Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  qqhcn Unicode version

Theorem qqhcn 24328
Description: The QQHom homomorphism is a continuous function. (Contributed by Thierry Arnoux, 9-Nov-2017.)
Hypotheses
Ref Expression
qqhcn.q  |-  Q  =  (flds  QQ )
qqhcn.j  |-  J  =  ( TopOpen `  Q )
qqhcn.z  |-  Z  =  ( ZMod `  R
)
qqhcn.k  |-  K  =  ( TopOpen `  R )
Assertion
Ref Expression
qqhcn  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( J  Cn  K ) )

Proof of Theorem qqhcn
Dummy variables  e 
d  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3522 . . . . . . . 8  |-  (NrmRing  i^i  DivRing )  C_  DivRing
21sseli 3304 . . . . . . 7  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e.  DivRing )
323ad2ant1 978 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e.  DivRing )
4 simp3 959 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (chr `  R
)  =  0 )
5 eqid 2404 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2404 . . . . . . 7  |-  (/r `  R
)  =  (/r `  R
)
7 eqid 2404 . . . . . . 7  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
85, 6, 7qqhf 24323 . . . . . 6  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> ( Base `  R ) )
93, 4, 8syl2anc 643 . . . . 5  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> ( Base `  R ) )
10 simpr 448 . . . . . . 7  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  -> 
e  e.  RR+ )
11 qsscn 10541 . . . . . . . . . . . . . 14  |-  QQ  C_  CC
12 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  q  e.  QQ )
1311, 12sseldi 3306 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  q  e.  CC )
14 0cn 9040 . . . . . . . . . . . . . . 15  |-  0  e.  CC
15 eqid 2404 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1615cnmetdval 18758 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  CC  /\  q  e.  CC )  ->  ( 0 ( abs 
o.  -  ) q
)  =  ( abs `  ( 0  -  q
) ) )
1714, 16mpan 652 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  (
0 ( abs  o.  -  ) q )  =  ( abs `  (
0  -  q ) ) )
18 df-neg 9250 . . . . . . . . . . . . . . . 16  |-  -u q  =  ( 0  -  q )
1918fveq2i 5690 . . . . . . . . . . . . . . 15  |-  ( abs `  -u q )  =  ( abs `  (
0  -  q ) )
2019a1i 11 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  ( abs `  -u q )  =  ( abs `  (
0  -  q ) ) )
21 absneg 12037 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  ( abs `  -u q )  =  ( abs `  q
) )
2217, 20, 213eqtr2d 2442 . . . . . . . . . . . . 13  |-  ( q  e.  CC  ->  (
0 ( abs  o.  -  ) q )  =  ( abs `  q
) )
2313, 22syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
0 ( abs  o.  -  ) q )  =  ( abs `  q
) )
24 zssq 10537 . . . . . . . . . . . . . . 15  |-  ZZ  C_  QQ
25 0z 10249 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
2624, 25sselii 3305 . . . . . . . . . . . . . 14  |-  0  e.  QQ
2726a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  0  e.  QQ )
2827, 12ovresd 6173 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
0 ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( 0 ( abs  o.  -  ) q ) )
29 eqid 2404 . . . . . . . . . . . . . 14  |-  ( norm `  R )  =  (
norm `  R )
30 qqhcn.z . . . . . . . . . . . . . 14  |-  Z  =  ( ZMod `  R
)
3129, 30qqhnm 24327 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  ( ( norm `  R
) `  ( (QQHom `  R ) `  q
) )  =  ( abs `  q ) )
3231adantlr 696 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
( norm `  R ) `  ( (QQHom `  R
) `  q )
)  =  ( abs `  q ) )
3323, 28, 323eqtr4d 2446 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
