Theorem cycsubmel 19170

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 2833	. 2 ⊢ (𝑋 ∈ 𝐶 ↔ 𝑋 ∈ ran 𝐹)
3		ovex 7393	. . . 4 ⊢ (𝑥 · 𝐴) ∈ V
4		cycsubm.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))
5	3, 4	fnmpti 6632	. . 3 ⊢ 𝐹 Fn ℕ0
6		fvelrnb 6891	. . 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 7367	. . . . . 6 ⊢ (𝑥 = 𝑖 → (𝑥 · 𝐴) = (𝑖 · 𝐴))
9		ovex 7393	. . . . . 6 ⊢ (𝑖 · 𝐴) ∈ V
10	8, 4, 9	fvmpt 6939	. . . . 5 ⊢ (𝑖 ∈ ℕ0 → (𝐹‘𝑖) = (𝑖 · 𝐴))
11	10	eqeq1d 2743	. . . 4 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋))
12		eqcom 2748	. . . 4 ⊢ ((𝑖 · 𝐴) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴))
13	11, 12	bitrdi 289	. . 3 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴)))
14	13	rexbiia 3086	. 2 ⊢ (∃𝑖 ∈ ℕ0 (𝐹‘𝑖) = 𝑋 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))
15	2, 7, 14	3bitri 299	1 ⊢ (𝑋 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))

Syntax hints:   ↔ wb 208   = wceq 1548   ∈ wcel 2121  ∃wrex 3065   ↦ cmpt 5156  ran crn 5622   Fn wfn 6484  ‘cfv 6489  (class class class)co 7360  ℕ₀cn0 12432  Basecbs 17174  ._gcmg 19038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497  df-ov 7363

This theorem is referenced by:  cycsubmcl  19171  cycsubm  19172  cycsubmcom  19174