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Theorem nmmul 23934
Description: The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
nmmul.x 𝑋 = (Baseβ€˜π‘…)
nmmul.n 𝑁 = (normβ€˜π‘…)
nmmul.t Β· = (.rβ€˜π‘…)
Assertion
Ref Expression
nmmul ((𝑅 ∈ NrmRing ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴 Β· 𝐡)) = ((π‘β€˜π΄) Β· (π‘β€˜π΅)))

Proof of Theorem nmmul
StepHypRef Expression
1 nmmul.n . . 3 𝑁 = (normβ€˜π‘…)
2 eqid 2736 . . 3 (AbsValβ€˜π‘…) = (AbsValβ€˜π‘…)
31, 2nrgabv 23931 . 2 (𝑅 ∈ NrmRing β†’ 𝑁 ∈ (AbsValβ€˜π‘…))
4 nmmul.x . . 3 𝑋 = (Baseβ€˜π‘…)
5 nmmul.t . . 3 Β· = (.rβ€˜π‘…)
62, 4, 5abvmul 20195 . 2 ((𝑁 ∈ (AbsValβ€˜π‘…) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴 Β· 𝐡)) = ((π‘β€˜π΄) Β· (π‘β€˜π΅)))
73, 6syl3an1 1162 1 ((𝑅 ∈ NrmRing ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴 Β· 𝐡)) = ((π‘β€˜π΄) Β· (π‘β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  β€˜cfv 6479  (class class class)co 7337   Β· cmul 10977  Basecbs 17009  .rcmulr 17060  AbsValcabv 20182  normcnm 23838  NrmRingcnrg 23841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-map 8688  df-abv 20183  df-nrg 23847
This theorem is referenced by:  nrgdsdi  23935  nrgdsdir  23936  nminvr  23939  nmdvr  23940  nrginvrcnlem  23961
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