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Theorem qqhucn 27789
Description: The QQHom homomorphism is uniformly continuous. (Contributed by Thierry Arnoux, 28-Jan-2018.)
Hypotheses
Ref Expression
qqhucn.b  |-  B  =  ( Base `  R
)
qqhucn.q  |-  Q  =  (flds  QQ )
qqhucn.u  |-  U  =  (UnifSt `  Q )
qqhucn.v  |-  V  =  (metUnif `  ( ( dist `  R )  |`  ( B  X.  B
) ) )
qqhucn.z  |-  Z  =  ( ZMod `  R
)
qqhucn.1  |-  ( ph  ->  R  e. NrmRing )
qqhucn.2  |-  ( ph  ->  R  e.  DivRing )
qqhucn.3  |-  ( ph  ->  Z  e. NrmMod )
qqhucn.4  |-  ( ph  ->  (chr `  R )  =  0 )
Assertion
Ref Expression
qqhucn  |-  ( ph  ->  (QQHom `  R )  e.  ( U Cnu V ) )

Proof of Theorem qqhucn
Dummy variables  e 
d  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qqhucn.2 . . . . 5  |-  ( ph  ->  R  e.  DivRing )
2 qqhucn.4 . . . . 5  |-  ( ph  ->  (chr `  R )  =  0 )
3 qqhucn.b . . . . . 6  |-  B  =  ( Base `  R
)
4 eqid 2467 . . . . . 6  |-  (/r `  R
)  =  (/r `  R
)
5 eqid 2467 . . . . . 6  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
63, 4, 5qqhf 27783 . . . . 5  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )
71, 2, 6syl2anc 661 . . . 4  |-  ( ph  ->  (QQHom `  R ) : QQ --> B )
8 simpr 461 . . . . . 6  |-  ( (
ph  /\  e  e.  RR+ )  ->  e  e.  RR+ )
9 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  p  e.  QQ )
10 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  q  e.  QQ )
119, 10ovresd 6438 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
p ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( p ( abs  o.  -  ) q ) )
12 qsscn 11205 . . . . . . . . . . . . . . . 16  |-  QQ  C_  CC
1312, 9sseldi 3507 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  p  e.  CC )
1412, 10sseldi 3507 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  q  e.  CC )
15 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1615cnmetdval 21146 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  CC  /\  q  e.  CC )  ->  ( p ( abs 
o.  -  ) q
)  =  ( abs `  ( p  -  q
) ) )
1713, 14, 16syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
p ( abs  o.  -  ) q )  =  ( abs `  (
p  -  q ) ) )
1811, 17eqtrd 2508 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
p ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( abs `  ( p  -  q ) ) )
19 abssub 13139 . . . . . . . . . . . . . 14  |-  ( ( q  e.  CC  /\  p  e.  CC )  ->  ( abs `  (
q  -  p ) )  =  ( abs `  ( p  -  q
) ) )
2014, 13, 19syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  ( abs `  ( q  -  p ) )  =  ( abs `  (
p  -  q ) ) )
2118, 20eqtr4d 2511 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
p ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( abs `  ( q  -  p ) ) )
227ffvelrnda 6032 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  p  e.  QQ )  ->  ( (QQHom `  R ) `  p
)  e.  B )
2322adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  p )  e.  B
)
247adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  p  e.  QQ )  ->  (QQHom `  R ) : QQ --> B )
2524ffvelrnda 6032 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  q )  e.  B
)
2623, 25ovresd 6438 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  p )
( ( dist `  R
)  |`  ( B  X.  B ) ) ( (QQHom `  R ) `  q ) )  =  ( ( (QQHom `  R ) `  p
) ( dist `  R
) ( (QQHom `  R ) `  q
) ) )
