MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmmul Structured version   Unicode version

Theorem nmmul 18692
Description: The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
nmmul.x  |-  X  =  ( Base `  R
)
nmmul.n  |-  N  =  ( norm `  R
)
nmmul.t  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
nmmul  |-  ( ( R  e. NrmRing  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A  .x.  B ) )  =  ( ( N `  A )  x.  ( N `  B )
) )

Proof of Theorem nmmul
StepHypRef Expression
1 nmmul.n . . 3  |-  N  =  ( norm `  R
)
2 eqid 2435 . . 3  |-  (AbsVal `  R )  =  (AbsVal `  R )
31, 2nrgabv 18689 . 2  |-  ( R  e. NrmRing  ->  N  e.  (AbsVal `  R ) )
4 nmmul.x . . 3  |-  X  =  ( Base `  R
)
5 nmmul.t . . 3  |-  .x.  =  ( .r `  R )
62, 4, 5abvmul 15909 . 2  |-  ( ( N  e.  (AbsVal `  R )  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A  .x.  B ) )  =  ( ( N `  A )  x.  ( N `  B )
) )
73, 6syl3an1 1217 1  |-  ( ( R  e. NrmRing  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A  .x.  B ) )  =  ( ( N `  A )  x.  ( N `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073    x. cmul 8987   Basecbs 13461   .rcmulr 13522  AbsValcabv 15896   normcnm 18616  NrmRingcnrg 18619
This theorem is referenced by:  nrgdsdi  18693  nrgdsdir  18694  nminvr  18697  nmdvr  18698  nrginvrcnlem  18718
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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-mo 2285  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-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-abv 15897  df-nrg 18625
  Copyright terms: Public domain W3C validator