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Theorem qqhucn 26420
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 2442 . . . . . 6  |-  (/r `  R
)  =  (/r `  R
)
5 eqid 2442 . . . . . 6  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
63, 4, 5qqhf 26414 . . . . 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 6230 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
p ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( p ( abs  o.  -  ) q ) )
12 qsscn 10963 . . . . . . . . . . . . . . . 16  |-  QQ  C_  CC
1312, 9sseldi 3353 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  p  e.  CC )
1412, 10sseldi 3353 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  q  e.  CC )
15 eqid 2442 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1615cnmetdval 20349 . . . . . . . . . . . . . . 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 2474 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
p ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( abs `  ( p  -  q ) ) )
19 abssub 12813 . . . . . . . . . . . . . 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 2477 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
p ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( abs `  ( q  -  p ) ) )
227ffvelrnda 5842 . . . . . . . . . . . . . . 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 5842 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  q )  e.  B
)
2623, 25ovresd 6230 . . . . . . . . . . . . 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 20242 . . . . . . . . . . . . . . . . 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 2442 . . . . . . . . . . . . . . . 16  |-  ( norm `  R )  =  (
norm `  R )
32 eqid 2442 . . . . . . . . . . . . . . . 16  |-  ( -g `  R )  =  (
-g `  R )
33 eqid 2442 . . . . . . . . . . . . . . . 16  |-  ( dist `  R )  =  (
dist `  R )
3431, 3, 32, 33ngpdsr 20195 . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . 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 17864 . . . . . . . . . . . . . . . . . . . . 21  |-  ( QQ  e.  (SubRing ` fld )  /\  (flds  QQ )  e.  DivRing )
3837simpli 458 . . . . . . . . . . . . . . . . . . . 20  |-  QQ  e.  (SubRing ` fld )
39 subrgsubg 16870 . . . . . . . . . . . . . . . . . . . 20  |-  ( QQ  e.  (SubRing ` fld )  ->  QQ  e.  (SubGrp ` fld ) )
4038, 39ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  QQ  e.  (SubGrp ` fld )
41 cnfldsub 17843 . . . . . . . . . . . . . . . . . . . 20  |-  -  =  ( -g ` fld )
42 qqhucn.q . . . . . . . . . . . . . . . . . . . 20  |-  Q  =  (flds  QQ )
43 eqid 2442 . . . . . . . . . . . . . . . . . . . 20  |-  ( -g `  Q )  =  (
-g `  Q )
4441, 42, 43subgsub 15692 . . . . . . . . . . . . . . . . . . 19  |-  ( ( QQ  e.  (SubGrp ` fld )  /\  q  e.  QQ  /\  p  e.  QQ )  ->  ( q  -  p )  =  ( q ( -g `  Q
) p ) )
4540, 44mp3an1 1301 . . . . . . . . . . . . . . . . . 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 5694 . . . . . . . . . . . . . . . 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 26416 . . . . . . . . . . . . . . . . . . 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 22867 . . . . . . . . . . . . . . . . . 18  |-  QQ  =  ( Base `  Q )
5352, 43, 32ghmsub 15754 . . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  ( q ( -g `  Q ) p ) )  =  ( ( (QQHom `  R ) `  q ) ( -g `  R ) ( (QQHom `  R ) `  p
) ) )
5547, 54eqtr2d 2475 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  q )
( -g `  R ) ( (QQHom `  R
) `  p )
)  =  ( (QQHom `  R ) `  (
q  -  p ) ) )
5655fveq2d 5694 . . . . . . . . . . . . . 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 3539 . . . . . . . . . . . . . . . 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 10971 . . . . . . . . . . . . . . . 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 26418 . . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( norm `  R ) `  ( (QQHom `  R
) `  ( q  -  p ) ) )  =  ( abs `  (
q  -  p ) ) )
6735, 56, 663eqtrd 2478 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  p )
( dist `  R )
( (QQHom `  R
) `  q )
)  =  ( abs `  ( q  -  p
) ) )
6826, 67eqtrd 2474 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  p )
( ( dist `  R
)  |`  ( B  X.  B ) ) ( (QQHom `  R ) `  q ) )  =  ( abs `  (
q  -  p ) ) )
6921, 68eqtr4d 2477 . . . . . . . . . . 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 4301 . . . . . . . . . 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 2798 . . . . . . . 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 2798 . . . . . . 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 4295 . . . . . . . . 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 2756 . . . . . . 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 3072 . . . . . 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 2798 . . . 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 2442 . . . 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 10656 . . . . . . 7  |-  0  e.  ZZ
85 zq 10958 . . . . . . 7  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
8684, 85ax-mp 5 . . . . . 6  |-  0  e.  QQ
87 ne0i 3642 . . . . . 6  |-  ( 0  e.  QQ  ->  QQ  =/=  (/) )
8886, 87ax-mp 5 . . . . 5  |-  QQ  =/=  (/)
8988a1i 11 . . . 4  |-  ( ph  ->  QQ  =/=  (/) )
90 drngrng 16838 . . . . 5  |-  ( R  e.  DivRing  ->  R  e.  Ring )
91 eqid 2442 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
923, 91rngidcl 16664 . . . . 5  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  B )
93 ne0i 3642 . . . . 5  |-  ( ( 1r `  R )  e.  B  ->  B  =/=  (/) )
941, 90, 92, 934syl 21 . . . 4  |-  ( ph  ->  B  =/=  (/) )
95 cnfldxms 20355 . . . . . . . 8  |-fld  e.  *MetSp
96 qex 10964 . . . . . . . 8  |-  QQ  e.  _V
97 ressxms 20099 . . . . . . . 8  |-  ( (fld  e. 
