Theorem exfinfldd 43519

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 7459	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑃↑𝑛) = (𝑃↑𝑁))
2	1	eqeq2d 2774	. . . 4 ⊢ (𝑛 = 𝑁 → ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ↔ (♯‘(Base‘𝑘)) = (𝑃↑𝑁)))
3	2	anbi1d 643	. . 3 ⊢ (𝑛 = 𝑁 → (((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃) ↔ ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃)))
4	3	rexbidv 3190	. 2 ⊢ (𝑛 = 𝑁 → (∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃) ↔ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃)))
5		oveq1 7458	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝↑𝑛) = (𝑃↑𝑛))
6	5	eqeq2d 2774	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ↔ (♯‘(Base‘𝑘)) = (𝑃↑𝑛)))
7		eqeq2 2775	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((chr‘𝑘) = 𝑝 ↔ (chr‘𝑘) = 𝑃))
8	6, 7	anbi12d 644	. . . . 5 ⊢ (𝑝 = 𝑃 → (((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝) ↔ ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃)))
9	8	rexbidv 3190	. . . 4 ⊢ (𝑝 = 𝑃 → (∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝) ↔ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃)))
10	9	ralbidv 3189	. . 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 43518	. . . 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 3581	. 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 3581	1 ⊢ (𝜑 → ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ‘cfv 6566  (class class class)co 7451  ℕcn 12375  ↑cexp 14244  ♯chash 14515  ℙcprime 16890  Basecbs 17431  Fieldcfield 21032  chrcchr 21862

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  ax-exfinfld 43518

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-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-iota 6520  df-fv 6574  df-ov 7454

This theorem is used by:  aks5  43520