Theorem nmmul 22468

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

Step	Hyp	Ref	Expression
1		nmmul.n	. . 3 |- N = ( norm ` R )
2		eqid 2622	. . 3 |- (AbsVal ` R ) = (AbsVal ` R )
3	1, 2	nrgabv 22465	. 2 |- ( R e. NrmRing -> N e. (AbsVal ` R ) )
4		nmmul.x	. . 3 |- X = ( Base ` R )
5		nmmul.t	. . 3 |- .x. = ( .r ` R )
6	2, 4, 5	abvmul 18829	. 2 |- ( ( N e. (AbsVal ` R ) /\ A e. X /\ B e. X ) -> ( N ` ( A .x. B ) ) = ( ( N ` A ) x. ( N ` B ) ) )
7	3, 6	syl3an1 1359	1 |- ( ( R e. NrmRing /\ A e. X /\ B e. X ) -> ( N ` ( A .x. B ) ) = ( ( N ` A ) x. ( N ` B ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   x. cmul 9941   Basecbs 15857   .rcmulr 15942  AbsValcabv 18816   normcnm 22381  NrmRingcnrg 22384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-abv 18817  df-nrg 22390

This theorem is referenced by:  nrgdsdi  22469  nrgdsdir  22470  nminvr  22473  nmdvr  22474  nrginvrcnlem  22495