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Theorem qqhcn 24377
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 3564 . . . . . . . 8  |-  (NrmRing  i^i  DivRing )  C_  DivRing
21sseli 3346 . . . . . . 7  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e.  DivRing )
323ad2ant1 979 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e.  DivRing )
4 simp3 960 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (chr `  R
)  =  0 )
5 eqid 2438 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2438 . . . . . . 7  |-  (/r `  R
)  =  (/r `  R
)
7 eqid 2438 . . . . . . 7  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
85, 6, 7qqhf 24372 . . . . . 6  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> ( Base `  R ) )
93, 4, 8syl2anc 644 . . . . 5  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> ( Base `  R ) )
10 simpr 449 . . . . . . 7  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  -> 
e  e.  RR+ )
11 qsscn 10587 . . . . . . . . . . . . . 14  |-  QQ  C_  CC
12 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  q  e.  QQ )
1311, 12sseldi 3348 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  q  e.  CC )
14 0cn 9086 . . . . . . . . . . . . . . 15  |-  0  e.  CC
15 eqid 2438 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1615cnmetdval 18807 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  CC  /\  q  e.  CC )  ->  ( 0 ( abs 
o.  -  ) q
)  =  ( abs `  ( 0  -  q
) ) )
1714, 16mpan 653 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  (
0 ( abs  o.  -  ) q )  =  ( abs `  (
0  -  q ) ) )
18 df-neg 9296 . . . . . . . . . . . . . . . 16  |-  -u q  =  ( 0  -  q )
1918fveq2i 5733 . . . . . . . . . . . . . . 15  |-  ( abs `  -u q )  =  ( abs `  (
0  -  q ) )
2019a1i 11 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  ( abs `  -u q )  =  ( abs `  (
0  -  q ) ) )
21 absneg 12084 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  ( abs `  -u q )  =  ( abs `  q
) )
2217, 20, 213eqtr2d 2476 . . . . . . . . . . . . 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 10583 . . . . . . . . . . . . . . 15  |-  ZZ  C_  QQ
25 0z 10295 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
2624, 25sselii 3347 . . . . . . . . . . . . . 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 6216 . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . 14  |-  ( norm `  R )  =  (
norm `  R )
30 qqhcn.z . . . . . . . . . . . . . 14  |-  Z  =  ( ZMod `  R
)
3129, 30qqhnm 24376 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  q  e.  QQ )  ->  ( ( norm `  R
) `  ( (QQHom `  R ) `  q
) )  =  ( abs `  q ) )
3231adantlr 697 . . . . . . . . . . . 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 2480 . . . . . . . . . . 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 708 . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . 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 6216 . . . . . . . . . . . 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 3563 . . . . . . . . . . . . . . . . 17  |-  (NrmRing  i^i  DivRing )  C_ NrmRing
3938sseli 3346 . . . . . . . . . . . . . . . 16  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e. NrmRing )
40393ad2ant1 979 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e. NrmRing )
4140ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e. NrmRing )
42 nrgngp 18700 . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . 14  |-  ( -g `  R )  =  (
-g `  R )
45 eqid 2438 . . . . . . . . . . . . . 14  |-  ( dist `  R )  =  (
dist `  R )
4629, 5, 44, 45ngpdsr 18653 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  R  e.  DivRing )
494ad2antrr 708 . . . . . . . . . . . . . . . 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 24370 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R ) `  0
)  =  ( 0g
`  R ) )
5148, 49, 50syl2anc 644 . . . . . . . . . . . . . . 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 6099 . . . . . . . . . . . . . 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 18648 . . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  R )  =  ( 0g `  R
)
565, 55, 44grpsubid1 14876 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  Grp  /\  ( (QQHom `  R ) `  q )  e.  (
Base `  R )
)  ->  ( (
(QQHom `  R ) `  q ) ( -g `  R ) ( 0g
`  R ) )  =  ( (QQHom `  R ) `  q
) )
5754, 36, 56syl2anc 644 . . . . . . . . . . . . . 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 2470 . . . . . . . . . . . . 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 5734 . . . . . . . . . . . 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 2474 . . . . . . . . . . 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 2473 . . . . . . . . . 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 4224 . . . . . . . . 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 200 . . . . . . . 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 2791 . . . . . . 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 4218 . . . . . . . . . 10  |-  ( d  =  e  ->  (
( 0 ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  <->  ( 0 ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  e
) )
6665imbi1d 310 . . . . . . . . 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 2727 . . . . . . . 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 3054 . . . . . . 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 644 . . . . . 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 2791 . . . . 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 18813 . . . . . . . . 9  |-fld  e.  * MetSp
73 qex 10588 . . . . . . . . 9  |-  QQ  e.  _V
74 ressxms 18557 . . . . . . . . 9  |-  ( (fld  e. 
* MetSp  /\  QQ  e.  _V )  ->  (flds  QQ )  e.  * MetSp )
7572, 73, 74mp2an 655 . . . . . . . 8  |-  (flds  QQ )  e.  * MetSp
7671, 75eqeltri 2508 . . . . . . 7  |-  Q  e. 
