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Theorem qqhnm 26441
Description: The norm of the image by QQHom of a rational number in a topological division ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
Hypotheses
Ref Expression
qqhnm.n  |-  N  =  ( norm `  R
)
qqhnm.z  |-  Z  =  ( ZMod `  R
)
Assertion
Ref Expression
qqhnm  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( N `  (
(QQHom `  R ) `  Q ) )  =  ( abs `  Q
) )

Proof of Theorem qqhnm
StepHypRef Expression
1 simpr 461 . . 3  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  Q  e.  QQ )
2 qeqnumdivden 13845 . . . 4  |-  ( Q  e.  QQ  ->  Q  =  ( (numer `  Q )  /  (denom `  Q ) ) )
32fveq2d 5716 . . 3  |-  ( Q  e.  QQ  ->  ( abs `  Q )  =  ( abs `  (
(numer `  Q )  /  (denom `  Q )
) ) )
41, 3syl 16 . 2  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( abs `  Q
)  =  ( abs `  ( (numer `  Q
)  /  (denom `  Q ) ) ) )
5 qnumcl 13839 . . . . 5  |-  ( Q  e.  QQ  ->  (numer `  Q )  e.  ZZ )
61, 5syl 16 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (numer `  Q )  e.  ZZ )
76zcnd 10769 . . 3  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (numer `  Q )  e.  CC )
8 qdencl 13840 . . . . 5  |-  ( Q  e.  QQ  ->  (denom `  Q )  e.  NN )
91, 8syl 16 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (denom `  Q )  e.  NN )
109nncnd 10359 . . 3  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (denom `  Q )  e.  CC )
11 nnne0 10375 . . . 4  |-  ( (denom `  Q )  e.  NN  ->  (denom `  Q )  =/=  0 )
121, 8, 113syl 20 . . 3  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (denom `  Q )  =/=  0 )
137, 10, 12absdivd 12962 . 2  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( abs `  (
(numer `  Q )  /  (denom `  Q )
) )  =  ( ( abs `  (numer `  Q ) )  / 
( abs `  (denom `  Q ) ) ) )
14 inss2 3592 . . . . 5  |-  (NrmRing  i^i  DivRing )  C_  DivRing
15 simpl1 991 . . . . 5  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  R  e.  (NrmRing  i^i  DivRing ) )
1614, 15sseldi 3375 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  R  e.  DivRing )
17 simpl3 993 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (chr `  R )  =  0 )
18 eqid 2443 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
19 eqid 2443 . . . . . 6  |-  (/r `  R
)  =  (/r `  R
)
20 eqid 2443 . . . . . 6  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
2118, 19, 20qqhvval 26434 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (
(QQHom `  R ) `  Q )  =  ( ( ( ZRHom `  R ) `  (numer `  Q ) ) (/r `  R ) ( ( ZRHom `  R ) `  (denom `  Q )
) ) )
2221fveq2d 5716 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( N `  ( (QQHom `  R ) `  Q
) )  =  ( N `  ( ( ( ZRHom `  R
) `  (numer `  Q
) ) (/r `  R
) ( ( ZRHom `  R ) `  (denom `  Q ) ) ) ) )
2316, 17, 1, 22syl21anc 1217 . . 3  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( N `  (
(QQHom `  R ) `  Q ) )  =  ( N `  (
( ( ZRHom `  R ) `  (numer `  Q ) ) (/r `  R ) ( ( ZRHom `  R ) `  (denom `  Q )
) ) ) )
24 inss1 3591 . . . . 5  |-  (NrmRing  i^i  DivRing )  C_ NrmRing
2524, 15sseldi 3375 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  R  e. NrmRing )
26 drngnzr 17366 . . . . 5  |-  ( R  e.  DivRing  ->  R  e. NzRing )
2716, 26syl 16 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  R  e. NzRing )
28 drngrng 16861 . . . . . 6  |-  ( R  e.  DivRing  ->  R  e.  Ring )
2920zrhrhm 17965 . . . . . 6  |-  ( R  e.  Ring  ->  ( ZRHom `  R )  e.  (ring RingHom  R
) )
30 zringbas 17911 . . . . . . 7  |-  ZZ  =  ( Base ` ring )
3130, 18rhmf 16838 . . . . . 6  |-  ( ( ZRHom `  R )  e.  (ring RingHom  R )  ->  ( ZRHom `  R ) : ZZ --> ( Base `  R
) )
3216, 28, 29, 314syl 21 . . . . 5  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( ZRHom `  R
) : ZZ --> ( Base `  R ) )
3332, 6ffvelrnd 5865 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( ( ZRHom `  R ) `  (numer `  Q ) )  e.  ( Base `  R
) )
349nnzd 10767 . . . . 5  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (denom `  Q )  e.  ZZ )
35 eqid 2443 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
3618, 20, 35elzrhunit 26430 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  Q )  e.  ZZ  /\  (denom `  Q )  =/=  0
) )  ->  (
( ZRHom `  R
) `  (denom `  Q
) )  e.  (Unit `  R ) )
