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Theorem qqhcn 28133
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 3715 . . . . . . . 8  |-  (NrmRing  i^i  DivRing )  C_  DivRing
21sseli 3495 . . . . . . 7  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e.  DivRing )
323ad2ant1 1017 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e.  DivRing )
4 simp3 998 . . . . . 6  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (chr `  R
)  =  0 )
5 eqid 2457 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
6 eqid 2457 . . . . . . 7  |-  (/r `  R
)  =  (/r `  R
)
7 eqid 2457 . . . . . . 7  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
85, 6, 7qqhf 28128 . . . . . 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 11218 . . . . . . . . . . . . . 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 3497 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  e  e.  RR+ )  /\  q  e.  QQ )  ->  q  e.  CC )
14 0cn 9605 . . . . . . . . . . . . . . 15  |-  0  e.  CC
15 eqid 2457 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1615cnmetdval 21404 . . . . . . . . . . . . . . 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 9827 . . . . . . . . . . . . . . . 16  |-  -u q  =  ( 0  -  q )
1918fveq2i 5875 . . . . . . . . . . . . . . 15  |-  ( abs `  -u q )  =  ( abs `  (
0  -  q ) )
2019a1i 11 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  ( abs `  -u q )  =  ( abs `  (
0  -  q ) ) )
21 absneg 13122 . . . . . . . . . . . . . 14  |-  ( q  e.  CC  ->  ( abs `  -u q )  =  ( abs `  q
) )
2217, 20, 213eqtr2d 2504 . . . . . . . . . . . . 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 11214 . . . . . . . . . . . . . . 15  |-  ZZ  C_  QQ
25 0z 10896 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
2624, 25sselii 3496 . . . . . . . . . . . . . 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 6442 . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . 14  |-  ( norm `  R )  =  (
norm `  R )
30 qqhcn.z . . . . . . . . . . . . . 14  |-  Z  =  ( ZMod `  R
)
3129, 30qqhnm 28132 . . . . . . . . . . . . 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 2508 . . . . . . . . . . 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 6033 . . . . . . . . . . . . 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 6033 . . . . . . . . . . . . 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 6442 . . . . . . . . . . . 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 3714 . . . . . . . . . . . . . . . . 17  |-  (NrmRing  i^i  DivRing )  C_ NrmRing
3938sseli 3495 . . . . . . . . . . . . . . . 16  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e. NrmRing )
40393ad2ant1 1017 . . . . . . . . . . . . . . 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 21297 . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . 14  |-  ( -g `  R )  =  (
-g `  R )
45 eqid 2457 . . . . . . . . . . . . . 14  |-  ( dist `  R )  =  (
dist `  R )
4629, 5, 44, 45ngpdsr 21250 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 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 28126 . . . . . . . . . . . . . . . 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 6312 . . . . . . . . . . . . . 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 21245 . . . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  R )  =  ( 0g `  R
)
565, 55, 44grpsubid1 16250 . . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . . 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 5876 . . . . . . . . . . . 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 2502 . . . . . . . . . . 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 2501 . . . . . . . . . 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 4466 . . . . . . . . 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 2871 . . . . . . 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 4460 . . . . . . . . . 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 2896 . . . . . . . 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 3210 . . . . . . 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 2871 . . . . 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 21410 . . . . . . . . 9  |-fld  e.  *MetSp
73 qex 11219 . . . . . . . . 9  |-  QQ  e.  _V
74 ressxms 21154 . . . . . . . . 9  |-  ( (fld  e. 
*MetSp  /\  QQ  e.  _V )  ->  (flds  QQ )  e.  *MetSp )
7572, 73, 74mp2an 672 . . . . . . . 8  |-  (flds  QQ )  e.  *MetSp
7671, 75eqeltri 2541 . . . . . . 7  |-  Q  e. 