27 qqhucn.1 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  e. NrmRing )
28 nrgngp 21039 . . . . . . . . . . . . . . . . 17  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
2927, 28syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  R  e. NrmGrp )
3029ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  R  e. NrmGrp )
31 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( norm `  R )  =  (
norm `  R )
32 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( -g `  R )  =  (
-g `  R )
33 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( dist `  R )  =  (
dist `  R )
3431, 3, 32, 33ngpdsr 20992 . . . . . . . . . . . . . . 15  |-  ( ( R  e. NrmGrp  /\  (
(QQHom `  R ) `  p )  e.  B  /\  ( (QQHom `  R
) `  q )  e.  B )  ->  (
( (QQHom `  R
) `  p )
( dist `  R )
( (QQHom `  R
) `  q )
)  =  ( (
norm `  R ) `  ( ( (QQHom `  R ) `  q
) ( -g `  R
) ( (QQHom `  R ) `  p
) ) ) )
3530, 23, 25, 34syl3anc 1228 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  p )
( dist `  R )
( (QQHom `  R
) `  q )
)  =  ( (
norm `  R ) `  ( ( (QQHom `  R ) `  q
) ( -g `  R
) ( (QQHom `  R ) `  p
) ) ) )
3610, 9jca 532 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
q  e.  QQ  /\  p  e.  QQ )
)
37 qsubdrg 18340 . . . . . . . . . . . . . . . . . . . . 21  |-  ( QQ  e.  (SubRing ` fld )  /\  (flds  QQ )  e.  DivRing )
3837simpli 458 . . . . . . . . . . . . . . . . . . . 20  |-  QQ  e.  (SubRing ` fld )
39 subrgsubg 17306 . . . . . . . . . . . . . . . . . . . 20  |-  ( QQ  e.  (SubRing ` fld )  ->  QQ  e.  (SubGrp ` fld ) )
4038, 39ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  QQ  e.  (SubGrp ` fld )
41 cnfldsub 18316 . . . . . . . . . . . . . . . . . . . 20  |-  -  =  ( -g ` fld )
42 qqhucn.q . . . . . . . . . . . . . . . . . . . 20  |-  Q  =  (flds  QQ )
43 eqid 2467 . . . . . . . . . . . . . . . . . . . 20  |-  ( -g `  Q )  =  (
-g `  Q )
4441, 42, 43subgsub 16085 . . . . . . . . . . . . . . . . . . 19  |-  ( ( QQ  e.  (SubGrp ` fld )  /\  q  e.  QQ  /\  p  e.  QQ )  ->  ( q  -  p )  =  ( q ( -g `  Q
) p ) )
4540, 44mp3an1 1311 . . . . . . . . . . . . . . . . . 18  |-  ( ( q  e.  QQ  /\  p  e.  QQ )  ->  ( q  -  p
)  =  ( q ( -g `  Q
) p ) )
4636, 45syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
q  -  p )  =  ( q (
-g `  Q )
p ) )
4746fveq2d 5876 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  ( q  -  p
) )  =  ( (QQHom `  R ) `  ( q ( -g `  Q ) p ) ) )
48 simpll 753 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  ph )
493, 4, 5, 42qqhghm 27785 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )
501, 2, 49syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  (QQHom `  R )  e.  ( Q  GrpHom  R ) )
5148, 50syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (QQHom `  R )  e.  ( Q  GrpHom  R ) )
5242qrngbas 23670 . . . . . . . . . . . . . . . . . 18  |-  QQ  =  ( Base `  Q )
5352, 43, 32ghmsub 16147 . . . . . . . . . . . . . . . . 17  |-  ( ( (QQHom `  R )  e.  ( Q  GrpHom  R )  /\  q  e.  QQ  /\  p  e.  QQ )  ->  ( (QQHom `  R ) `  (
q ( -g `  Q
) p ) )  =  ( ( (QQHom `  R ) `  q
) ( -g `  R
) ( (QQHom `  R ) `  p
) ) )
5451, 10, 9, 53syl3anc 1228 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  ( q ( -g `  Q ) p ) )  =  ( ( (QQHom `  R ) `  q ) ( -g `  R ) ( (QQHom `  R ) `  p
