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Theorem qqhnm 28124
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 14290 . . . 4  |-  ( Q  e.  QQ  ->  Q  =  ( (numer `  Q )  /  (denom `  Q ) ) )
32fveq2d 5876 . . 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 14284 . . . . 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 10991 . . 3  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (numer `  Q )  e.  CC )
8 qdencl 14285 . . . . 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 10572 . . 3  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (denom `  Q )  e.  CC )
11 nnne0 10589 . . . 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 13297 . 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 3715 . . . . 5  |-  (NrmRing  i^i  DivRing )  C_  DivRing
15 simpl1 999 . . . . 5  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  R  e.  (NrmRing  i^i  DivRing ) )
1614, 15sseldi 3497 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  R  e.  DivRing )
17 simpl3 1001 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (chr `  R )  =  0 )
18 eqid 2457 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
19 eqid 2457 . . . . . 6  |-  (/r `  R
)  =  (/r `  R
)
20 eqid 2457 . . . . . 6  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
2118, 19, 20qqhvval 28117 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (
(QQHom `  R ) `  Q )  =  ( ( ( ZRHom `  R ) `  (numer `  Q ) ) (/r `  R ) ( ( ZRHom `  R ) `  (denom `  Q )
) ) )
2221fveq2d 5876 . . . 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 1227 . . 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 3714 . . . . 5  |-  (NrmRing  i^i  DivRing )  C_ NrmRing
2524, 15sseldi 3497 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  R  e. NrmRing )
26 drngnzr 18036 . . . . 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 drngring 17529 . . . . . 6  |-  ( R  e.  DivRing  ->  R  e.  Ring )
2920zrhrhm 18675 . . . . . 6  |-  ( R  e.  Ring  ->  ( ZRHom `  R )  e.  (ring RingHom  R
) )
30 zringbas 18620 . . . . . . 7  |-  ZZ  =  ( Base ` ring )
3130, 18rhmf 17501 . . . . . 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 6033 . . . 4  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  ( ( ZRHom `  R ) `  (numer `  Q ) )  e.  ( Base `  R
) )
349nnzd 10989 . . . . 5  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  Q  e.  QQ )  ->  (denom `  Q )  e.  ZZ )
35 eqid 2457 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
3618, 20, 35elzrhunit 28113 . . . . 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 1229 . . . 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 2457 . . . . 5  |-  (Unit `  R )  =  (Unit `  R )
4018, 38, 39, 19nmdvr 21304 . . . 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 1229 . . 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 1000 . . . . 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 28101 . . . . . 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 28103 . . . . 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 1231 . . . 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 28103 . . . . 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 1231 . . . 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 6314 . . 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 2503 . 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 2503 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652    i^i cin 3470   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509    / cdiv 10227   NNcn 10556   ZZcz 10885   QQcq 11207   abscabs 13078  numercnumer 14277  denomcdenom 14278   Basecbs 14643   0gc0g 14856   Ringcrg 17324  Unitcui 17414  /rcdvr 17457   RingHom crh 17487   DivRingcdr 17522  NzRingcnzr 18031  ℤringzring 18614   ZRHomczrh 18663   ZModczlm 18664  chrcchr 18665   normcnm 21222  NrmRingcnrg 21225  NrmModcnlm 21226  QQHomcqqh 28106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ico 11560  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-dvds 13998  df-gcd 14156  df-numer 14279  df-denom 14280  df-gz 14459  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-rest 14839  df-topn 14840  df-0g 14858  df-topgen 14860  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-grp 16183  df-minusg 16184  df-sbg 16185  df-mulg 16186  df-subg 16324  df-ghm 16391  df-od 16679  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-cring 17327  df-oppr 17398  df-dvdsr 17416  df-unit 17417  df-invr 17447  df-dvr 17458  df-rnghom 17490  df-drng 17524  df-subrg 17553  df-abv 17592  df-lmod 17640  df-nzr 18032  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-cnfld 18547  df-zring 18615  df-zrh 18667  df-zlm 18668  df-chr 18669  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-xms 20948  df-ms 20949  df-nm 21228  df-ngp 21229  df-nrg 21231  df-nlm 21232  df-qqh 28107
This theorem is referenced by:  qqhcn  28125  qqhucn  28126
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