0 ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( ( norm `  R
) `  ( (QQHom `  R ) `  q
) ) )
349ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (QQHom `  R ) : QQ --> ( Base `  R )
)
3534, 27ffvelrnd 5830 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  0 )  e.  ( Base `  R
) )
3634, 12ffvelrnd 5830 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  q )  e.  (
Base `  R )
)
3735, 36ovresd 6173 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  0 )
( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ( (QQHom `  R ) `  q
) )  =  ( ( (QQHom `  R
) `  0 )
( dist `  R )
( (QQHom `  R
) `  q )
) )
38 inss1 3521 . . . . . . . . . . . . . . . . 17  |-  (NrmRing  i^i  DivRing )  C_ NrmRing
3938sseli 3304 . . . . . . . . . . . . . . . 16  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e. NrmRing )
40393ad2ant1 978 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e. NrmRing )
4140ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e. NrmRing )
42 nrgngp 18651 . . . . . . . . . . . . . 14  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
4341, 42syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e. NrmGrp )
44 eqid 2404 . . . . . . . . . . . . . 14  |-  ( -g `  R )  =  (
-g `  R )
45 eqid 2404 . . . . . . . . . . . . . 14  |-  ( dist `  R )  =  (
dist `  R )
4629, 5, 44, 45ngpdsr 18604 . . . . . . . . . . . . 13  |-  ( ( R  e. NrmGrp  /\  (
(QQHom `  R ) `  0 )  e.  ( Base `  R
)  /\  ( (QQHom `  R ) `  q
)  e.  ( Base `  R ) )  -> 
( ( (QQHom `  R ) `  0
) ( dist `  R
) ( (QQHom `  R ) `  q
) )  =  ( ( norm `  R
) `  ( (
(QQHom `  R ) `  q ) ( -g `  R ) ( (QQHom `  R ) `  0
) ) ) )
4743, 35, 36, 46syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  0 )
( dist `  R )
( (QQHom `  R
) `  q )
)  =  ( (
norm `  R ) `  ( ( (QQHom `  R ) `  q
) ( -g `  R
) ( (QQHom `  R ) `  0
) ) ) )
483ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e.  DivRing )
494ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (chr `  R )  =  0 )
505, 6, 7qqh0 24321 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R ) `  0
)  =  ( 0g
`  R ) )
5148, 49, 50syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  0 )  =  ( 0g `  R
) )
5251oveq2d 6056 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  q )
( -g `  R ) ( (QQHom `  R
) `  0 )
)  =  ( ( (QQHom `  R ) `  q ) ( -g `  R ) ( 0g
`  R ) ) )
53 ngpgrp 18599 . . . . . . . . . . . . . . . 16  |-  ( R  e. NrmGrp  ->  R  e.  Grp )
5443, 53syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e.  Grp )
55 eqid 2404 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  R )  =  ( 0g `  R
)
565, 55, 44grpsubid1 14829 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Grp  /\  ( (QQHom `  R ) `  q )  e.  (
Base `  R )
)  ->  ( (
(QQHom `  R ) `  q ) ( -g `  R ) ( 0g
`  R ) )  =  ( (QQHom `  R ) `  q
) )
5754, 36, 56syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  q )
( -g `  R ) ( 0g `  R
) )  =  ( (QQHom `  R ) `  q ) )
5852, 57eqtrd 2436 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  q )
( -g `  R ) ( (QQHom `  R
) `  0 )
)  =  ( (QQHom `  R ) `  q
) )
5958fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
( norm `  R ) `  ( ( (QQHom `  R ) `  q
) ( -g `  R
) ( (QQHom `  R ) `  0
) ) )  =  ( ( norm `  R
) `  ( (QQHom `  R ) `  q
) ) )
6037, 47, 593eqtrd 2440 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  0 )
( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ( (QQHom `  R ) `  q
) )  =  ( ( norm `  R
) `  ( (QQHom `  R ) `  q
) ) )