) ) )
5547, 54eqtr2d 2509 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  q )
( -g `  R ) ( (QQHom `  R
) `  p )
)  =  ( (QQHom `  R ) `  (
q  -  p ) ) )
5655fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( norm `  R ) `  ( ( (QQHom `  R ) `  q
) ( -g `  R
) ( (QQHom `  R ) `  p
) ) )  =  ( ( norm `  R
) `  ( (QQHom `  R ) `  (
q  -  p ) ) ) )
5727, 1elind 3693 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  R  e.  (NrmRing  i^i  DivRing ) )
5857ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  R  e.  (NrmRing  i^i  DivRing ) )
59 qqhucn.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  Z  e. NrmMod )
6059ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  Z  e. NrmMod )
612ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (chr `  R )  =  0 )
62 qsubcl 11213 . . . . . . . . . . . . . . . 16  |-  ( ( q  e.  QQ  /\  p  e.  QQ )  ->  ( q  -  p
)  e.  QQ )
6336, 62syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
q  -  p )  e.  QQ )
64 qqhucn.z . . . . . . . . . . . . . . . 16  |-  Z  =  ( ZMod `  R
)
6531, 64qqhnm 27787 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  (NrmRing  i^i 
DivRing )  /\  Z  e. NrmMod  /\  (chr `  R )  =  0 )  /\  ( q  -  p
)  e.  QQ )  ->  ( ( norm `  R ) `  (
(QQHom `  R ) `  ( q  -  p
) ) )  =  ( abs `  (
q  -  p ) ) )
6658, 60, 61, 63, 65syl31anc 1231 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( norm `  R ) `  ( (QQHom `  R
) `  ( q  -  p ) ) )  =  ( abs `  (
q  -  p ) ) )
6735, 56, 663eqtrd 2512 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  p )
( dist `  R )
( (QQHom `  R
) `  q )
)  =  ( abs `  ( q  -  p
) ) )
6826, 67eqtrd 2508 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  p )
( ( dist `  R
)  |`  ( B  X.  B ) ) ( (QQHom `  R ) `  q ) )  =  ( abs `  (
q  -  p ) ) )
6921, 68eqtr4d 2511 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
p ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( ( (QQHom `  R
) `  p )
( ( dist `  R
)  |`  ( B  X.  B ) ) ( (QQHom `  R ) `  q ) ) )
7069breq1d 4463 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( p ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  e  <->  ( (
(QQHom `  R ) `  p ) ( (
dist `  R )  |`  ( B  X.  B
) ) ( (QQHom `  R ) `  q
) )  <  e
) )
7170biimpd 207 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( p ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  e  ->  (
( (QQHom `  R
) `  p )
( ( dist `  R
)  |`  ( B  X.  B ) ) ( (QQHom `  R ) `  q ) )  < 
e ) )
7271ralrimiva 2881 . . . . . . . 8  |-  ( (
ph  /\  p  e.  QQ )  ->  A. q  e.  QQ  ( ( p ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  e  ->  ( ( (QQHom `  R ) `  p
) ( ( dist `  R )  |`  ( B  X.  B ) ) ( (QQHom `  R
) `  q )
)  <  e )
)
7372ralrimiva 2881 . . . . . . 7  |-  ( ph  ->  A. p  e.  QQ  A. q  e.  QQ  (
( p ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  e  ->  (
( (QQHom `  R
) `  p )
( ( dist `  R
)  |`  ( B  X.  B ) ) ( (QQHom `  R ) `  q ) )  < 
e ) )
7473adantr 465 . . . . . 6  |-  ( (
ph  /\  e  e.  RR+ )  ->  A. p  e.  QQ  A. q  e.  QQ  ( ( p ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  e  ->  ( ( (QQHom `  R ) `  p
) ( ( dist `  R )  |`  ( B  X.  B ) ) ( (QQHom `  R
) `  q )
)  <  e )
)
75 breq2 4457 . . . . . . . . 9  |-  ( d  =  e  ->  (
