Theorem isatl 39297

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 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		fveq2 6826	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2		isatlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2782	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4		fveq2 6826	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
5		isatlat.g	. . . . . . 7 ⊢ 𝐺 = (glb‘𝐾)
6	4, 5	eqtr4di 2782	. . . . . 6 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
7	6	dmeqd 5852	. . . . 5 ⊢ (𝑘 = 𝐾 → dom (glb‘𝑘) = dom 𝐺)
8	3, 7	eleq12d 2822	. . . 4 ⊢ (𝑘 = 𝐾 → ((Base‘𝑘) ∈ dom (glb‘𝑘) ↔ 𝐵 ∈ dom 𝐺))
9		fveq2 6826	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
10		isatlat.z	. . . . . . . 8 ⊢ 0 = (0.‘𝐾)
11	9, 10	eqtr4di 2782	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
12	11	neeq2d 2985	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥 ≠ (0.‘𝑘) ↔ 𝑥 ≠ 0 ))
13		fveq2 6826	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
14		isatlat.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
15	13, 14	eqtr4di 2782	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
16		fveq2 6826	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
17		isatlat.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
18	16, 17	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
19	18	breqd 5106	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑥 ↔ 𝑦 ≤ 𝑥))
20	15, 19	rexeqbidv 3311	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
21	12, 20	imbi12d 344	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))
22	3, 21	raleqbidv 3310	. . . 4 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))
23	8, 22	anbi12d 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 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))))
24		df-atl 39296	. . 3 ⊢ AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥))}
25	23, 24	elrab2 3653	. 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))))
26		3anass 1094	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) ↔ (𝐾 ∈ Lat ∧ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))))
27	25, 26	bitr4i 278	1 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   class class class wbr 5095  dom cdm 5623  ‘cfv 6486  Basecbs 17139  lecple 17187  glbcglb 18235  0.cp0 18346  Latclat 18356  Atomscatm 39261  AtLatcal 39262

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-dm 5633  df-iota 6442  df-fv 6494  df-atl 39296

This theorem is referenced by:  atllat  39298  atl0dm  39300  atlex  39314