Theorem isatl 40106

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 6885	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2		isatlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2818	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4		fveq2 6885	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
5		isatlat.g	. . . . . . 7 ⊢ 𝐺 = (glb‘𝐾)
6	4, 5	eqtr4di 2818	. . . . . 6 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
7	6	dmeqd 5897	. . . . 5 ⊢ (𝑘 = 𝐾 → dom (glb‘𝑘) = dom 𝐺)
8	3, 7	eleq12d 2859	. . . 4 ⊢ (𝑘 = 𝐾 → ((Base‘𝑘) ∈ dom (glb‘𝑘) ↔ 𝐵 ∈ dom 𝐺))
9		fveq2 6885	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
10		isatlat.z	. . . . . . . 8 ⊢ 0 = (0.‘𝐾)
11	9, 10	eqtr4di 2818	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
12	11	neeq2d 3020	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥 ≠ (0.‘𝑘) ↔ 𝑥 ≠ 0 ))
13		fveq2 6885	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
14		isatlat.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
15	13, 14	eqtr4di 2818	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
16		fveq2 6885	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
17		isatlat.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
18	16, 17	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
19	18	breqd 5122	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑥 ↔ 𝑦 ≤ 𝑥))
20	15, 19	rexeqbidv 3341	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
21	12, 20	imbi12d 347	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))
22	3, 21	raleqbidv 3340	. . . 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 644	. . 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 40105	. . 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 3656	. 2 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ (𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))))
26		3anass 1111	. 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 281	1 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   class class class wbr 5111  dom cdm 5663  ‘cfv 6540  Basecbs 17287  lecple 17335  glbcglb 18384  0.cp0 18495  Latclat 18505  Atomscatm 40070  AtLatcal 40071

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-dm 5673  df-iota 6496  df-fv 6548  df-atl 40105

This theorem is used by:  atllat  40107  atl0dm  40109  atlex  40123