Theorem exfinfldd 42688

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 7364	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑃↑𝑛) = (𝑃↑𝑁))
2	1	eqeq2d 2750	. . . 4 ⊢ (𝑛 = 𝑁 → ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ↔ (♯‘(Base‘𝑘)) = (𝑃↑𝑁)))
3	2	anbi1d 637	. . 3 ⊢ (𝑛 = 𝑁 → (((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃) ↔ ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃)))
4	3	rexbidv 3163	. 2 ⊢ (𝑛 = 𝑁 → (∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃) ↔ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃)))
5		oveq1 7363	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝↑𝑛) = (𝑃↑𝑛))
6	5	eqeq2d 2750	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ↔ (♯‘(Base‘𝑘)) = (𝑃↑𝑛)))
7		eqeq2 2751	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((chr‘𝑘) = 𝑝 ↔ (chr‘𝑘) = 𝑃))
8	6, 7	anbi12d 638	. . . . 5 ⊢ (𝑝 = 𝑃 → (((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝) ↔ ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃)))
9	8	rexbidv 3163	. . . 4 ⊢ (𝑝 = 𝑃 → (∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝) ↔ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃)))
10	9	ralbidv 3162	. . 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 42687	. . . 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 3561	. 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 3561	1 ⊢ (𝜑 → ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  ‘cfv 6485  (class class class)co 7356  ℕcn 12165  ↑cexp 14014  ♯chash 14283  ℙcprime 16631  Basecbs 17170  Fieldcfield 20702  chrcchr 21476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-exfinfld 42687

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359

This theorem is referenced by:  aks5  42689