( p ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  <->  ( p
( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  e
) )
7675imbi1d 317 . . . . . . . 8  |-  ( d  =  e  ->  (
( ( p ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  ->  (
( (QQHom `  R
) `  p )
( ( dist `  R
)  |`  ( B  X.  B ) ) ( (QQHom `  R ) `  q ) )  < 
e )  <->  ( (
p ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  e  ->  ( ( (QQHom `  R ) `  p
) ( ( dist `  R )  |`  ( B  X.  B ) ) ( (QQHom `  R
) `  q )
)  <  e )
) )
77762ralbidv 2911 . . . . . . 7  |-  ( d  =  e  ->  ( A. p  e.  QQ  A. q  e.  QQ  (
( p ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  ->  (
( (QQHom `  R
) `  p )
( ( dist `  R
)  |`  ( B  X.  B ) ) ( (QQHom `  R ) `  q ) )  < 
e )  <->  A. p  e.  QQ  A. q  e.  QQ  ( ( p ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  e  ->  ( ( (QQHom `  R ) `  p
) ( ( dist `  R )  |`  ( B  X.  B ) ) ( (QQHom `  R
) `  q )
)  <  e )
) )
7877rspcev 3219 . . . . . 6  |-  ( ( e  e.  RR+  /\  A. p  e.  QQ  A. q  e.  QQ  ( ( p ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  e  ->  ( ( (QQHom `  R ) `  p
) ( ( dist `  R )  |`  ( B  X.  B ) ) ( (QQHom `  R
) `  q )
)  <  e )
)  ->  E. d  e.  RR+  A. p  e.  QQ  A. q  e.  QQ  ( ( p ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  d  ->  ( ( (QQHom `  R ) `  p
) ( ( dist `  R )  |`  ( B  X.  B ) ) ( (QQHom `  R
) `  q )
)  <  e )
)
798, 74, 78syl2anc 661 . . . . 5  |-  ( (
ph  /\  e  e.  RR+ )  ->  E. d  e.  RR+  A. p  e.  QQ  A. q  e.  QQ  ( ( p ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  d  ->  ( ( (QQHom `  R ) `  p
) ( ( dist `  R )  |`  ( B  X.  B ) ) ( (QQHom `  R
) `  q )
)  <  e )
)
8079ralrimiva 2881 . . . 4  |-  ( ph  ->  A. e  e.  RR+  E. d  e.  RR+  A. p  e.  QQ  A. q  e.  QQ  ( ( p ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  d  ->  ( ( (QQHom `  R ) `  p
) ( ( dist `  R )  |`  ( B  X.  B ) ) ( (QQHom `  R
) `  q )
)  <  e )
)
817, 80jca 532 . . 3  |-  ( ph  ->  ( (QQHom `  R
) : QQ --> B  /\  A. e  e.  RR+  E. d  e.  RR+  A. p  e.  QQ  A. q  e.  QQ  ( ( p ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  d  ->  ( ( (QQHom `  R ) `  p
) ( ( dist `  R )  |`  ( B  X.  B ) ) ( (QQHom `  R
) `  q )
)  <  e )
) )
82 eqid 2467 . . . 4  |-  (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) )  =  (metUnif `  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) )
83 qqhucn.v . . . 4  |-  V  =  (metUnif `  ( ( dist `  R )  |`  ( B  X.  B
) ) )
84 0z 10887 . . . . . . 7  |-  0  e.  ZZ
85 zq 11200 . . . . . . 7  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
8684, 85ax-mp 5 . . . . . 6  |-  0  e.  QQ
87 ne0i 3796 . . . . . 6  |-  ( 0  e.  QQ  ->  QQ  =/=  (/) )
8886, 87ax-mp 5 . . . . 5  |-  QQ  =/=  (/)
8988a1i 11 . . . 4  |-  ( ph  ->  QQ  =/=  (/) )
90 drngrng 17274 . . . . 5  |-  ( R  e.  DivRing  ->  R  e.  Ring )
91 eqid 2467 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
923, 91ringidcl 17091 . . . . 5  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  B )
93 ne0i 3796 . . . . 5  |-  ( ( 1r `  R )  e.  B  ->  B  =/=  (/) )
941, 90, 92, 934syl 21 . . . 4  |-  ( ph  ->  B  =/=  (/) )
95 cnfldxms 21152 . . . . . . . 8  |-fld  e.  *MetSp
96 qex 11206 . . . . . . . 8  |-  QQ  e.  _V
97 ressxms 20896 . . . . . . . 8  |-  ( (fld  e. 