6133, 60eqtr4d 2439 . . . . . . . . . 10  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
0 ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( ( (QQHom `  R
) `  0 )
( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ( (QQHom `  R ) `  q
) ) )
6261breq1d 4182 . . . . . . . . 9  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  e  <->  ( (
(QQHom `  R ) `  0 ) ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ( (QQHom `  R ) `  q
) )  <  e
) )
6362biimpd 199 . . . . . . . 8  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  (
( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  e  ->  (
( (QQHom `  R
) `  0 )
( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ( (QQHom `  R ) `  q
) )  <  e
) )
6463ralrimiva 2749 . . . . . . 7  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  ->  A. q  e.  QQ  ( ( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  e  ->  (
( (QQHom `  R
) `  0 )
( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ( (QQHom `  R ) `  q
) )  <  e
) )
65 breq2 4176 . . . . . . . . . 10  |-  ( d  =  e  ->  (
( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  <->  ( 0 ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  e
) )
6665imbi1d 309 . . . . . . . . 9  |-  ( d  =  e  ->  (
( ( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  ->  (
( (QQHom `  R
) `  0 )
( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ( (QQHom `  R ) `  q
) )  <  e
)  <->  ( ( 0 ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  e  ->  ( ( (QQHom `  R ) `  0
) ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) ( (QQHom `  R ) `  q ) )  < 
e ) ) )
6766ralbidv 2686 . . . . . . . 8  |-  ( d  =  e  ->  ( A. q  e.  QQ  ( ( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  ->  (
( (QQHom `  R
) `  0 )
( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ( (QQHom `  R ) `  q
) )  <  e
)  <->  A. q  e.  QQ  ( ( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  e  ->  (
( (QQHom `  R
) `  0 )
( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ( (QQHom `  R ) `  q
) )  <  e
) ) )
6867rspcev 3012 . . . . . . 7  |-  ( ( e  e.  RR+  /\  A. q  e.  QQ  (
( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  e  ->  (
( (QQHom `  R
) `  0 )
( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ( (QQHom `  R ) `  q
) )  <  e
) )  ->  E. d  e.  RR+  A. q  e.  QQ  ( ( 0 ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  d  ->  ( ( (QQHom `  R ) `  0
) ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) ( (QQHom `  R ) `  q ) )  < 
e ) )
6910, 64, 68syl2anc 643 . . . . . 6  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  ->  E. d  e.  RR+  A. q  e.  QQ  ( ( 0 ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  d  ->  ( ( (QQHom `  R ) `  0
) ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) ( (QQHom `  R ) `  q ) )  < 
e ) )
7069ralrimiva 2749 . . . . 5  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  A. e  e.  RR+  E. d  e.  RR+  A. q  e.  QQ  ( ( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  ->  (
( (QQHom `  R
) `  0 )
( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ( (QQHom `  R ) `  q
) )  <  e
) )
71 qqhcn.q . . . . . . . 8  |-  Q  =  (flds  QQ )
72 cnfldxms 18764 . . . . . . . . 9  |-fld  e.  * MetSp
73 qex 10542 . . . . . . . . 9  |-  QQ  e.  _V
74 ressxms 18508 . . . . . . . . 9  |-  ( (fld  e. 
* MetSp  /\  QQ  e.  _V )  ->  (flds  QQ )  e.  * MetSp )
7572, 73, 74mp2an 654 . . . . . . . 8  |-  (flds  QQ )  e.  * MetSp
7671, 75eqeltri 2474 . . . . . . 7  |-  Q  e. 
