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Theorem fnmgp 19232
Description: The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
Assertion
Ref Expression
fnmgp mulGrp Fn V

Proof of Theorem fnmgp
StepHypRef Expression
1 ovex 7173 . 2 (𝑥 sSet ⟨(+g‘ndx), (.r𝑥)⟩) ∈ V
2 df-mgp 19231 . 2 mulGrp = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(+g‘ndx), (.r𝑥)⟩))
31, 2fnmpti 6471 1 mulGrp Fn V
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3469  cop 4545   Fn wfn 6329  cfv 6334  (class class class)co 7140  ndxcnx 16471   sSet csts 16472  +gcplusg 16556  .rcmulr 16557  mulGrpcmgp 19230
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342  df-ov 7143  df-mgp 19231
This theorem is referenced by:  ringidval  19244  mgpf  19303  prdsmgp  19354  prdscrngd  19357  pws1  19360  pwsmgp  19362
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