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Theorem cycsubmel 18819
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 2830 . 2 (𝑋𝐶𝑋 ∈ ran 𝐹)
3 ovex 7308 . . . 4 (𝑥 · 𝐴) ∈ V
4 cycsubm.f . . . 4 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))
53, 4fnmpti 6576 . . 3 𝐹 Fn ℕ0
6 fvelrnb 6830 . . 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 7282 . . . . . 6 (𝑥 = 𝑖 → (𝑥 · 𝐴) = (𝑖 · 𝐴))
9 ovex 7308 . . . . . 6 (𝑖 · 𝐴) ∈ V
108, 4, 9fvmpt 6875 . . . . 5 (𝑖 ∈ ℕ0 → (𝐹𝑖) = (𝑖 · 𝐴))
1110eqeq1d 2740 . . . 4 (𝑖 ∈ ℕ0 → ((𝐹𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋))
12 eqcom 2745 . . . 4 ((𝑖 · 𝐴) = 𝑋𝑋 = (𝑖 · 𝐴))
1311, 12bitrdi 287 . . 3 (𝑖 ∈ ℕ0 → ((𝐹𝑖) = 𝑋𝑋 = (𝑖 · 𝐴)))
1413rexbiia 3180 . 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 205   = wceq 1539  wcel 2106  wrex 3065  cmpt 5157  ran crn 5590   Fn wfn 6428  cfv 6433  (class class class)co 7275  0cn0 12233  Basecbs 16912  .gcmg 18700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-ov 7278
This theorem is referenced by:  cycsubmcl  18820  cycsubm  18821  cycsubmcom  18823
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