* MetSp
7771qrngbas 21266 . . . . . . . 8  |-  QQ  =  ( Base `  Q )
78 cnfldds 16668 . . . . . . . . . 10  |-  ( abs 
o.  -  )  =  ( dist ` fld )
7971, 78ressds 13596 . . . . . . . . 9  |-  ( QQ  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  Q
) )
8073, 79ax-mp 8 . . . . . . . 8  |-  ( abs 
o.  -  )  =  ( dist `  Q )
8177, 80xmsxmet2 18442 . . . . . . 7  |-  ( Q  e.  * MetSp  ->  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  e.  ( * Met `  QQ ) )
8276, 81mp1i 12 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  e.  ( * Met `  QQ ) )
83 ngpxms 18601 . . . . . . . . 9  |-  ( R  e. NrmGrp  ->  R  e.  * MetSp )
8439, 42, 833syl 19 . . . . . . . 8  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e.  * MetSp )
85843ad2ant1 978 . . . . . . 7  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e.  *
MetSp )
865, 45xmsxmet2 18442 . . . . . . 7  |-  ( R  e.  * MetSp  ->  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  e.  ( * Met `  ( Base `  R
) ) )
8785, 86syl 16 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  e.  ( * Met `  ( Base `  R
) ) )
8826a1i 11 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  0  e.  QQ )
89 qqhcn.j . . . . . . . . 9  |-  J  =  ( TopOpen `  Q )
9080reseq1i 5101 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  =  ( ( dist `  Q
)  |`  ( QQ  X.  QQ ) )
9189, 77, 90xmstopn 18434 . . . . . . . 8  |-  ( Q  e.  * MetSp  ->  J  =  ( MetOpen `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) )
9276, 91ax-mp 8 . . . . . . 7  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) )
93 eqid 2404 . . . . . . 7  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) )  =  (
MetOpen `  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) )
9492, 93metcnp 18524 . . . . . 6  |-  ( ( ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) )  e.  ( * Met `  QQ )  /\  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  e.  ( * Met `  ( Base `  R
) )  /\  0  e.  QQ )  ->  (
(QQHom `  R )  e.  ( ( J  CnP  ( MetOpen `  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) ) `  0
)  <->  ( (QQHom `  R ) : QQ --> ( Base `  R )  /\  A. e  e.  RR+  E. d  e.  RR+  A. q  e.  QQ  ( ( 0 ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  d  ->  ( ( (QQHom `  R ) `  0
) ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) ( (QQHom `  R ) `  q ) )  < 
e ) ) ) )
9582, 87, 88, 94syl3anc 1184 . . . . 5  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R )  e.  ( ( J  CnP  ( MetOpen
`  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) ) ) `  0 )  <-> 
( (QQHom `  R
) : QQ --> ( Base `  R )  /\  A. e  e.  RR+  E. d  e.  RR+  A. q  e.  QQ  ( ( 0 ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  d  ->  ( ( (QQHom `  R ) `  0
) ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) ( (QQHom `  R ) `  q ) )  < 
e ) ) ) )
969, 70, 95mpbir2and 889 . . . 4  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( ( J  CnP  ( MetOpen `  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ) ) ` 
0 ) )
97 qqhcn.k . . . . . . . 8  |-  K  =  ( TopOpen `  R )
98 eqid 2404 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  =  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) )
9997, 5, 98xmstopn 18434 . . . . . . 7  |-  ( R  e.  * MetSp  ->  K  =  ( MetOpen `  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
10085, 99syl 16 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  K  =  ( MetOpen `  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
101100oveq2d 6056 . . . . 5  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  ( J  CnP  K )  =  ( J  CnP  ( MetOpen `  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) ) ) )
102101fveq1d 5689 . . . 4  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  ( ( J  CnP  K ) ` 
0 )  =  ( ( J  CnP  ( MetOpen
`  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) ) ) `  0 ) )
10396, 102eleqtrrd 2481 . . 3  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( ( J  CnP  K ) `
 0 ) )
104 cnfldtgp 18852 . . . . . 6  |-fld  e.  TopGrp
105 qsubdrg 16706 . . . . . . . 8  |-  ( QQ  e.  (SubRing ` fld )  /\  (flds  QQ )  e.  DivRing )
106105simpli 445 . . . . . . 7  |-  QQ  e.  (SubRing ` fld )
107 subrgsubg 15829 . . . . . . 7  |-  ( QQ  e.  (SubRing ` fld )  ->  QQ  e.  (SubGrp ` fld ) )
108106, 107ax-mp 8 . . . . . 6  |-  QQ  e.  (SubGrp ` fld )
10971subgtgp 18088 . . . . . 6  |-  ( (fld  e. 
