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Theorem nmmul 24574
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 2728 . . 3 (AbsValβ€˜π‘…) = (AbsValβ€˜π‘…)
31, 2nrgabv 24571 . 2 (𝑅 ∈ NrmRing β†’ 𝑁 ∈ (AbsValβ€˜π‘…))
4 nmmul.x . . 3 𝑋 = (Baseβ€˜π‘…)
5 nmmul.t . . 3 Β· = (.rβ€˜π‘…)
62, 4, 5abvmul 20702 . 2 ((𝑁 ∈ (AbsValβ€˜π‘…) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴 Β· 𝐡)) = ((π‘β€˜π΄) Β· (π‘β€˜π΅)))
73, 6syl3an1 1161 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 1085   = wceq 1534   ∈ wcel 2099  β€˜cfv 6542  (class class class)co 7414   Β· cmul 11137  Basecbs 17173  .rcmulr 17227  AbsValcabv 20689  normcnm 24478  NrmRingcnrg 24481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8840  df-abv 20690  df-nrg 24487
This theorem is referenced by:  nrgdsdi  24575  nrgdsdir  24576  nminvr  24579  nmdvr  24580  nrginvrcnlem  24601
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