Theorem cycsubmel 19277

Description: Characterization of an element of the set of nonnegative integer powers of an element 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.)

Hypotheses

Ref	Expression
cycsubm.b	⊢ 𝐵 = (Base‘𝐺)
cycsubm.t	⊢ · = (.g‘𝐺)
cycsubm.f	⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))
cycsubm.c	⊢ 𝐶 = ran 𝐹

Assertion

Ref	Expression
cycsubmel	⊢ (𝑋 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑖,𝐹   𝑖,𝑋   𝑥,𝑖   𝑥, ·

Allowed substitution hints:   𝐴(𝑖)   𝐵(𝑥, 𝑖)   𝐶(𝑥, 𝑖)   · (𝑖)   𝐹(𝑥)   𝐺(𝑥, 𝑖)   𝑋(𝑥)

Proof of Theorem cycsubmel

Step	Hyp	Ref	Expression
1		cycsubm.c	. . 3 ⊢ 𝐶 = ran 𝐹
2	1	eleq2i 2854	. 2 ⊢ (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ ran 𝐹)
3		ovex 7445	. . . 4 ⊢ (𝑥 · 𝐴) ∈ V
4		cycsubm.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))
5	3, 4	fnmpti 6678	. . 3 ⊢ 𝐹 Fn ℕ0
6		fvelrnb 6941	. . 3 ⊢ (𝐹 Fn ℕ0 → (𝑋 ∈ ran 𝐹 ↔ ∃𝑖 ∈ ℕ0 (𝐹‘𝑖) = 𝑋))
7	5, 6	ax-mp 5	. 2 ⊢ (𝑋 ∈ ran 𝐹 ↔ ∃𝑖 ∈ ℕ0 (𝐹‘𝑖) = 𝑋)
8		oveq1 7419	. . . . . 6 ⊢ (𝑥 = 𝑖 → (𝑥 · 𝐴) = (𝑖 · 𝐴))
9		ovex 7445	. . . . . 6 ⊢ (𝑖 · 𝐴) ∈ V
10	8, 4, 9	fvmpt 6989	. . . . 5 ⊢ (𝑖 ∈ ℕ0 → (𝐹‘𝑖) = (𝑖 · 𝐴))
11	10	eqeq1d 2764	. . . 4 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋))
12		eqcom 2769	. . . 4 ⊢ ((𝑖 · 𝐴) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴))
13	11, 12	bitrdi 290	. . 3 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴)))
14	13	rexbiia 3109	. 2 ⊢ (∃𝑖 ∈ ℕ0 (𝐹‘𝑖) = 𝑋 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))
15	2, 7, 14	3bitri 300	1 ⊢ (𝑋 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2142  ∃wrex 3088   ↦ cmpt 5191  ran crn 5661   Fn wfn 6531  ‘cfv 6536  (class class class)co 7412  ℕ₀cn0 12510  Basecbs 17275  ._gcmg 19139

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-ov 7415

This theorem is used by:  cycsubmcl  19278  cycsubm  19279  cycsubmcom  19281