3716, 17, 34, 12, 36syl22anc 1219 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( ( ZRHom `  R ) `  (denom `  Q ) )  e.  (Unit `  R )
)
38 qqhnm.n . . . . 5  |-  N  =  ( norm `  R
)
39 eqid 2443 . . . . 5  |-  (Unit `  R )  =  (Unit `  R )
4018, 38, 39, 19nmdvr 20273 . . . 4  |-  ( ( ( R  e. NrmRing  /\  R  e. NzRing )  /\  ( ( ( ZRHom `  R
) `  (numer `  Q
) )  e.  (
Base `  R )  /\  ( ( ZRHom `  R ) `  (denom `  Q ) )  e.  (Unit `  R )
) )  ->  ( N `  ( (
( ZRHom `  R
) `  (numer `  Q
) ) (/r `  R
) ( ( ZRHom `  R ) `  (denom `  Q ) ) ) )  =  ( ( N `  ( ( ZRHom `  R ) `  (numer `  Q )
) )  /  ( N `  ( ( ZRHom `  R ) `  (denom `  Q ) ) ) ) )
4125, 27, 33, 37, 40syl22anc 1219 . . 3  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( N `  (
( ( ZRHom `  R ) `  (numer `  Q ) ) (/r `  R ) ( ( ZRHom `  R ) `  (denom `  Q )
) ) )  =  ( ( N `  ( ( ZRHom `  R ) `  (numer `  Q ) ) )  /  ( N `  ( ( ZRHom `  R ) `  (denom `  Q ) ) ) ) )
42 simpl2 992 . . . . 5  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  Z  e. NrmMod )
43 qqhnm.z . . . . . . 7  |-  Z  =  ( ZMod `  R
)
4443zhmnrg 26418 . . . . . 6  |-  ( R  e. NrmRing  ->  Z  e. NrmRing )
4525, 44syl 16 . . . . 5  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  Z  e. NrmRing )
4618, 38, 43, 20zrhnm 26420 . . . . 5  |-  ( ( ( Z  e. NrmMod  /\  Z  e. NrmRing  /\  R  e. NzRing )  /\  (numer `  Q )  e.  ZZ )  ->  ( N `  ( ( ZRHom `  R ) `  (numer `  Q ) ) )  =  ( abs `  (numer `  Q )
) )
4742, 45, 27, 6, 46syl31anc 1221 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( N `  (
( ZRHom `  R
) `  (numer `  Q
) ) )  =  ( abs `  (numer `  Q ) ) )
4818, 38, 43, 20zrhnm 26420 . . . . 5  |-  ( ( ( Z  e. NrmMod  /\  Z  e. NrmRing  /\  R  e. NzRing )  /\  (denom `  Q )  e.  ZZ )  ->  ( N `  ( ( ZRHom `  R ) `  (denom `  Q ) ) )  =  ( abs `  (denom `  Q )
) )
4942, 45, 27, 34, 48syl31anc 1221 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( N `  (
( ZRHom `  R
) `  (denom `  Q
) ) )  =  ( abs `  (denom `  Q ) ) )
5047, 49oveq12d 6130 . . 3  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( ( N `  ( ( ZRHom `  R ) `  (numer `  Q ) ) )  /  ( N `  ( ( ZRHom `  R ) `  (denom `  Q ) ) ) )  =  ( ( abs `  (numer `  Q ) )  / 
( abs `  (denom `  Q ) ) ) )
5123, 41, 503eqtrrd 2480 . 2  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( ( abs `  (numer `  Q ) )  / 
( abs `  (denom `  Q ) ) )  =  ( N `  ( (QQHom `  R ) `  Q ) ) )
524, 13, 513eqtrrd 2480 1  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( N `  (
(QQHom `  R ) `  Q ) )  =  ( abs `  Q
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620    i^i cin 3348   -->wf 5435   ` cfv 5439  (class class class)co 6112   0cc0 9303    / cdiv 10014   NNcn 10343   ZZcz 10667   QQcq 10974   abscabs 12744  numercnumer 13832  denomcdenom 13833   Basecbs 14195   0gc0g 14399   Ringcrg 16667  Unitcui 16753  /rcdvr 16796   RingHom crh 16826   DivRingcdr 16854  NzRingcnzr 17361  ℤringzring 17905   ZRHomczrh 17953   ZModczlm 17954  chrcchr 17955   normcnm 20191  NrmRingcnrg 20194  NrmModcnlm 20195  QQHomcqqh 26423
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 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-ico 11327  df-fz 11459  df-fzo 11570  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-dvds 13557  df-gcd 13712  df-numer 13834  df-denom 13835  df-gz 14012  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-sca 14275  df-vsca 14276  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-rest 14382  df-topn 14383  df-0g 14401  df-topgen 14403  df-mnd 15436  df-mhm 15485  df-grp 15566  df-minusg 15567  df-sbg 15568  df-mulg 15569  df-subg 15699  df-ghm 15766  df-od 16053  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-cring 16670  df-oppr 16737  df-dvdsr 16755  df-unit 16756  df-invr 16786  df-dvr 16797  df-rnghom 16828  df-drng 16856  df-subrg 16885  df-abv 16924  df-lmod 16972  df-nzr 17362  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-cnfld 17841  df-zring 17906  df-zrh 17957  df-zlm 17958  df-chr 17959  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-xms 19917  df-ms 19918  df-nm 20197  df-ngp 20198  df-nrg 20200  df-nlm 20201  df-qqh 26424
This theorem is referenced by:  qqhcn  26442  qqhucn  26443
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