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Theorem cycsubmel 18332
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 2907 . 2 (𝑋𝐶𝑋 ∈ ran 𝐹)
3 ovex 7171 . . . 4 (𝑥 · 𝐴) ∈ V
4 cycsubm.f . . . 4 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))
53, 4fnmpti 6472 . . 3 𝐹 Fn ℕ0
6 fvelrnb 6707 . . 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 7145 . . . . . 6 (𝑥 = 𝑖 → (𝑥 · 𝐴) = (𝑖 · 𝐴))
9 ovex 7171 . . . . . 6 (𝑖 · 𝐴) ∈ V
108, 4, 9fvmpt 6749 . . . . 5 (𝑖 ∈ ℕ0 → (𝐹𝑖) = (𝑖 · 𝐴))
1110eqeq1d 2826 . . . 4 (𝑖 ∈ ℕ0 → ((𝐹𝑖) = 𝑋 ↔ (𝑖 · 𝐴) = 𝑋))
12 eqcom 2831 . . . 4 ((𝑖 · 𝐴) = 𝑋𝑋 = (𝑖 · 𝐴))
1311, 12syl6bb 290 . . 3 (𝑖 ∈ ℕ0 → ((𝐹𝑖) = 𝑋𝑋 = (𝑖 · 𝐴)))
1413rexbiia 3240 . 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 1538  wcel 2115  wrex 3133  cmpt 5127  ran crn 5537   Fn wfn 6331  cfv 6336  (class class class)co 7138  0cn0 11883  Basecbs 16472  .gcmg 18213
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344  df-ov 7141
This theorem is referenced by:  cycsubmcl  18333  cycsubm  18334  cycsubmcom  18336
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