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Theorem qqhcn 26435
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 3586 . . . . . . . 8  |-  (NrmRing  i^i  DivRing )  C_  DivRing
21sseli 3367 . . . . . . 7  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e.  DivRing )
323ad2ant1 1009 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e.  DivRing )
4 simp3 990 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (chr `  R
)  =  0 )
5 eqid 2443 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2443 . . . . . . 7  |-  (/r `  R
)  =  (/r `  R
)
7 eqid 2443 . . . . . . 7  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
85, 6, 7qqhf 26430 . . . . . 6  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> ( Base `  R ) )
93, 4, 8syl2anc 661 . . . . 5  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> ( Base `  R ) )
10 simpr 461 . . . . . . 7  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  -> 
e  e.  RR+ )
11 qsscn 10979 . . . . . . . . . . . . . 14  |-  QQ  C_  CC
12 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  q  e.  QQ )
1311, 12sseldi 3369 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  q  e.  CC )
14 0cn 9393 . . . . . . . . . . . . . . 15  |-  0  e.  CC
15 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1615cnmetdval 20365 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  CC  /\  q  e.  CC )  ->  ( 0 ( abs 
o.  -  ) q
)  =  ( abs `  ( 0  -  q
) ) )
1714, 16mpan 670 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  (
0 ( abs  o.  -  ) q )  =  ( abs `  (
0  -  q ) ) )
18 df-neg 9613 . . . . . . . . . . . . . . . 16  |-  -u q  =  ( 0  -  q )
1918fveq2i 5709 . . . . . . . . . . . . . . 15  |-  ( abs `  -u q )  =  ( abs `  (
0  -  q ) )
2019a1i 11 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  ( abs `  -u q )  =  ( abs `  (
0  -  q ) ) )
21 absneg 12781 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  ( abs `  -u q )  =  ( abs `  q
) )
2217, 20, 213eqtr2d 2481 . . . . . . . . . . . . 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 10975 . . . . . . . . . . . . . . 15  |-  ZZ  C_  QQ
25 0z 10672 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
2624, 25sselii 3368 . . . . . . . . . . . . . 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 6246 . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . 14  |-  ( norm `  R )  =  (
norm `  R )
30 qqhcn.z . . . . . . . . . . . . . 14  |-  Z  =  ( ZMod `  R
)
3129, 30qqhnm 26434 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  ( ( norm `  R
) `  ( (QQHom `  R ) `  q
) )  =  ( abs `  q ) )
3231adantlr 714 . . . . . . . . . . . 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 2485 . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 5859 . . . . . . . . . . . . 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 5859 . . . . . . . . . . . . 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 6246 . . . . . . . . . . . 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 3585 . . . . . . . . . . . . . . . . 17  |-  (NrmRing  i^i  DivRing )  C_ NrmRing
3938sseli 3367 . . . . . . . . . . . . . . . 16  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e. NrmRing )
40393ad2ant1 1009 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e. NrmRing )
4140ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e. NrmRing )
42 nrgngp 20258 . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . 14  |-  ( -g `  R )  =  (
-g `  R )
45 eqid 2443 . . . . . . . . . . . . . 14  |-  ( dist `  R )  =  (
dist `  R )
4629, 5, 44, 45ngpdsr 20211 . . . . . . . . . . . . 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 1218 . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e.  DivRing )
494ad2antrr 725 . . . . . . . . . . . . . . . 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 26428 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R ) `  0
)  =  ( 0g
`  R ) )
5148, 49, 50syl2anc 661 . . . . . . . . . . . . . . 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 6122 . . . . . . . . . . . . . 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 20206 . . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  R )  =  ( 0g `  R
)
565, 55, 44grpsubid1 15626 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Grp  /\  ( (QQHom `  R ) `  q )  e.  (
Base `  R )
)  ->  ( (
(QQHom `  R ) `  q ) ( -g `  R ) ( 0g
`  R ) )  =  ( (QQHom `  R ) `  q
) )
5754, 36, 56syl2anc 661 . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . 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 5710 . . . . . . . . . . . 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 2479 . . . . . . . . . . 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 2478 . . . . . . . . . 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 4317 . . . . . . . . 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 207 . . . . . . . 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 2814 . . . . . . 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 4311 . . . . . . . . . 10  |-  ( d  =  e  ->  (
( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  <->  ( 0 ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  e
) )
6665imbi1d 317 . . . . . . . . 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 2750 . . . . . . . 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 3088 . . . . . . 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 661 . . . . . 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 2814 . . . . 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 20371 . . . . . . . . 9  |-fld  e.  *MetSp
73 qex 10980 . . . . . . . . 9  |-  QQ  e.  _V
74 ressxms 20115 . . . . . . . . 9  |-  ( (fld  e. 
