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Theorem qqhcn 28844
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 3665 . . . . . . . 8  |-  (NrmRing  i^i  DivRing )  C_  DivRing
21sseli 3440 . . . . . . 7  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e.  DivRing )
323ad2ant1 1035 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e.  DivRing )
4 simp3 1016 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (chr `  R
)  =  0 )
5 eqid 2462 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2462 . . . . . . 7  |-  (/r `  R
)  =  (/r `  R
)
7 eqid 2462 . . . . . . 7  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
85, 6, 7qqhf 28839 . . . . . 6  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> ( Base `  R ) )
93, 4, 8syl2anc 671 . . . . 5  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> ( Base `  R ) )
10 simpr 467 . . . . . . 7  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  -> 
e  e.  RR+ )
11 qsscn 11304 . . . . . . . . . . . . . 14  |-  QQ  C_  CC
12 simpr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  q  e.  QQ )
1311, 12sseldi 3442 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  q  e.  CC )
14 0cn 9661 . . . . . . . . . . . . . . 15  |-  0  e.  CC
15 eqid 2462 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1615cnmetdval 21840 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  CC  /\  q  e.  CC )  ->  ( 0 ( abs 
o.  -  ) q
)  =  ( abs `  ( 0  -  q
) ) )
1714, 16mpan 681 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  (
0 ( abs  o.  -  ) q )  =  ( abs `  (
0  -  q ) ) )
18 df-neg 9889 . . . . . . . . . . . . . . . 16  |-  -u q  =  ( 0  -  q )
1918fveq2i 5891 . . . . . . . . . . . . . . 15  |-  ( abs `  -u q )  =  ( abs `  (
0  -  q ) )
2019a1i 11 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  ( abs `  -u q )  =  ( abs `  (
0  -  q ) ) )
21 absneg 13389 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  ( abs `  -u q )  =  ( abs `  q
) )
2217, 20, 213eqtr2d 2502 . . . . . . . . . . . . 13  |-  ( q  e.  CC  ->  (
0 ( abs  o.  -  ) q )  =  ( abs `  q
) )
2313, 22syl 17 . . . . . . . . . . . 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 11300 . . . . . . . . . . . . . . 15  |-  ZZ  C_  QQ
25 0z 10977 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
2624, 25sselii 3441 . . . . . . . . . . . . . 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 6464 . . . . . . . . . . . 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 2462 . . . . . . . . . . . . . 14  |-  ( norm `  R )  =  (
norm `  R )
30 qqhcn.z . . . . . . . . . . . . . 14  |-  Z  =  ( ZMod `  R
)
3129, 30qqhnm 28843 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  ( ( norm `  R
) `  ( (QQHom `  R ) `  q
) )  =  ( abs `  q ) )
3231adantlr 726 . . . . . . . . . . . 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 2506 . . . . . . . . . . 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 737 . . . . . . . . . . . . . 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 6046 . . . . . . . . . . . . 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 6046 . . . . . . . . . . . . 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 6464 . . . . . . . . . . . 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 3664 . . . . . . . . . . . . . . . . 17  |-  (NrmRing  i^i  DivRing )  C_ NrmRing
3938sseli 3440 . . . . . . . . . . . . . . . 16  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e. NrmRing )
40393ad2ant1 1035 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e. NrmRing )
4140ad2antrr 737 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e. NrmRing )
42 nrgngp 21714 . . . . . . . . . . . . . 14  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
4341, 42syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e. NrmGrp )
44 eqid 2462 . . . . . . . . . . . . . 14  |-  ( -g `  R )  =  (
-g `  R )
45 eqid 2462 . . . . . . . . . . . . . 14  |-  ( dist `  R )  =  (
dist `  R )
4629, 5, 44, 45ngpdsr 21667 . . . . . . . . . . . . 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 1276 . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e.  DivRing )
494ad2antrr 737 . . . . . . . . . . . . . . . 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 28837 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R ) `  0
)  =  ( 0g
`  R ) )
5148, 49, 50syl2anc 671 . . . . . . . . . . . . . . 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 6331 . . . . . . . . . . . . . 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 21662 . . . . . . . . . . . . . . . 16  |-  ( R  e. NrmGrp  ->  R  e.  Grp )
5443, 53syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e.  Grp )
55 eqid 2462 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  R )  =  ( 0g `  R
)
565, 55, 44grpsubid1 16788 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Grp  /\  ( (QQHom `  R ) `  q )  e.  (
Base `  R )
)  ->  ( (
(QQHom `  R ) `  q ) ( -g `  R ) ( 0g
`  R ) )  =  ( (QQHom `  R ) `  q
) )
5754, 36, 56syl2anc 671 . . . . . . . . . . . . . 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 2496 . . . . . . . . . . . . 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 5892 . . . . . . . . . . . 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 2500 . . . . . . . . . . 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 2499 . . . . . . . . . 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 4426 . . . . . . . . 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 212 . . . . . . . 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 4420 . . . . . . . . . 10  |-  ( d  =  e  ->  (
( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  <->  ( 0 ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  e
) )
6665imbi1d 323 . . . . . . . . 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 2839 . . . . . . . 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 3162 . . . . . . 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 671 . . . . . 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 21846 . . . . . . . . 9  |-fld  e.  *MetSp
73 qex 11305 . . . . . . . . 9  |-  QQ  e.  _V
74 ressxms 21589 . . . . . . . . 9  |-  ( (fld  e. 
