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Theorem nmdvr 20231
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 753 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e. NrmRing )
2 simprl 755 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  A  e.  X )
3 nrgrng 20224 . . . . . 6  |-  ( R  e. NrmRing  ->  R  e.  Ring )
43ad2antrr 725 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e.  Ring )
5 simprr 756 . . . . 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 2438 . . . . . 6  |-  ( invr `  R )  =  (
invr `  R )
8 nmdvr.x . . . . . 6  |-  X  =  ( Base `  R
)
96, 7, 8rnginvcl 16758 . . . . 5  |-  ( ( R  e.  Ring  /\  B  e.  U )  ->  (
( invr `  R ) `  B )  e.  X
)
104, 5, 9syl2anc 661 . . . 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 2438 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
138, 11, 12nmmul 20225 . . . 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 1218 . . 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 754 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e. NzRing )
1611, 6, 7nminvr 20230 . . . . 5  |-  ( ( R  e. NrmRing  /\  R  e. NzRing  /\  B  e.  U
)  ->  ( N `  ( ( invr `  R
) `  B )
)  =  ( 1  /  ( N `  B ) ) )
171, 15, 5, 16syl3anc 1218 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  (
( invr `  R ) `  B ) )  =  ( 1  /  ( N `  B )
) )
1817oveq2d 6102 . . 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 2470 . 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 16767 . . . 4  |-  ( ( A  e.  X  /\  B  e.  U )  ->  ( A  ./  B
)  =  ( A ( .r `  R
) ( ( invr `  R ) `  B
) ) )
2221adantl 466 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( A  ./  B
)  =  ( A ( .r `  R
) ( ( invr `  R ) `  B
) ) )
2322fveq2d 5690 . 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 20223 . . . . . 6  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
2524ad2antrr 725 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  R  e. NrmGrp )
268, 11nmcl 20187 . . . . 5  |-  ( ( R  e. NrmGrp  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
2725, 2, 26syl2anc 661 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  A
)  e.  RR )
2827recnd 9404 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  A
)  e.  CC )
298, 6unitss 16742 . . . . . 6  |-  U  C_  X
3029, 5sseldi 3349 . . . . 5  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  ->  B  e.  X )
318, 11nmcl 20187 . . . . 5  |-  ( ( R  e. NrmGrp  /\  B  e.  X )  ->  ( N `  B )  e.  RR )
3225, 30, 31syl2anc 661 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  B
)  e.  RR )
3332recnd 9404 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  B
)  e.  CC )
3411, 6unitnmn0 20229 . . . . 5  |-  ( ( R  e. NrmRing  /\  R  e. NzRing  /\  B  e.  U
)  ->  ( N `  B )  =/=  0
)
35343expa 1187 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  B  e.  U )  ->  ( N `  B )  =/=  0 )
3635adantrl 715 . . 3  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( A  e.  X  /\  B  e.  U ) )  -> 
( N `  B
)  =/=  0 )
3728, 33, 36divrecd 10102 . 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 2480 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 369    = wceq 1369    e. wcel 1756    =/= wne 2601   ` cfv 5413  (class class class)co 6086   RRcr 9273   0cc0 9274   1c1 9275    x. cmul 9279    / cdiv 9985   Basecbs 14166   .rcmulr 14231   Ringcrg 16635  Unitcui 16721   invrcinvr 16753  /rcdvr 16764  NzRingcnzr 17319   normcnm 20149  NrmGrpcngp 20150  NrmRingcnrg 20152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ico 11298  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-0g 14372  df-topgen 14374  df-mnd 15407  df-grp 15536  df-minusg 15537  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-abv 16882  df-nzr 17320  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-xms 19875  df-ms 19876  df-nm 20155  df-ngp 20156  df-nrg 20158
This theorem is referenced by:  qqhnm  26388
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