TopGrp  /\  QQ  e.  (SubGrp ` fld ) )  ->  Q  e.  TopGrp )
110104, 108, 109mp2an 654 . . . . 5  |-  Q  e. 
TopGrp
111 tgptmd 18062 . . . . 5  |-  ( Q  e.  TopGrp  ->  Q  e. TopMnd )
112110, 111mp1i 12 . . . 4  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  Q  e. TopMnd )
113 nrgtrg 18678 . . . . 5  |-  ( R  e. NrmRing  ->  R  e.  TopRing )
114 trgtmd2 18151 . . . . 5  |-  ( R  e.  TopRing  ->  R  e. TopMnd )
11540, 113, 1143syl 19 . . . 4  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e. TopMnd )
1165, 6, 7, 71qqhghm 24325 . . . . 5  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )
1173, 4, 116syl2anc 643 . . . 4  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )
11877, 89, 97ghmcnp 18097 . . . 4  |-  ( ( Q  e. TopMnd  /\  R  e. TopMnd  /\  (QQHom `  R )  e.  ( Q  GrpHom  R ) )  ->  ( (QQHom `  R )  e.  ( ( J  CnP  K
) `  0 )  <->  ( 0  e.  QQ  /\  (QQHom `  R )  e.  ( J  Cn  K
) ) ) )
119112, 115, 117, 118syl3anc 1184 . . 3  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R )  e.  ( ( J  CnP  K
) `  0 )  <->  ( 0  e.  QQ  /\  (QQHom `  R )  e.  ( J  Cn  K
) ) ) )
120103, 119mpbid 202 . 2  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  ( 0  e.  QQ  /\  (QQHom `  R )  e.  ( J  Cn  K ) ) )
121120simprd 450 1  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( J  Cn  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    i^i cin 3279   class class class wbr 4172    X. cxp 4835    |` cres 4839    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946    < clt 9076    - cmin 9247   -ucneg 9248   ZZcz 10238   QQcq 10530   RR+crp 10568   abscabs 11994   Basecbs 13424   ↾s cress 13425   distcds 13493   TopOpenctopn 13604   0gc0g 13678   Grpcgrp 14640   -gcsg 14643  SubGrpcsubg 14893    GrpHom cghm 14958  /rcdvr 15742   DivRingcdr 15790  SubRingcsubrg 15819   * Metcxmt 16641   MetOpencmopn 16646  ℂfldccnfld 16658   ZRHomczrh 16733   ZModczlm 16734  chrcchr 16735    Cn ccn 17242    CnP ccnp 17243  TopMndctmd 18053   TopGrpctgp 18054   TopRingctrg 18138   * MetSpcxme 18300   normcnm 18577  NrmGrpcngp 18578  NrmRingcnrg 18580  NrmModcnlm 18581  QQHomcqqh 24309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-numer 13082  df-denom 13083  df-gz 13253  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-plusf 14646  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-od 15122  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774  df-drng 15792  df-subrg 15821  df-abv 15860  df-lmod 15907  df-scaf 15908  df-sra 16199  df-rgmod 16200  df-nzr 16284  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-zrh 16737  df-zlm 16738  df-chr 16739  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cn 17245  df-cnp 17246  df-tx 17547  df-hmeo 17740  df-tmd 18055  df-tgp 18056  df-trg 18142  df-xms 18303  df-ms 18304  df-tms 18305  df-nm 18583  df-ngp 18584  df-nrg 18586  df-nlm 18587  df-qqh 24310
  Copyright terms: Public domain W3C validator