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Theorem nmdvr 18197
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 18190 . . . . . 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 2296 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
8 nmdvr.x . . . . . 6  |-  X  =  ( Base `  R
)
96, 7, 8rnginvcl 15474 . . . . 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 2296 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
138, 11, 12nmmul 18191 . . . 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 18196 . . . . 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 5890 . . 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 2328 . 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 15483 . . . 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 5545 . 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 18189 . . . . . 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 18153 . . . . 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 8877 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  A
)  e.  CC )
298, 6unitss 15458 . . . . . 6  |-  U  C_  X
3029, 5sseldi 3191 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  B  e.  X )
318, 11nmcl 18153 . . . . 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 8877 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  B
)  e.  CC )
3411, 6unitnmn0 18195 . . . . 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 9555 . 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 2338 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 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    / cdiv 9439   Basecbs 13164   .rcmulr 13225   Ringcrg 15353  Unitcui 15437   invrcinvr 15469  /rcdvr 15480  NzRingcnzr 16025   normcnm 18115  NrmGrpcngp 18116  NrmRingcnrg 18118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ico 10678  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-topgen 13360  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-abv 15598  df-nzr 16026  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901  df-ms 17902  df-nm 18121  df-ngp 18122  df-nrg 18124
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