*MetSp  /\  QQ  e.  _V )  ->  (flds  QQ )  e.  *MetSp )
9895, 96, 97mp2an 672 . . . . . . 7  |-  (flds  QQ )  e.  *MetSp
9942, 98eqeltri 2551 . . . . . 6  |-  Q  e. 
*MetSp
100 cnfldds 18300 . . . . . . . . 9  |-  ( abs 
o.  -  )  =  ( dist ` fld )
10142, 100ressds 14686 . . . . . . . 8  |-  ( QQ  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  Q
) )
10296, 101ax-mp 5 . . . . . . 7  |-  ( abs 
o.  -  )  =  ( dist `  Q )
10352, 102xmsxmet2 20830 . . . . . 6  |-  ( Q  e.  *MetSp  ->  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  e.  ( *Met `  QQ ) )
10499, 103mp1i 12 . . . . 5  |-  ( ph  ->  ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) )  e.  ( *Met `  QQ ) )
105 xmetpsmet 20719 . . . . 5  |-  ( ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  e.  ( *Met `  QQ )  ->  ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) )  e.  (PsMet `  QQ ) )
106104, 105syl 16 . . . 4  |-  ( ph  ->  ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) )  e.  (PsMet `  QQ ) )
107 ngpxms 20989 . . . . . 6  |-  ( R  e. NrmGrp  ->  R  e.  *MetSp )
1083, 33xmsxmet2 20830 . . . . . 6  |-  ( R  e.  *MetSp  ->  (
( dist `  R )  |`  ( B  X.  B
) )  e.  ( *Met `  B
) )
10927, 28, 107, 1084syl 21 . . . . 5  |-  ( ph  ->  ( ( dist `  R
)  |`  ( B  X.  B ) )  e.  ( *Met `  B ) )
110 xmetpsmet 20719 . . . . 5  |-  ( ( ( dist `  R
)  |`  ( B  X.  B ) )  e.  ( *Met `  B )  ->  (
( dist `  R )  |`  ( B  X.  B
) )  e.  (PsMet `  B ) )
111109, 110syl 16 . . . 4  |-  ( ph  ->  ( ( dist `  R
)  |`  ( B  X.  B ) )  e.  (PsMet `  B )
)
11282, 83, 89, 94, 106, 111metucn 20960 . . 3  |-  ( ph  ->  ( (QQHom `  R
)  e.  ( (metUnif `  ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) ) Cnu V )  <->  ( (QQHom `  R ) : QQ --> B  /\  A. e  e.  RR+  E. d  e.  RR+  A. p  e.  QQ  A. q  e.  QQ  (
( p ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  ->  (
( (QQHom `  R
) `  p )
( ( dist `  R
)  |`  ( B  X.  B ) ) ( (QQHom `  R ) `  q ) )  < 
e ) ) ) )
11381, 112mpbird 232 . 2  |-  ( ph  ->  (QQHom `  R )  e.  ( (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) Cnu V ) )
114 qqhucn.u . . . . . 6  |-  U  =  (UnifSt `  Q )
11542fveq2i 5875 . . . . . 6  |-  (UnifSt `  Q )  =  (UnifSt `  (flds  QQ ) )
116 ressuss 20634 . . . . . . 7  |-  ( QQ  e.  _V  ->  (UnifSt `  (flds  QQ ) )  =  ( (UnifSt ` fld )t  ( QQ  X.  QQ ) ) )
11796, 116ax-mp 5 . . . . . 6  |-  (UnifSt `  (flds  QQ ) )  =  ( (UnifSt ` fld )t  ( QQ  X.  QQ ) )
118114, 115, 1173eqtri 2500 . . . . 5  |-  U  =  ( (UnifSt ` fld )t  ( QQ  X.  QQ ) )
119 eqid 2467 . . . . . . 7  |-  (UnifSt ` fld )  =  (UnifSt ` fld )
120119cnflduss 21664 . . . . . 6  |-  (UnifSt ` fld )  =  (metUnif `  ( abs  o. 