*MetSp  /\  QQ  e.  _V )  ->  (flds  QQ )  e.  *MetSp )
7572, 73, 74mp2an 683 . . . . . . . 8  |-  (flds  QQ )  e.  *MetSp
7671, 75eqeltri 2536 . . . . . . 7  |-  Q  e. 
*MetSp
7771qrngbas 24506 . . . . . . . 8  |-  QQ  =  ( Base `  Q )
78 cnfldds 19029 . . . . . . . . . 10  |-  ( abs 
o.  -  )  =  ( dist ` fld )
7971, 78ressds 15360 . . . . . . . . 9  |-  ( QQ  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  Q
) )
8073, 79ax-mp 5 . . . . . . . 8  |-  ( abs 
o.  -  )  =  ( dist `  Q )
8177, 80xmsxmet2 21523 . . . . . . 7  |-  ( Q  e.  *MetSp  ->  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  e.  ( *Met `  QQ ) )
8276, 81mp1i 13 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  e.  ( *Met `  QQ ) )
83 ngpxms 21664 . . . . . . . . 9  |-  ( R  e. NrmGrp  ->  R  e.  *MetSp )
8439, 42, 833syl 18 . . . . . . . 8  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e.  *MetSp )
85843ad2ant1 1035 . . . . . . 7  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e.  *MetSp )
865, 45xmsxmet2 21523 . . . . . . 7  |-  ( R  e.  *MetSp  ->  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  e.  ( *Met `  ( Base `  R
) ) )
8785, 86syl 17 . . . . . 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 5120 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  =  ( ( dist `  Q
)  |`  ( QQ  X.  QQ ) )
9189, 77, 90xmstopn 21515 . . . . . . . 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 2462 . . . . . . 7  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) )  =  (
MetOpen `  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) )
9492, 93metcnp 21605 . . . . . 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 1276 . . . . 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 938 . . . 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 2462 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  =  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) )
9997, 5, 98xmstopn 21515 . . . . . . 7  |-  ( R  e.  *MetSp  ->  K  =  ( MetOpen `  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
10085, 99syl 17 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  K  =  ( MetOpen `  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
101100oveq2d 6331 . . . . 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 5890 . . . 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 2543 . . 3  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( ( J  CnP  K ) `
 0 ) )
104 cnfldtgp 21950 . . . . . 6  |-fld  e.  TopGrp
105 qsubdrg 19069 . . . . . . . 8  |-  ( QQ  e.  (SubRing ` fld )  /\  (flds  QQ )  e.  DivRing )
106105simpli 464 . . . . . . 7  |-  QQ  e.  (SubRing ` fld )
107 subrgsubg 18063 . . . . . . 7  |-  ( QQ  e.  (SubRing ` fld )  ->  QQ  e.  (SubGrp ` fld ) )
108106, 107ax-mp 5 . . . . . 6  |-  QQ  e.  (SubGrp ` fld )
10971subgtgp 21169 . . . . . 6  |-  ( (fld  e. 
TopGrp  /\  QQ  e.  (SubGrp ` fld ) )  ->  Q  e.  TopGrp )
110104, 108, 109mp2an 683 . . . . 5  |-  Q  e. 
