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Theorem nmdvr 18698
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 18691 . . . . . 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 2435 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
8 nmdvr.x . . . . . 6  |-  X  =  ( Base `  R
)
96, 7, 8rnginvcl 15773 . . . . 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 2435 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
138, 11, 12nmmul 18692 . . . 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 18697 . . . . 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 6089 . . 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 2467 . 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 15782 . . . 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 5724 . 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 18690 . . . . . 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 18654 . . . . 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 9106 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  A
)  e.  CC )
298, 6unitss 15757 . . . . . 6  |-  U  C_  X
3029, 5sseldi 3338 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  B  e.  X )
318, 11nmcl 18654 . . . . 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 9106 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  B
)  e.  CC )
3411, 6unitnmn0 18696 . . . . 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 9785 . 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 2477 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 1652    e. wcel 1725    =/= wne 2598   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    / cdiv 9669   Basecbs 13461   .rcmulr 13522   Ringcrg 15652  Unitcui 15736   invrcinvr 15768  /rcdvr 15779  NzRingcnzr 16320   normcnm 18616  NrmGrpcngp 18617  NrmRingcnrg 18619
This theorem is referenced by:  qqhnm  24366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ico 10914  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-topgen 13659  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-abv 15897  df-nzr 16321  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-xms 18342  df-ms 18343  df-nm 18622  df-ngp 18623  df-nrg 18625
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