Theorem isatl 40622

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 6916	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2		isatlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2816	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4		fveq2 6916	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
5		isatlat.g	. . . . . . 7 ⊢ 𝐺 = (glb‘𝐾)
6	4, 5	eqtr4di 2816	. . . . . 6 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
7	6	dmeqd 5899	. . . . 5 ⊢ (𝑘 = 𝐾 → dom (glb‘𝑘) = dom 𝐺)
8	3, 7	eleq12d 2858	. . . 4 ⊢ (𝑘 = 𝐾 → ((Base‘𝑘) ∈ dom (glb‘𝑘) ↔ 𝐵 ∈ dom 𝐺))
9		fveq2 6916	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
10		isatlat.z	. . . . . . . 8 ⊢ 0 = (0.‘𝐾)
11	9, 10	eqtr4di 2816	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
12	11	neeq2d 3019	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥 ≠ (0.‘𝑘) ↔ 𝑥 ≠ 0 ))
13		fveq2 6916	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
14		isatlat.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
15	13, 14	eqtr4di 2816	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
16		fveq2 6916	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
17		isatlat.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
18	16, 17	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
19	18	breqd 5119	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑥 ↔ 𝑦 ≤ 𝑥))
20	15, 19	rexeqbidv 3339	. . . . . 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 3338	. . . 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 40621	. . 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 3652	. 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 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   class class class wbr 5108  dom cdm 5658  ‘cfv 6566  Basecbs 17431  lecple 17479  glbcglb 18534  0.cp0 18645  Latclat 18655  Atomscatm 40586  AtLatcal 40587

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-dm 5668  df-iota 6520  df-fv 6574  df-atl 40621

This theorem is used by:  atllat  40623  atl0dm  40625  atlex  40639