*MetSp
7771qrngbas 23930 . . . . . . . 8  |-  QQ  =  ( Base `  Q )
78 cnfldds 18557 . . . . . . . . . 10  |-  ( abs 
o.  -  )  =  ( dist ` fld )
7971, 78ressds 14830 . . . . . . . . 9  |-  ( QQ  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  Q
) )
8073, 79ax-mp 5 . . . . . . . 8  |-  ( abs 
o.  -  )  =  ( dist `  Q )
8177, 80xmsxmet2 21088 . . . . . . 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 21247 . . . . . . . . 9  |-  ( R  e. NrmGrp  ->  R  e.  *MetSp )
8439, 42, 833syl 20 . . . . . . . 8  |-  ( R  e.  (NrmRing  i^i  DivRing )  ->  R  e.  *MetSp )
85843ad2ant1 1017 . . . . . . 7  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  R  e.  *MetSp )
865, 45xmsxmet2 21088 . . . . . . 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 5279 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  =  ( ( dist `  Q
)  |`  ( QQ  X.  QQ ) )
9189, 77, 90xmstopn 21080 . . . . . . . 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 2457 . . . . . . 7  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) )  =  (
MetOpen `  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) ) )
9492, 93metcnp 21170 . . . . . 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 1228 . . . . 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 922 . . . 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 2457 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  =  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) )
9997, 5, 98xmstopn 21080 . . . . . . 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 6312 . . . . 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 5874 . . . 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 2548 . . 3  |-  ( ( R  e.  (NrmRing  i^i  DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( ( J  CnP  K ) `
 0 ) )
104 cnfldtgp 21499 . . . . . 6  |-fld  e.  TopGrp
105 qsubdrg 18597 . . . . . . . 8  |-  ( QQ  e.  (SubRing ` fld )  /\  (flds  QQ )  e.  DivRing )
106105simpli 458 . . . . . . 7  |-  QQ  e.  (SubRing ` fld )
107 subrgsubg 17562 . . . . . . 7  |-  ( QQ  e.  (SubRing ` fld )  ->  QQ  e.  (SubGrp ` fld ) )
108106, 107ax-mp 5 . . . . . 6  |-  QQ  e.  (SubGrp ` fld )
10971subgtgp 20730 . . . . . 6  |-  ( (fld  e. 
TopGrp  /\  QQ  e.  (SubGrp ` fld ) )  ->  Q  e.  TopGrp )
110104, 108, 109mp2an 672 . . . . 5  |-  Q  e. 
TopGrp
111 tgptmd 20704 . . . . 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 21324 . . . . 5  |-  ( R  e. NrmRing  ->  R  e.  TopRing )
114 trgtmd2 20797 . . . . 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 28130 . . . . 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 20739 . . . 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 1228 . . 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109    i^i cin 3470   class class class wbr 4456    X. cxp 5006    |` cres 5010    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509    < clt 9645    - cmin 9824   -ucneg 9825   ZZcz 10885   QQcq 11207   RR+crp 11245   abscabs 13079   Basecbs 14644   ↾s cress 14645   distcds 14721   TopOpenctopn 14839   0gc0g 14857   Grpcgrp 16180   -gcsg 16182  SubGrpcsubg 16322    GrpHom cghm 16391  /rcdvr 17458   DivRingcdr 17523  SubRingcsubrg 17552   *Metcxmt 18530   MetOpencmopn 18535  ℂfldccnfld 18547   ZRHomczrh 18664   ZModczlm 18665  chrcchr 18666    Cn ccn 19852    CnP ccnp 19853  TopMndctmd 20695   TopGrpctgp 20696   TopRingctrg 20784   *MetSpcxme 20946   normcnm 21223  NrmGrpcngp 21224  NrmRingcnrg 21226  NrmModcnlm 21227  QQHomcqqh 28114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-numer 14280  df-denom 14281  df-gz 14460  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-plusf 15998  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-ghm 16392  df-cntz 16482  df-od 16680  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-ring 17327  df-cring 17328  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-dvr 17459  df-rnghom 17491  df-drng 17525  df-subrg 17554  df-abv 17593  df-lmod 17641  df-scaf 17642  df-sra 17945  df-rgmod 17946  df-nzr 18033  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-cnfld 18548  df-zring 18616  df-zrh 18668  df-zlm 18669  df-chr 18670  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cn 19855  df-cnp 19856  df-tx 20189  df-hmeo 20382  df-tmd 20697  df-tgp 20698  df-trg 20788  df-xms 20949  df-ms 20950  df-tms 20951  df-nm 21229  df-ngp 21230  df-nrg 21232  df-nlm 21233  df-qqh 28115
This theorem is referenced by: (None)
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