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Theorem isatl 34063
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 6148 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isatlat.b . . . . . 6 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2673 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6148 . . . . . . 7 (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
5 isatlat.g . . . . . . 7 𝐺 = (glb‘𝐾)
64, 5syl6eqr 2673 . . . . . 6 (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
76dmeqd 5286 . . . . 5 (𝑘 = 𝐾 → dom (glb‘𝑘) = dom 𝐺)
83, 7eleq12d 2692 . . . 4 (𝑘 = 𝐾 → ((Base‘𝑘) ∈ dom (glb‘𝑘) ↔ 𝐵 ∈ dom 𝐺))
9 fveq2 6148 . . . . . . . 8 (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
10 isatlat.z . . . . . . . 8 0 = (0.‘𝐾)
119, 10syl6eqr 2673 . . . . . . 7 (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
1211neeq2d 2850 . . . . . 6 (𝑘 = 𝐾 → (𝑥 ≠ (0.‘𝑘) ↔ 𝑥0 ))
13 fveq2 6148 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
14 isatlat.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
1513, 14syl6eqr 2673 . . . . . . 7 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
16 fveq2 6148 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
17 isatlat.l . . . . . . . . 9 = (le‘𝐾)
1816, 17syl6eqr 2673 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = )
1918breqd 4624 . . . . . . 7 (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑥𝑦 𝑥))
2015, 19rexeqbidv 3142 . . . . . 6 (𝑘 = 𝐾 → (∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥 ↔ ∃𝑦𝐴 𝑦 𝑥))
2112, 20imbi12d 334 . . . . 5 (𝑘 = 𝐾 → ((𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
223, 21raleqbidv 3141 . . . 4 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥)))
238, 22anbi12d 746 . . 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 34062 . . 3 AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥))}
2523, 24elrab2 3348 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥𝐵 (𝑥0 → ∃𝑦𝐴 𝑦 𝑥))))
26 3anass 1040 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908   class class class wbr 4613  dom cdm 5074  cfv 5847  Basecbs 15781  lecple 15869  glbcglb 16864  0.cp0 16958  Latclat 16966  Atomscatm 34027  AtLatcal 34028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-dm 5084  df-iota 5810  df-fv 5855  df-atl 34062
This theorem is referenced by:  atllat  34064  atl0dm  34066  atlex  34080
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