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Theorem nmmul 24024
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 2736 . . 3 (AbsVal‘𝑅) = (AbsVal‘𝑅)
31, 2nrgabv 24021 . 2 (𝑅 ∈ NrmRing → 𝑁 ∈ (AbsVal‘𝑅))
4 nmmul.x . . 3 𝑋 = (Base‘𝑅)
5 nmmul.t . . 3 · = (.r𝑅)
62, 4, 5abvmul 20284 . 2 ((𝑁 ∈ (AbsVal‘𝑅) ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁𝐴) · (𝑁𝐵)))
73, 6syl3an1 1163 1 ((𝑅 ∈ NrmRing ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁𝐴) · (𝑁𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  cfv 6494  (class class class)co 7354   · cmul 11053  Basecbs 17080  .rcmulr 17131  AbsValcabv 20271  normcnm 23928  NrmRingcnrg 23931
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-ov 7357  df-oprab 7358  df-mpo 7359  df-map 8764  df-abv 20272  df-nrg 23937
This theorem is referenced by:  nrgdsdi  24025  nrgdsdir  24026  nminvr  24029  nmdvr  24030  nrginvrcnlem  24051
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