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Theorem nmdvr 18181
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 730 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e. NrmRing )
2 simprl 732 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  A  e.  X )
3 nrgrng 18174 . . . . . 6  |-  ( R  e. NrmRing  ->  R  e.  Ring )
43ad2antrr 706 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e.  Ring )
5 simprr 733 . . . . 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 2283 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
8 nmdvr.x . . . . . 6  |-  X  =  ( Base `  R
)
96, 7, 8rnginvcl 15458 . . . . 5  |-  ( ( R  e.  Ring  /\  B  e.  U )  ->  (
( invr `  R ) `  B )  e.  X
)
104, 5, 9syl2anc 642 . . . 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 2283 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
138, 11, 12nmmul 18175 . . . 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 1182 . . 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 731 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e. NzRing )
1611, 6, 7nminvr 18180 . . . . 5  |-  ( ( R  e. NrmRing  /\  R  e. NzRing  /\  B  e.  U
)  ->  ( N `  ( ( invr `  R
) `  B )
)  =  ( 1  /  ( N `  B ) ) )
171, 15, 5, 16syl3anc 1182 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  (
( invr `  R ) `  B ) )  =  ( 1  /  ( N `  B )
) )
1817oveq2d 5874 . . 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 2315 . 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 15467 . . . 4  |-  ( ( A  e.  X  /\  B  e.  U )  ->  ( A  ./  B
)  =  ( A ( .r `  R
) ( ( invr `  R ) `  B
) ) )
2221adantl 452 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( A  ./  B
)  =  ( A ( .r `  R
) ( ( invr `  R ) `  B
) ) )
2322fveq2d 5529 . 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 18173 . . . . . 6  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
2524ad2antrr 706 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e. NrmGrp )
268, 11nmcl 18137 . . . . 5  |-  ( ( R  e. NrmGrp  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
2725, 2, 26syl2anc 642 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  A
)  e.  RR )
2827recnd 8861 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  A
)  e.  CC )
298, 6unitss 15442 . . . . . 6  |-  U  C_  X
3029, 5sseldi 3178 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  B  e.  X )
318, 11nmcl 18137 . . . . 5  |-  ( ( R  e. NrmGrp  /\  B  e.  X )  ->  ( N `  B )  e.  RR )
3225, 30, 31syl2anc 642 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  B
)  e.  RR )
3332recnd 8861 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  B
)  e.  CC )
3411, 6unitnmn0 18179 . . . . 5  |-  ( ( R  e. NrmRing  /\  R  e. NzRing  /\  B  e.  U
)  ->  ( N `  B )  =/=  0
)
35343expa 1151 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  B  e.  U )  ->  ( N `  B )  =/=  0 )
3635adantrl 696 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  B
)  =/=  0 )
3728, 33, 36divrecd 9539 . 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 2325 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 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    / cdiv 9423   Basecbs 13148   .rcmulr 13209   Ringcrg 15337  Unitcui 15421   invrcinvr 15453  /rcdvr 15464  NzRingcnzr 16009   normcnm 18099  NrmGrpcngp 18100  NrmRingcnrg 18102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-topgen 13344  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-abv 15582  df-nzr 16010  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885  df-ms 17886  df-nm 18105  df-ngp 18106  df-nrg 18108
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