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Theorem nmmul 23273
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 · = (.r𝑅)
Assertion
Ref Expression
nmmul ((𝑅 ∈ NrmRing ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁𝐴) · (𝑁𝐵)))

Proof of Theorem nmmul
StepHypRef Expression
1 nmmul.n . . 3 𝑁 = (norm‘𝑅)
2 eqid 2824 . . 3 (AbsVal‘𝑅) = (AbsVal‘𝑅)
31, 2nrgabv 23270 . 2 (𝑅 ∈ NrmRing → 𝑁 ∈ (AbsVal‘𝑅))
4 nmmul.x . . 3 𝑋 = (Base‘𝑅)
5 nmmul.t . . 3 · = (.r𝑅)
62, 4, 5abvmul 19600 . 2 ((𝑁 ∈ (AbsVal‘𝑅) ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁𝐴) · (𝑁𝐵)))
73, 6syl3an1 1160 1 ((𝑅 ∈ NrmRing ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁𝐴) · (𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  cfv 6343  (class class class)co 7149   · cmul 10540  Basecbs 16483  .rcmulr 16566  AbsValcabv 19587  normcnm 23186  NrmRingcnrg 23189
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-abv 19588  df-nrg 23195
This theorem is referenced by:  nrgdsdi  23274  nrgdsdir  23275  nminvr  23278  nmdvr  23279  nrginvrcnlem  23300
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