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Theorem nmdvr 18570
Description: The norm of a division in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
nmdvr.x  |-  X  =  ( Base `  R
)
nmdvr.n  |-  N  =  ( norm `  R
)
nmdvr.u  |-  U  =  (Unit `  R )
nmdvr.d  |-  ./  =  (/r
`  R )
Assertion
Ref Expression
nmdvr  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  ( A  ./  B ) )  =  ( ( N `
 A )  / 
( N `  B
) ) )

Proof of Theorem nmdvr
StepHypRef Expression
1 simpll 731 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e. NrmRing )
2 simprl 733 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  A  e.  X )
3 nrgrng 18563 . . . . . 6  |-  ( R  e. NrmRing  ->  R  e.  Ring )
43ad2antrr 707 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e.  Ring )
5 simprr 734 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  B  e.  U )
6 nmdvr.u . . . . . 6  |-  U  =  (Unit `  R )
7 eqid 2380 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
8 nmdvr.x . . . . . 6  |-  X  =  ( Base `  R
)
96, 7, 8rnginvcl 15701 . . . . 5  |-  ( ( R  e.  Ring  /\  B  e.  U )  ->  (
( invr `  R ) `  B )  e.  X
)
104, 5, 9syl2anc 643 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( ( invr `  R
) `  B )  e.  X )
11 nmdvr.n . . . . 5  |-  N  =  ( norm `  R
)
12 eqid 2380 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
138, 11, 12nmmul 18564 . . . 4  |-  ( ( R  e. NrmRing  /\  A  e.  X  /\  ( (
invr `  R ) `  B )  e.  X
)  ->  ( N `  ( A ( .r
`  R ) ( ( invr `  R
) `  B )
) )  =  ( ( N `  A
)  x.  ( N `
 ( ( invr `  R ) `  B
) ) ) )
141, 2, 10, 13syl3anc 1184 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  ( A ( .r `  R ) ( (
invr `  R ) `  B ) ) )  =  ( ( N `
 A )  x.  ( N `  (
( invr `  R ) `  B ) ) ) )
15 simplr 732 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e. NzRing )
1611, 6, 7nminvr 18569 . . . . 5  |-  ( ( R  e. NrmRing  /\  R  e. NzRing  /\  B  e.  U
)  ->  ( N `  ( ( invr `  R
) `  B )
)  =  ( 1  /  ( N `  B ) ) )
171, 15, 5, 16syl3anc 1184 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  (
( invr `  R ) `  B ) )  =  ( 1  /  ( N `  B )
) )
1817oveq2d 6029 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( ( N `  A )  x.  ( N `  ( ( invr `  R ) `  B ) ) )  =  ( ( N `
 A )  x.  ( 1  /  ( N `  B )
) ) )
1914, 18eqtrd 2412 . 2  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  ( A ( .r `  R ) ( (
invr `  R ) `  B ) ) )  =  ( ( N `
 A )  x.  ( 1  /  ( N `  B )
) ) )
20 nmdvr.d . . . . 5  |-  ./  =  (/r
`  R )
218, 12, 6, 7, 20dvrval 15710 . . . 4  |-  ( ( A  e.  X  /\  B  e.  U )  ->  ( A  ./  B
)  =  ( A ( .r `  R
) ( ( invr `  R ) `  B
) ) )
2221adantl 453 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( A  ./  B
)  =  ( A ( .r `  R
) ( ( invr `  R ) `  B
) ) )
2322fveq2d 5665 . 2  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  ( A  ./  B ) )  =  ( N `  ( A ( .r `  R ) ( (
invr `  R ) `  B ) ) ) )
24 nrgngp 18562 . . . . . 6  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
2524ad2antrr 707 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e. NrmGrp )
268, 11nmcl 18526 . . . . 5  |-  ( ( R  e. NrmGrp  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
2725, 2, 26syl2anc 643 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  A
)  e.  RR )
2827recnd 9040 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  A
)  e.  CC )
298, 6unitss 15685 . . . . . 6  |-  U  C_  X
3029, 5sseldi 3282 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  B  e.  X )
318, 11nmcl 18526 . . . . 5  |-  ( ( R  e. NrmGrp  /\  B  e.  X )  ->  ( N `  B )  e.  RR )
3225, 30, 31syl2anc 643 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  B
)  e.  RR )
3332recnd 9040 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  B
)  e.  CC )
3411, 6unitnmn0 18568 . . . . 5  |-  ( ( R  e. NrmRing  /\  R  e. NzRing  /\  B  e.  U
)  ->  ( N `  B )  =/=  0
)
35343expa 1153 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  B  e.  U )  ->  ( N `  B )  =/=  0 )
3635adantrl 697 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  B
)  =/=  0 )
3728, 33, 36divrecd 9718 . 2  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( ( N `  A )  /  ( N `  B )
)  =  ( ( N `  A )  x.  ( 1  / 
( N `  B
) ) ) )
3819, 23, 373eqtr4d 2422 1  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  ( A  ./  B ) )  =  ( ( N `
 A )  / 
( N `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   ` cfv 5387  (class class class)co 6013   RRcr 8915   0cc0 8916   1c1 8917    x. cmul 8921    / cdiv 9602   Basecbs 13389   .rcmulr 13450   Ringcrg 15580  Unitcui 15664   invrcinvr 15696  /rcdvr 15707  NzRingcnzr 16248   normcnm 18488  NrmGrpcngp 18489  NrmRingcnrg 18491
This theorem is referenced by:  qqhnm  24166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-tpos 6408  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-ico 10847  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-topgen 13587  df-0g 13647  df-mnd 14610  df-grp 14732  df-minusg 14733  df-mgp 15569  df-rng 15583  df-ur 15585  df-oppr 15648  df-dvdsr 15666  df-unit 15667  df-invr 15697  df-dvr 15708  df-abv 15825  df-nzr 16249  df-xmet 16612  df-met 16613  df-bl 16614  df-mopn 16615  df-top 16879  df-bases 16881  df-topon 16882  df-topsp 16883  df-xms 18252  df-ms 18253  df-nm 18494  df-ngp 18495  df-nrg 18497
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