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Theorem nmmul 18191
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 2296 . . 3  |-  (AbsVal `  R )  =  (AbsVal `  R )
31, 2nrgabv 18188 . 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 15610 . 2  |-  ( ( N  e.  (AbsVal `  R )  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A  .x.  B ) )  =  ( ( N `  A )  x.  ( N `  B )
) )
73, 6syl3an1 1215 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 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874    x. cmul 8758   Basecbs 13164   .rcmulr 13225  AbsValcabv 15597   normcnm 18115  NrmRingcnrg 18118
This theorem is referenced by:  nrgdsdi  18192  nrgdsdir  18193  nminvr  18196  nmdvr  18197  nrginvrcnlem  18217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-abv 15598  df-nrg 18124
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