Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cycsubmel Structured version   Visualization version   GIF version

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  ℕ0cn0 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
 Copyright terms: Public domain W3C validator