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Theorem isatl 38169
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 𝐡 = (Baseβ€˜πΎ)
isatlat.g 𝐺 = (glbβ€˜πΎ)
isatlat.l ≀ = (leβ€˜πΎ)
isatlat.z 0 = (0.β€˜πΎ)
isatlat.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
isatl (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐡 ∈ dom 𝐺 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)))
Distinct variable groups:   𝑦,𝐴   π‘₯,𝐡   π‘₯,𝑦,𝐾
Allowed substitution hints:   𝐴(π‘₯)   𝐡(𝑦)   𝐺(π‘₯,𝑦)   ≀ (π‘₯,𝑦)   0 (π‘₯,𝑦)

Proof of Theorem isatl
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6892 . . . . . 6 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
2 isatlat.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
31, 2eqtr4di 2791 . . . . 5 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
4 fveq2 6892 . . . . . . 7 (π‘˜ = 𝐾 β†’ (glbβ€˜π‘˜) = (glbβ€˜πΎ))
5 isatlat.g . . . . . . 7 𝐺 = (glbβ€˜πΎ)
64, 5eqtr4di 2791 . . . . . 6 (π‘˜ = 𝐾 β†’ (glbβ€˜π‘˜) = 𝐺)
76dmeqd 5906 . . . . 5 (π‘˜ = 𝐾 β†’ dom (glbβ€˜π‘˜) = dom 𝐺)
83, 7eleq12d 2828 . . . 4 (π‘˜ = 𝐾 β†’ ((Baseβ€˜π‘˜) ∈ dom (glbβ€˜π‘˜) ↔ 𝐡 ∈ dom 𝐺))
9 fveq2 6892 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (0.β€˜π‘˜) = (0.β€˜πΎ))
10 isatlat.z . . . . . . . 8 0 = (0.β€˜πΎ)
119, 10eqtr4di 2791 . . . . . . 7 (π‘˜ = 𝐾 β†’ (0.β€˜π‘˜) = 0 )
1211neeq2d 3002 . . . . . 6 (π‘˜ = 𝐾 β†’ (π‘₯ β‰  (0.β€˜π‘˜) ↔ π‘₯ β‰  0 ))
13 fveq2 6892 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
14 isatlat.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
1513, 14eqtr4di 2791 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
16 fveq2 6892 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = (leβ€˜πΎ))
17 isatlat.l . . . . . . . . 9 ≀ = (leβ€˜πΎ)
1816, 17eqtr4di 2791 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = ≀ )
1918breqd 5160 . . . . . . 7 (π‘˜ = 𝐾 β†’ (𝑦(leβ€˜π‘˜)π‘₯ ↔ 𝑦 ≀ π‘₯))
2015, 19rexeqbidv 3344 . . . . . 6 (π‘˜ = 𝐾 β†’ (βˆƒπ‘¦ ∈ (Atomsβ€˜π‘˜)𝑦(leβ€˜π‘˜)π‘₯ ↔ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯))
2112, 20imbi12d 345 . . . . 5 (π‘˜ = 𝐾 β†’ ((π‘₯ β‰  (0.β€˜π‘˜) β†’ βˆƒπ‘¦ ∈ (Atomsβ€˜π‘˜)𝑦(leβ€˜π‘˜)π‘₯) ↔ (π‘₯ β‰  0 β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)))
223, 21raleqbidv 3343 . . . 4 (π‘˜ = 𝐾 β†’ (βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)(π‘₯ β‰  (0.β€˜π‘˜) β†’ βˆƒπ‘¦ ∈ (Atomsβ€˜π‘˜)𝑦(leβ€˜π‘˜)π‘₯) ↔ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)))
238, 22anbi12d 632 . . 3 (π‘˜ = 𝐾 β†’ (((Baseβ€˜π‘˜) ∈ dom (glbβ€˜π‘˜) ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)(π‘₯ β‰  (0.β€˜π‘˜) β†’ βˆƒπ‘¦ ∈ (Atomsβ€˜π‘˜)𝑦(leβ€˜π‘˜)π‘₯)) ↔ (𝐡 ∈ dom 𝐺 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯))))
24 df-atl 38168 . . 3 AtLat = {π‘˜ ∈ Lat ∣ ((Baseβ€˜π‘˜) ∈ dom (glbβ€˜π‘˜) ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)(π‘₯ β‰  (0.β€˜π‘˜) β†’ βˆƒπ‘¦ ∈ (Atomsβ€˜π‘˜)𝑦(leβ€˜π‘˜)π‘₯))}
2523, 24elrab2 3687 . 2 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ (𝐡 ∈ dom 𝐺 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯))))
26 3anass 1096 . 2 ((𝐾 ∈ Lat ∧ 𝐡 ∈ dom 𝐺 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)) ↔ (𝐾 ∈ Lat ∧ (𝐡 ∈ dom 𝐺 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯))))
2725, 26bitr4i 278 1 (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐡 ∈ dom 𝐺 ∧ βˆ€π‘₯ ∈ 𝐡 (π‘₯ β‰  0 β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071   class class class wbr 5149  dom cdm 5677  β€˜cfv 6544  Basecbs 17144  lecple 17204  glbcglb 18263  0.cp0 18376  Latclat 18384  Atomscatm 38133  AtLatcal 38134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-dm 5687  df-iota 6496  df-fv 6552  df-atl 38168
This theorem is referenced by:  atllat  38170  atl0dm  38172  atlex  38186
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