Theorem exfinfldd 42316

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 7360	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑃↑𝑛) = (𝑃↑𝑁))
2	1	eqeq2d 2744	. . . 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 3157	. 2 ⊢ (𝑛 = 𝑁 → (∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃) ↔ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃)))
5		oveq1 7359	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑝↑𝑛) = (𝑃↑𝑛))
6	5	eqeq2d 2744	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ↔ (♯‘(Base‘𝑘)) = (𝑃↑𝑛)))
7		eqeq2 2745	. . . . . 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 3157	. . . 4 ⊢ (𝑝 = 𝑃 → (∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝) ↔ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑛) ∧ (chr‘𝑘) = 𝑃)))
10	9	ralbidv 3156	. . 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 42315	. . . 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 3574	. 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 3574	1 ⊢ (𝜑 → ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  ‘cfv 6486  (class class class)co 7352  ℕcn 12132  ↑cexp 13970  ♯chash 14239  ℙcprime 16584  Basecbs 17122  Fieldcfield 20647  chrcchr 21440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-exfinfld 42315

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-ov 7355

This theorem is referenced by:  aks5  42317