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Theorem nlmnrg 18707
 Description: The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypothesis
Ref Expression
nlmnrg.1 Scalar
Assertion
Ref Expression
nlmnrg NrmMod NrmRing

Proof of Theorem nlmnrg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . 4
2 eqid 2435 . . . 4
3 eqid 2435 . . . 4
4 nlmnrg.1 . . . 4 Scalar
5 eqid 2435 . . . 4
6 eqid 2435 . . . 4
71, 2, 3, 4, 5, 6isnlm 18703 . . 3 NrmMod NrmGrp NrmRing
87simplbi 447 . 2 NrmMod NrmGrp NrmRing
98simp3d 971 1 NrmMod NrmRing
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 936   wceq 1652   wcel 1725  wral 2697  cfv 5446  (class class class)co 6073   cmul 8987  cbs 13461  Scalarcsca 13524  cvsca 13525  clmod 15942  cnm 18616  NrmGrpcngp 18617  NrmRingcnrg 18619  NrmModcnlm 18620 This theorem is referenced by:  nlmngp2  18708  nlmtlm  18721  nvctvc  18727  lssnlm  18728  sitgclbn  24649 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-nlm 18626
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