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Theorem cycsubmel 18473
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 · = (.g𝐺)
cycsubm.f 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))
cycsubm.c 𝐶 = ran 𝐹
Assertion
Ref Expression
cycsubmel (𝑋𝐶 ↔ ∃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
StepHypRef Expression
1 cycsubm.c . . 3 𝐶 = ran 𝐹
21eleq2i 2825 . 2 (𝑋𝐶𝑋 ∈ ran 𝐹)
3 ovex 7215 . . . 4 (𝑥 · 𝐴) ∈ V
4 cycsubm.f . . . 4 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))
53, 4fnmpti 6490 . . 3 𝐹 Fn ℕ0
6 fvelrnb 6742 . . 3 (𝐹 Fn ℕ0 → (𝑋 ∈ ran 𝐹 ↔ ∃𝑖 ∈ ℕ0 (𝐹𝑖) = 𝑋))
75, 6ax-mp 5 . 2 (𝑋 ∈ ran 𝐹 ↔ ∃𝑖 ∈ ℕ0 (𝐹𝑖) = 𝑋)
8 oveq1 7189 . . . . . 6 (𝑥 = 𝑖 → (𝑥 · 𝐴) = (𝑖 · 𝐴))
9 ovex 7215 . . . . . 6 (𝑖 · 𝐴) ∈ V
108, 4, 9fvmpt 6787 . . . . 5 (𝑖 ∈ ℕ0 → (𝐹𝑖) = (𝑖 · 𝐴))
1110eqeq1d 2741 . . . 4 (𝑖 ∈ ℕ0 → ((𝐹𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋))
12 eqcom 2746 . . . 4 ((𝑖 · 𝐴) = 𝑋𝑋 = (𝑖 · 𝐴))
1311, 12bitrdi 290 . . 3 (𝑖 ∈ ℕ0 → ((𝐹𝑖) = 𝑋𝑋 = (𝑖 · 𝐴)))
1413rexbiia 3161 . 2 (∃𝑖 ∈ ℕ0 (𝐹𝑖) = 𝑋 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))
152, 7, 143bitri 300 1 (𝑋𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1542  wcel 2114  wrex 3055  cmpt 5120  ran crn 5536   Fn wfn 6344  cfv 6349  (class class class)co 7182  0cn0 11988  Basecbs 16598  .gcmg 18354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-iota 6307  df-fun 6351  df-fn 6352  df-fv 6357  df-ov 7185
This theorem is referenced by:  cycsubmcl  18474  cycsubm  18475  cycsubmcom  18477
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