* MetSp
7771qrngbas 21315 . . . . . . . 8  |-  QQ  =  ( Base `  Q )
78 cnfldds 16715 . . . . . . . . . 10  |-  ( abs 
o.  -  )  =  ( dist ` fld )
7971, 78ressds 13643 . . . . . . . . 9  |-  ( QQ  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  Q
) )
8073, 79ax-mp 8 . . . . . . . 8  |-  ( abs 
o.  -  )  =  ( dist `  Q )
8177, 80xmsxmet2 18491 . . . . . . 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 18650 . . . . . . . . 9  |-  ( R  e. NrmGrp  ->  R  e.  * MetSp )
8439, 42, 833syl 19 . . . . . . . 8  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e.  * MetSp )
85843ad2ant1 979 . . . . . . 7  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e.  *
MetSp )
865, 45xmsxmet2 18491 . . . . . . 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 5144 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  =  ( ( dist `  Q
)  |`  ( QQ  X.  QQ ) )
9189, 77, 90xmstopn 18483 . . . . . . . 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 2438 . . . . . . 7  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) )  =  (
MetOpen `  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) )
9492, 93metcnp 18573 . . . . . 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 1185 . . . . 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 890 . . . 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 2438 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  =  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) )
9997, 5, 98xmstopn 18483 . . . . . . 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 6099 . . . . 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 5732 . . . 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 2515 . . 3  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( ( J  CnP  K ) `
 0 ) )
104 cnfldtgp 18901 . . . . . 6  |-fld  e.  TopGrp
105 qsubdrg 16753 . . . . . . . 8  |-  ( QQ  e.  (SubRing ` fld )  /\  (flds  QQ )  e.  DivRing )
106105simpli 446 . . . . . . 7  |-  QQ  e.  (SubRing ` fld )
107 subrgsubg 15876 . . . . . . 7  |-  ( QQ  e.  (SubRing ` fld )  ->  QQ  e.  (SubGrp ` fld ) )
108106, 107ax-mp 8 . . . . . 6  |-  QQ  e.  (SubGrp ` fld )
10971subgtgp 18137 . . . . . 6  |-  ( (fld  e. 
TopGrp  /\  QQ  e.  (SubGrp ` fld ) )  ->  Q  e.  TopGrp )
110104, 108, 109mp2an 655 . . . . 5  |-  Q  e. 
TopGrp
111 tgptmd 18111 . . . . 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 18727 . . . . 5  |-  ( R  e. NrmRing  ->  R  e.  TopRing )
114 trgtmd2 18200 . . . . 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 24374 . . . . 5  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )
1173, 4, 116syl2anc 644 . . . 4  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )
11877, 89, 97ghmcnp 18146 . . . 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 1185 . . 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 203 . 2  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  ( 0  e.  QQ  /\  (QQHom `  R )  e.  ( J  Cn  K ) ) )
121120simprd 451 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    i^i cin 3321   class class class wbr 4214    X. cxp 4878    |` cres 4882    o. ccom 4884   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992    < clt 9122    - cmin 9293   -ucneg 9294   ZZcz 10284   QQcq 10576   RR+crp 10614   abscabs 12041   Basecbs 13471   ↾s cress 13472   distcds 13540   TopOpenctopn 13651   0gc0g 13725   Grpcgrp 14687   -gcsg 14690  SubGrpcsubg 14940    GrpHom cghm 15005  /rcdvr 15789   DivRingcdr 15837  SubRingcsubrg 15866   * Metcxmt 16688   MetOpencmopn 16693  ℂfldccnfld 16705   ZRHomczrh 16780   ZModczlm 16781  chrcchr 16782    Cn ccn 17290    CnP ccnp 17291  TopMndctmd 18102   TopGrpctgp 18103   TopRingctrg 18187   * MetSpcxme 18349   normcnm 18626  NrmGrpcngp 18627  NrmRingcnrg 18629  NrmModcnlm 18630  QQHomcqqh 24358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-gcd 13009  df-numer 13129  df-denom 13130  df-gz 13300  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-plusf 14693  df-mhm 14740  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mulg 14817  df-subg 14943  df-ghm 15006  df-cntz 15118  df-od 15169  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-dvr 15790  df-rnghom 15821  df-drng 15839  df-subrg 15868  df-abv 15907  df-lmod 15954  df-scaf 15955  df-sra 16246  df-rgmod 16247  df-nzr 16331  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-cnfld 16706  df-zrh 16784  df-zlm 16785  df-chr 16786  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cn 17293  df-cnp 17294  df-tx 17596  df-hmeo 17789  df-tmd 18104  df-tgp 18105  df-trg 18191  df-xms 18352  df-ms 18353  df-tms 18354  df-nm 18632  df-ngp 18633  df-nrg 18635  df-nlm 18636  df-qqh 24359
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