TopGrp
111 tgptmd 21143 . . . . 5  |-  ( Q  e.  TopGrp  ->  Q  e. TopMnd )
112110, 111mp1i 13 . . . 4  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  Q  e. TopMnd )
113 nrgtrg 21741 . . . . 5  |-  ( R  e. NrmRing  ->  R  e.  TopRing )
114 trgtmd2 21232 . . . . 5  |-  ( R  e.  TopRing  ->  R  e. TopMnd )
11540, 113, 1143syl 18 . . . 4  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e. TopMnd )
1165, 6, 7, 71qqhghm 28841 . . . . 5  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )
1173, 4, 116syl2anc 671 . . . 4  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )
11877, 89, 97ghmcnp 21178 . . . 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 1276 . . 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 215 . 2  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  ( 0  e.  QQ  /\  (QQHom `  R )  e.  ( J  Cn  K ) ) )
121120simprd 469 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   A.wral 2749   E.wrex 2750   _Vcvv 3057    i^i cin 3415   class class class wbr 4416    X. cxp 4851    |` cres 4855    o. ccom 4857   -->wf 5597   ` cfv 5601  (class class class)co 6315   CCcc 9563   0cc0 9565    < clt 9701    - cmin 9886   -ucneg 9887   ZZcz 10966   QQcq 11293   RR+crp 11331   abscabs 13346   Basecbs 15170   ↾s cress 15171   distcds 15248   TopOpenctopn 15369   0gc0g 15387   Grpcgrp 16718   -gcsg 16720  SubGrpcsubg 16860    GrpHom cghm 16929  /rcdvr 17959   DivRingcdr 18024  SubRingcsubrg 18053   *Metcxmt 19004   MetOpencmopn 19009  ℂfldccnfld 19019   ZRHomczrh 19120   ZModczlm 19121  chrcchr 19122    Cn ccn 20289    CnP ccnp 20290  TopMndctmd 21134   TopGrpctgp 21135   TopRingctrg 21219   *MetSpcxme 21381   normcnm 21640  NrmGrpcngp 21641  NrmRingcnrg 21643  NrmModcnlm 21644  QQHomcqqh 28825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643  ax-addf 9644  ax-mulf 9645
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-om 6720  df-1st 6820  df-2nd 6821  df-supp 6942  df-tpos 6999  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-er 7389  df-map 7500  df-ixp 7549  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fsupp 7910  df-fi 7951  df-sup 7982  df-inf 7983  df-oi 8051  df-card 8399  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-9 10703  df-10 10704  df-n0 10899  df-z 10967  df-dec 11081  df-uz 11189  df-q 11294  df-rp 11332  df-xneg 11438  df-xadd 11439  df-xmul 11440  df-ico 11670  df-icc 11671  df-fz 11814  df-fzo 11947  df-fl 12060  df-mod 12129  df-seq 12246  df-exp 12305  df-hash 12548  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-dvds 14355  df-gcd 14518  df-numer 14733  df-denom 14734  df-gz 14923  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-mulr 15253  df-starv 15254  df-sca 15255  df-vsca 15256  df-ip 15257  df-tset 15258  df-ple 15259  df-ds 15261  df-unif 15262  df-hom 15263  df-cco 15264  df-rest 15370  df-topn 15371  df-0g 15389  df-gsum 15390  df-topgen 15391  df-pt 15392  df-prds 15395  df-xrs 15449  df-qtop 15455  df-imas 15456  df-xps 15459  df-mre 15541  df-mrc 15542  df-acs 15544  df-plusf 16536  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-mhm 16631  df-submnd 16632  df-grp 16722  df-minusg 16723  df-sbg 16724  df-mulg 16725  df-subg 16863  df-ghm 16930  df-cntz 17020  df-od 17221  df-cmn 17481  df-abl 17482  df-mgp 17773  df-ur 17785  df-ring 17831  df-cring 17832  df-oppr 17900  df-dvdsr 17918  df-unit 17919  df-invr 17949  df-dvr 17960  df-rnghom 17992  df-drng 18026  df-subrg 18055  df-abv 18094  df-lmod 18142  df-scaf 18143  df-sra 18444  df-rgmod 18445  df-nzr 18531  df-psmet 19011  df-xmet 19012  df-met 19013  df-bl 19014  df-mopn 19015  df-cnfld 19020  df-zring 19089  df-zrh 19124  df-zlm 19125  df-chr 19126  df-top 19970  df-bases 19971  df-topon 19972  df-topsp 19973  df-cn 20292  df-cnp 20293  df-tx 20626  df-hmeo 20819  df-tmd 21136  df-tgp 21137  df-trg 21223  df-xms 21384  df-ms 21385  df-tms 21386  df-nm 21646  df-ngp 21647  df-nrg 21649  df-nlm 21650  df-qqh 28826
This theorem is referenced by:  rrhqima  28867
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