Theorem exfinfldd 42183

Description: For any prime 𝑃 and any positive integer 𝑁 there exists a field 𝑘 such that 𝑘 contains 𝑃↑𝑁 elements. (Contributed by metakunt, 13-Jul-2025.)

Hypotheses

Ref	Expression
exfinfldd.1	⊢ (𝜑 → 𝑃 ∈ ℙ)
exfinfldd.2	⊢ (𝜑 → 𝑁 ∈ ℕ)

Assertion

Ref	Expression
exfinfldd	⊢ (𝜑 → ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃))

Distinct variable groups:   𝑘,𝑁   𝑃,𝑘

Allowed substitution hint:   𝜑(𝑘)

Proof of Theorem exfinfldd

Dummy variables 𝑛 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7402	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑃↑𝑛) = (𝑃↑𝑁))
2	1	eqeq2d 2741	. . . 4 ⊢ (𝑛 = 𝑁 → ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ↔ (♯‘(Base‘𝑘)) = (𝑃↑𝑁)))
3	2	anbi1d 631	. . 3 ⊢ (𝑛 = 𝑁 → (((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃) ↔ ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃)))
4	3	rexbidv 3159	. 2 ⊢ (𝑛 = 𝑁 → (∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃) ↔ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃)))
5		oveq1 7401	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝↑𝑛) = (𝑃↑𝑛))
6	5	eqeq2d 2741	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ↔ (♯‘(Base‘𝑘)) = (𝑃↑𝑛)))
7		eqeq2 2742	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((chr‘𝑘) = 𝑝 ↔ (chr‘𝑘) = 𝑃))
8	6, 7	anbi12d 632	. . . . 5 ⊢ (𝑝 = 𝑃 → (((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝) ↔ ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃)))
9	8	rexbidv 3159	. . . 4 ⊢ (𝑝 = 𝑃 → (∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝) ↔ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃)))
10	9	ralbidv 3158	. . 3 ⊢ (𝑝 = 𝑃 → (∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝) ↔ ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃)))
11		ax-exfinfld 42182	. . . 4 ⊢ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝)
12	11	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝))
13		exfinfldd.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
14	10, 12, 13	rspcdva 3598	. 2 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃))
15		exfinfldd.2	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
16	4, 14, 15	rspcdva 3598	1 ⊢ (𝜑 → ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3046  ∃wrex 3055  ‘cfv 6519  (class class class)co 7394  ℕcn 12197  ↑cexp 14036  ♯chash 14305  ℙcprime 16647  Basecbs 17185  Fieldcfield 20645  chrcchr 21417

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  ax-exfinfld 42182

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-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-iota 6472  df-fv 6527  df-ov 7397

This theorem is referenced by:  aks5  42184