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Theorem cycsubmel 19218
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 2833 . 2 (𝑋𝐶𝑋 ∈ ran 𝐹)
3 ovex 7464 . . . 4 (𝑥 · 𝐴) ∈ V
4 cycsubm.f . . . 4 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))
53, 4fnmpti 6711 . . 3 𝐹 Fn ℕ0
6 fvelrnb 6969 . . 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 7438 . . . . . 6 (𝑥 = 𝑖 → (𝑥 · 𝐴) = (𝑖 · 𝐴))
9 ovex 7464 . . . . . 6 (𝑖 · 𝐴) ∈ V
108, 4, 9fvmpt 7016 . . . . 5 (𝑖 ∈ ℕ0 → (𝐹𝑖) = (𝑖 · 𝐴))
1110eqeq1d 2739 . . . 4 (𝑖 ∈ ℕ0 → ((𝐹𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋))
12 eqcom 2744 . . . 4 ((𝑖 · 𝐴) = 𝑋𝑋 = (𝑖 · 𝐴))
1311, 12bitrdi 287 . . 3 (𝑖 ∈ ℕ0 → ((𝐹𝑖) = 𝑋𝑋 = (𝑖 · 𝐴)))
1413rexbiia 3092 . 2 (∃𝑖 ∈ ℕ0 (𝐹𝑖) = 𝑋 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))
152, 7, 143bitri 297 1 (𝑋𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  wrex 3070  cmpt 5225  ran crn 5686   Fn wfn 6556  cfv 6561  (class class class)co 7431  0cn0 12526  Basecbs 17247  .gcmg 19085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434
This theorem is referenced by:  cycsubmcl  19219  cycsubm  19220  cycsubmcom  19222
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