333 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
17 isatlat.l . . . . . . . . 9 = (le‘𝐾)
1816, 17syl6eqr 2823 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1918breqd 4798 . . . . . . 7 (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑥𝑦 𝑥))
2015, 19rexeqbidv 3302 . . . . . 6 (𝑘 = 𝐾 → (∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥 ↔ ∃𝑦𝐴 𝑦 𝑥))
2112, 20imbi12d 333 . . . . 5 (𝑘 = 𝐾 → ((𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
223, 21raleqbidv 3301 . . . 4 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
238, 22anbi12d 610 . . 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 35108 . . 3 AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥))}
2523, 24elrab2 3519 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))))
26 3anass 1080 . 2 ((𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)) ↔ (𝐾 ∈ Lat ∧ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))))
2725, 26bitr4i 267 1 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062   class class class wbr 4787  dom cdm 5250  cfv 6032  Basecbs 16065  lecple 16157  glbcglb 17152  0.cp0 17246  Latclat 17254  Atomscatm 35073  AtLatcal 35074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-dm 5260  df-iota 5995  df-fv 6040  df-atl 35108
This theorem is referenced by:  atllat  35110  atl0dm  35112  atlex  35126
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