*MetSp  /\  QQ  e.  _V )  ->  (flds  QQ )  e.  *MetSp )
7572, 73, 74mp2an 672 . . . . . . . 8  |-  (flds  QQ )  e.  *MetSp
7671, 75eqeltri 2513 . . . . . . 7  |-  Q  e. 
*MetSp
7771qrngbas 22883 . . . . . . . 8  |-  QQ  =  ( Base `  Q )
78 cnfldds 17843 . . . . . . . . . 10  |-  ( abs 
o.  -  )  =  ( dist ` fld )
7971, 78ressds 14367 . . . . . . . . 9  |-  ( QQ  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  Q
) )
8073, 79ax-mp 5 . . . . . . . 8  |-  ( abs 
o.  -  )  =  ( dist `  Q )
8177, 80xmsxmet2 20049 . . . . . . 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 20208 . . . . . . . . 9  |-  ( R  e. NrmGrp  ->  R  e.  *MetSp )
8439, 42, 833syl 20 . . . . . . . 8  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e.  *MetSp )
85843ad2ant1 1009 . . . . . . 7  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e.  *MetSp )
865, 45xmsxmet2 20049 . . . . . . 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 5121 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  =  ( ( dist `  Q
)  |`  ( QQ  X.  QQ ) )
9189, 77, 90xmstopn 20041 . . . . . . . 8  |-  ( Q  e.  *MetSp  ->  J  =  ( MetOpen `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) )
9276, 91ax-mp 5 . . . . . . 7  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) )
93 eqid 2443 . . . . . . 7  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) )  =  (
MetOpen `  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) )
9492, 93metcnp 20131 . . . . . 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 1218 . . . . 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 913 . . . 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 2443 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  =  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) )
9997, 5, 98xmstopn 20041 . . . . . . 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 6122 . . . . 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 5708 . . . 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 2520 . . 3  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( ( J  CnP  K ) `
 0 ) )
104 cnfldtgp 20460 . . . . . 6  |-fld  e.  TopGrp
105 qsubdrg 17880 . . . . . . . 8  |-  ( QQ  e.  (SubRing ` fld )  /\  (flds  QQ )  e.  DivRing )
106105simpli 458 . . . . . . 7  |-  QQ  e.  (SubRing ` fld )
107 subrgsubg 16886 . . . . . . 7  |-  ( QQ  e.  (SubRing ` fld )  ->  QQ  e.  (SubGrp ` fld ) )
108106, 107ax-mp 5 . . . . . 6  |-  QQ  e.  (SubGrp ` fld )
10971subgtgp 19691 . . . . . 6  |-  ( (fld  e. 
TopGrp  /\  QQ  e.  (SubGrp ` fld ) )  ->  Q  e.  TopGrp )
110104, 108, 109mp2an 672 . . . . 5  |-  Q  e. 
