Theorem isatl 39884

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 6862	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2		isatlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2814	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4		fveq2 6862	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = (glb‘𝐾))
5		isatlat.g	. . . . . . 7 ⊢ 𝐺 = (glb‘𝐾)
6	4, 5	eqtr4di 2814	. . . . . 6 ⊢ (𝑘 = 𝐾 → (glb‘𝑘) = 𝐺)
7	6	dmeqd 5877	. . . . 5 ⊢ (𝑘 = 𝐾 → dom (glb‘𝑘) = dom 𝐺)
8	3, 7	eleq12d 2855	. . . 4 ⊢ (𝑘 = 𝐾 → ((Base‘𝑘) ∈ dom (glb‘𝑘) ↔ 𝐵 ∈ dom 𝐺))
9		fveq2 6862	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = (0.‘𝐾))
10		isatlat.z	. . . . . . . 8 ⊢ 0 = (0.‘𝐾)
11	9, 10	eqtr4di 2814	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (0.‘𝑘) = 0 )
12	11	neeq2d 3016	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥 ≠ (0.‘𝑘) ↔ 𝑥 ≠ 0 ))
13		fveq2 6862	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
14		isatlat.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
15	13, 14	eqtr4di 2814	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
16		fveq2 6862	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
17		isatlat.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
18	16, 17	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
19	18	breqd 5108	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑦(le‘𝑘)𝑥 ↔ 𝑦 ≤ 𝑥))
20	15, 19	rexeqbidv 3336	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥))
21	12, 20	imbi12d 346	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑥 ≠ (0.‘𝑘) → ∃𝑦 ∈ (Atoms‘𝑘)𝑦(le‘𝑘)𝑥) ↔ (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))
22	3, 21	raleqbidv 3335	. . . 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 641	. . 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 39883	. . 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 1105	. 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 280	1 ⊢ (𝐾 ∈ AtLat ↔ (𝐾 ∈ Lat ∧ 𝐵 ∈ dom 𝐺 ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≠ 0 → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃wrex 3085   class class class wbr 5097  dom cdm 5643  ‘cfv 6516  Basecbs 17236  lecple 17284  glbcglb 18333  0.cp0 18444  Latclat 18454  Atomscatm 39848  AtLatcal 39849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-dm 5653  df-iota 6472  df-fv 6524  df-atl 39883

This theorem is referenced by:  atllat  39885  atl0dm  39887  atlex  39901