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Theorem nmdvr 22474
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 790 . . . 4 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> R e. NrmRing )
2 simprl 794 . . . 4 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> A e. X )
3 nrgring 22467 . . . . . 6 |- ( R e. NrmRing -> R e. Ring )
43ad2antrr 762 . . . . 5 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> R e. Ring )
5 simprr 796 . . . . 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 2622 . . . . . 6 |- ( invr ` R ) = ( invr ` R )
8 nmdvr.x . . . . . 6 |- X = ( Base ` R )
96, 7, 8ringinvcl 18676 . . . . 5 |- ( ( R e. Ring /\ B e. U ) -> ( ( invr ` R ) ` B ) e. X )
104, 5, 9syl2anc 693 . . . 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 2622 . . . . 5 |- ( .r ` R ) = ( .r ` R )
138, 11, 12nmmul 22468 . . . 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 1326 . . 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 792 . . . . 5 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> R e. NzRing )
1611, 6, 7nminvr 22473 . . . . 5 |- ( ( R e. NrmRing /\ R e. NzRing /\ B e. U ) -> ( N ` ( ( invr ` R ) ` B ) ) = ( 1 / ( N ` B ) ) )
171, 15, 5, 16syl3anc 1326 . . . 4 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> ( N ` ( ( invr ` R ) ` B ) ) = ( 1 / ( N ` B ) ) )
1817oveq2d 6666 . . 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 2656 . 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 18685 . . . 4 |- ( ( A e. X /\ B e. U ) -> ( A ./ B ) = ( A ( .r ` R ) ( ( invr ` R ) ` B ) ) )
2221adantl 482 . . 3 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> ( A ./ B ) = ( A ( .r ` R ) ( ( invr ` R ) ` B ) ) )
2322fveq2d 6195 . 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 22466 . . . . . 6 |- ( R e. NrmRing -> R e. NrmGrp )
2524ad2antrr 762 . . . . 5 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> R e. NrmGrp )
268, 11nmcl 22420 . . . . 5 |- ( ( R e. NrmGrp /\ A e. X ) -> ( N ` A ) e. RR )
2725, 2, 26syl2anc 693 . . . 4 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> ( N ` A ) e. RR )
2827recnd 10068 . . 3 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> ( N ` A ) e. CC )
298, 6unitss 18660 . . . . . 6 |- U C_ X
3029, 5sseldi 3601 . . . . 5 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> B e. X )
318, 11nmcl 22420 . . . . 5 |- ( ( R e. NrmGrp /\ B e. X ) -> ( N ` B ) e. RR )
3225, 30, 31syl2anc 693 . . . 4 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> ( N ` B ) e. RR )
3332recnd 10068 . . 3 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> ( N ` B ) e. CC )
3411, 6unitnmn0 22472 . . . . 5 |- ( ( R e. NrmRing /\ R e. NzRing /\ B e. U ) -> ( N ` B ) =/= 0 )
35343expa 1265 . . . 4 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ B e. U ) -> ( N ` B ) =/= 0 )
3635adantrl 752 . . 3 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> ( N ` B ) =/= 0 )
3728, 33, 36divrecd 10804 . 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 2666 1 |- ( ( ( R e. NrmRing /\ R e. NzRing ) /\ ( A e. X /\ B e. U ) ) -> ( N ` ( A ./ B ) ) = ( ( N ` A ) / ( N ` B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   / cdiv 10684   Basecbs 15857   .rcmulr 15942   Ringcrg 18547  Unitcui 18639   invrcinvr 18671  /rcdvr 18682  NzRingcnzr 19257   normcnm 22381  NrmGrpcngp 22382  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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-abv 18817  df-nzr 19258  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388  df-nrg 22390
This theorem is referenced by:  qqhnm  30034
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