Theorem cycsubmel 19133

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 2829	. 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 6636	. . 3 ⊢ 𝐹 Fn ℕ0
6		fvelrnb 6895	. . 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 6942	. . . . 5 ⊢ (𝑖 ∈ ℕ0 → (𝐹‘𝑖) = (𝑖 · 𝐴))
11	10	eqeq1d 2739	. . . 4 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋))
12		eqcom 2744	. . . 4 ⊢ ((𝑖 · 𝐴) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴))
13	11, 12	bitrdi 287	. . 3 ⊢ (𝑖 ∈ ℕ0 → ((𝐹‘𝑖) = 𝑋 ↔ 𝑋 = (𝑖 · 𝐴)))
14	13	rexbiia 3082	. 2 ⊢ (∃𝑖 ∈ ℕ0 (𝐹‘𝑖) = 𝑋 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))
15	2, 7, 14	3bitri 297	1 ⊢ (𝑋 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ↦ cmpt 5180  ran crn 5626   Fn wfn 6488  ‘cfv 6493  (class class class)co 7360  ℕ₀cn0 12405  Basecbs 17140  ._gcmg 19001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-fv 6501  df-ov 7363

This theorem is referenced by:  cycsubmcl  19134  cycsubm  19135  cycsubmcom  19137