Theorem isatl 39299

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 6861	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2		isatlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2783	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4		fveq2 6861	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
5		isatlat.g	. . . . . . 7 ⊢ 𝐺 = (glb‘𝐾)
6	4, 5	eqtr4di 2783	. . . . . 6 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
7	6	dmeqd 5872	. . . . 5 ⊢ (𝑘 = 𝐾 → dom (glb‘𝑘) = dom 𝐺)
8	3, 7	eleq12d 2823	. . . 4 ⊢ (𝑘 = 𝐾 → ((Base‘𝑘) ∈ dom (glb‘𝑘) ↔ 𝐵 ∈ dom 𝐺))
9		fveq2 6861	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
10		isatlat.z	. . . . . . . 8 ⊢ 0 = (0.‘𝐾)
11	9, 10	eqtr4di 2783	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
12	11	neeq2d 2986	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥 ≠ (0.‘𝑘) ↔ 𝑥 ≠ 0 ))
13		fveq2 6861	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
14		isatlat.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
15	13, 14	eqtr4di 2783	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
16		fveq2 6861	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
17		isatlat.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
18	16, 17	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
19	18	breqd 5121	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑥 ↔ 𝑦 ≤ 𝑥))
20	15, 19	rexeqbidv 3322	. . . . . 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 3321	. . . 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 39298	. . 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 3665	. 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 2926  ∀wral 3045  ∃wrex 3054   class class class wbr 5110  dom cdm 5641  ‘cfv 6514  Basecbs 17186  lecple 17234  glbcglb 18278  0.cp0 18389  Latclat 18397  Atomscatm 39263  AtLatcal 39264

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-dm 5651  df-iota 6467  df-fv 6522  df-atl 39298

This theorem is referenced by:  atllat  39300  atl0dm  39302  atlex  39316