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Theorem fnmgp 18691
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 6841 . 2 (𝑥 sSet ⟨(+g‘ndx), (.r𝑥)⟩) ∈ V
2 df-mgp 18690 . 2 mulGrp = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(+g‘ndx), (.r𝑥)⟩))
31, 2fnmpti 6183 1 mulGrp Fn V
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3340  cop 4327   Fn wfn 6044  cfv 6049  (class class class)co 6813  ndxcnx 16056   sSet csts 16057  +gcplusg 16143  .rcmulr 16144  mulGrpcmgp 18689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-ov 6816  df-mgp 18690
This theorem is referenced by:  ringidval  18703  mgpf  18759  prdsmgp  18810  prdscrngd  18813  pws1  18816  pwsmgp  18818
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