TopGrp
111 tgptmd 19665 . . . . 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 20285 . . . . 5  |-  ( R  e. NrmRing  ->  R  e.  TopRing )
114 trgtmd2 19758 . . . . 5  |-  ( R  e.  TopRing  ->  R  e. TopMnd )
11540, 113, 1143syl 20 . . . 4  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e. TopMnd )
1165, 6, 7, 71qqhghm 26432 . . . . 5  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )
1173, 4, 116syl2anc 661 . . . 4  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )
11877, 89, 97ghmcnp 19700 . . . 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 1218 . . 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 210 . 2  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  ( 0  e.  QQ  /\  (QQHom `  R )  e.  ( J  Cn  K ) ) )
121120simprd 463 1  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2730   E.wrex 2731   _Vcvv 2987    i^i cin 3342   class class class wbr 4307    X. cxp 4853    |` cres 4857    o. ccom 4859   -->wf 5429   ` cfv 5433  (class class class)co 6106   CCcc 9295   0cc0 9297    < clt 9433    - cmin 9610   -ucneg 9611   ZZcz 10661   QQcq 10968   RR+crp 11006   abscabs 12738   Basecbs 14189   ↾s cress 14190   distcds 14262   TopOpenctopn 14375   0gc0g 14393   Grpcgrp 15425   -gcsg 15428  SubGrpcsubg 15690    GrpHom cghm 15759  /rcdvr 16789   DivRingcdr 16847  SubRingcsubrg 16876   *Metcxmt 17816   MetOpencmopn 17821  ℂfldccnfld 17833   ZRHomczrh 17946   ZModczlm 17947  chrcchr 17948    Cn ccn 18843    CnP ccnp 18844  TopMndctmd 19656   TopGrpctgp 19657   TopRingctrg 19745   *MetSpcxme 19907   normcnm 20184  NrmGrpcngp 20185  NrmRingcnrg 20187  NrmModcnlm 20188  QQHomcqqh 26416
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 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375  ax-addf 9376  ax-mulf 9377
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 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-of 6335  df-om 6492  df-1st 6592  df-2nd 6593  df-supp 6706  df-tpos 6760  df-recs 6847  df-rdg 6881  df-1o 6935  df-2o 6936  df-oadd 6939  df-er 7116  df-map 7231  df-ixp 7279  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-fsupp 7636  df-fi 7676  df-sup 7706  df-oi 7739  df-card 8124  df-cda 8352  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-q 10969  df-rp 11007  df-xneg 11104  df-xadd 11105  df-xmul 11106  df-ico 11321  df-icc 11322  df-fz 11453  df-fzo 11564  df-fl 11657  df-mod 11724  df-seq 11822  df-exp 11881  df-hash 12119  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-dvds 13551  df-gcd 13706  df-numer 13828  df-denom 13829  df-gz 14006  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-starv 14268  df-sca 14269  df-vsca 14270  df-ip 14271  df-tset 14272  df-ple 14273  df-ds 14275  df-unif 14276  df-hom 14277  df-cco 14278  df-rest 14376  df-topn 14377  df-0g 14395  df-gsum 14396  df-topgen 14397  df-pt 14398  df-prds 14401  df-xrs 14455  df-qtop 14460  df-imas 14461  df-xps 14463  df-mre 14539  df-mrc 14540  df-acs 14542  df-mnd 15430  df-plusf 15431  df-mhm 15479  df-submnd 15480  df-grp 15560  df-minusg 15561  df-sbg 15562  df-mulg 15563  df-subg 15693  df-ghm 15760  df-cntz 15850  df-od 16047  df-cmn 16294  df-abl 16295  df-mgp 16607  df-ur 16619  df-rng 16662  df-cring 16663  df-oppr 16730  df-dvdsr 16748  df-unit 16749  df-invr 16779  df-dvr 16790  df-rnghom 16821  df-drng 16849  df-subrg 16878  df-abv 16917  df-lmod 16965  df-scaf 16966  df-sra 17268  df-rgmod 17269  df-nzr 17355  df-psmet 17824  df-xmet 17825  df-met 17826  df-bl 17827  df-mopn 17828  df-cnfld 17834  df-zring 17899  df-zrh 17950  df-zlm 17951  df-chr 17952  df-top 18518  df-bases 18520  df-topon 18521  df-topsp 18522  df-cn 18846  df-cnp 18847  df-tx 19150  df-hmeo 19343  df-tmd 19658  df-tgp 19659  df-trg 19749  df-xms 19910  df-ms 19911  df-tms 19912  df-nm 20190  df-ngp 20191  df-nrg 20193  df-nlm 20194  df-qqh 26417
This theorem is referenced by: (None)
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