-  ) )
121120oveq1i 6305 . . . . 5  |-  ( (UnifSt ` fld )t  ( QQ  X.  QQ ) )  =  ( (metUnif `  ( abs  o. 
-  ) )t  ( QQ 
X.  QQ ) )
122 cnxmet 21148 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
123 xmetpsmet 20719 . . . . . . 7  |-  ( ( abs  o.  -  )  e.  ( *Met `  CC )  ->  ( abs 
o.  -  )  e.  (PsMet `  CC ) )
124122, 123ax-mp 5 . . . . . 6  |-  ( abs 
o.  -  )  e.  (PsMet `  CC )
125 restmetu 20958 . . . . . 6  |-  ( ( QQ  =/=  (/)  /\  ( abs  o.  -  )  e.  (PsMet `  CC )  /\  QQ  C_  CC )  ->  ( (metUnif `  ( abs  o.  -  ) )t  ( QQ  X.  QQ ) )  =  (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) )
12688, 124, 12, 125mp3an 1324 . . . . 5  |-  ( (metUnif `  ( abs  o.  -  ) )t  ( QQ  X.  QQ ) )  =  (metUnif `  ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) )
127118, 121, 1263eqtri 2500 . . . 4  |-  U  =  (metUnif `  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) )
128127a1i 11 . . 3  |-  ( ph  ->  U  =  (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) )
129128oveq1d 6310 . 2  |-  ( ph  ->  ( U Cnu V )  =  ( (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) Cnu V ) )
130113, 129eleqtrrd 2558 1  |-  ( ph  ->  (QQHom `  R )  e.  ( U Cnu V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   class class class wbr 4453    X. cxp 5003    |` cres 5007    o. ccom 5009   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504    < clt 9640    - cmin 9817   ZZcz 10876   QQcq 11194   RR+crp 11232   abscabs 13047   Basecbs 14507   ↾s cress 14508   distcds 14581   ↾t crest 14693   -gcsg 15927  SubGrpcsubg 16067    GrpHom cghm 16136   1rcur 17025   Ringcrg 17070  /rcdvr 17203   DivRingcdr 17267  SubRingcsubrg 17296  PsMetcpsmet 18272   *Metcxmt 18273  metUnifcmetu 18280  ℂfldccnfld 18290   ZRHomczrh 18406   ZModczlm 18407  chrcchr 18408  UnifStcuss 20624   Cnucucn 20646   *MetSpcxme 20688   normcnm 20965  NrmGrpcngp 20966  NrmRingcnrg 20968  NrmModcnlm 20969  QQHomcqqh 27769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ico 11547  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-numer 14144  df-denom 14145  df-gz 14324  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-rest 14695  df-topn 14696  df-0g 14714  df-topgen 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-ghm 16137  df-od 16426  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-cring 17073  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-rnghom 17236  df-drng 17269  df-subrg 17298  df-abv 17337  df-lmod 17385  df-nzr 17776  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-metu 18289  df-cnfld 18291  df-zring 18359  df-zrh 18410  df-zlm 18411  df-chr 18412  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-fil 20215  df-ust 20571  df-uss 20627  df-ucn 20647  df-xms 20691  df-ms 20692  df-nm 20971  df-ngp 20972  df-nrg 20974  df-nlm 20975  df-qqh 27770
This theorem is referenced by:  rrhcn  27794
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