*MetSp  /\  QQ  e.  _V )  ->  (flds  QQ )  e.  *MetSp )
9895, 96, 97mp2an 672 . . . . . . 7  |-  (flds  QQ )  e.  *MetSp
9942, 98eqeltri 2512 . . . . . 6  |-  Q  e. 
*MetSp
100 cnfldds 17827 . . . . . . . . 9  |-  ( abs 
o.  -  )  =  ( dist ` fld )
10142, 100ressds 14351 . . . . . . . 8  |-  ( QQ  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  Q
) )
10296, 101ax-mp 5 . . . . . . 7  |-  ( abs 
o.  -  )  =  ( dist `  Q )
10352, 102xmsxmet2 20033 . . . . . 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 19922 . . . . 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 20192 . . . . . 6  |-  ( R  e. NrmGrp  ->  R  e.  *MetSp )
1083, 33xmsxmet2 20033 . . . . . 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 19922 . . . . 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 20163 . . 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 5693 . . . . . 6  |-  (UnifSt `  Q )  =  (UnifSt `  (flds  QQ ) )
116 ressuss 19837 . . . . . . 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 2466 . . . . 5  |-  U  =  ( (UnifSt ` fld )t  ( QQ  X.  QQ ) )
119 eqid 2442 . . . . . . 7  |-  (UnifSt ` fld )  =  (UnifSt ` fld )
120119cnflduss 20867 . . . . . 6  |-  (UnifSt ` fld )  =  (metUnif `  ( abs  o. 
-  ) )
121120oveq1i 6100 . . . . 5  |-  ( (UnifSt ` fld )t  ( QQ  X.  QQ ) )  =  ( (metUnif `  ( abs  o. 
-  ) )t  ( QQ 
X.  QQ ) )
122 cnxmet 20351 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
123 xmetpsmet 19922 . . . . . . 7  |-  ( ( abs  o.  -  )  e.  ( *Met `  CC )  ->  ( abs 
o.  -  )  e.  (PsMet `  CC ) )
124122, 123ax-mp 5 . . . . . 6  |-  ( abs 
o.  -  )  e.  (PsMet `  CC )
125 restmetu 20161 . . . . . 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 1314 . . . . 5  |-  ( (metUnif `  ( abs  o.  -  ) )t  ( QQ  X.  QQ ) )  =  (metUnif `  ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) )
127118, 121, 1263eqtri 2466 . . . 4  |-  U  =  (metUnif `  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) )
128127a1i 11 . . 3  |-  ( ph  ->  U  =  (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) )
129128oveq1d 6105 . 2  |-  ( ph  ->  ( U Cnu V )  =  ( (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) Cnu V ) )
130113, 129eleqtrrd 2519 1  |-  ( ph  ->  (QQHom `  R )  e.  ( U Cnu V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   A.wral 2714   E.wrex 2715   _Vcvv 2971    i^i cin 3326    C_ wss 3327   (/)c0 3636   class class class wbr 4291    X. cxp 4837    |` cres 4841    o. ccom 4843   -->wf 5413   ` cfv 5417  (class class class)co 6090   CCcc 9279   0cc0 9281    < clt 9417    - cmin 9594   ZZcz 10645   QQcq 10952   RR+crp 10990   abscabs 12722   Basecbs 14173   ↾s cress 14174   distcds 14246   ↾t crest 14358   -gcsg 15412  SubGrpcsubg 15674    GrpHom cghm 15743   1rcur 16602   Ringcrg 16644  /rcdvr 16773   DivRingcdr 16831  SubRingcsubrg 16860  PsMetcpsmet 17799   *Metcxmt 17800  metUnifcmetu 17807  ℂfldccnfld 17817   ZRHomczrh 17930   ZModczlm 17931  chrcchr 17932  UnifStcuss 19827   Cnucucn 19849   *MetSpcxme 19891   normcnm 20168  NrmGrpcngp 20169  NrmRingcnrg 20171  NrmModcnlm 20172  QQHomcqqh 26400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ico 11305  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-dvds 13535  df-gcd 13690  df-numer 13812  df-denom 13813  df-gz 13990  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-rest 14360  df-topn 14361  df-0g 14379  df-topgen 14381  df-mnd 15414  df-mhm 15463  df-grp 15544  df-minusg 15545  df-sbg 15546  df-mulg 15547  df-subg 15677  df-ghm 15744  df-od 16031  df-cmn 16278  df-abl 16279  df-mgp 16591  df-ur 16603  df-rng 16646  df-cring 16647  df-oppr 16714  df-dvdsr 16732  df-unit 16733  df-invr 16763  df-dvr 16774  df-rnghom 16805  df-drng 16833  df-subrg 16862  df-abv 16901  df-lmod 16949  df-nzr 17339  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-fbas 17813  df-fg 17814  df-metu 17816  df-cnfld 17818  df-zring 17883  df-zrh 17934  df-zlm 17935  df-chr 17936  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-fil 19418  df-ust 19774  df-uss 19830  df-ucn 19850  df-xms 19894  df-ms 19895  df-nm 20174  df-ngp 20175  df-nrg 20177  df-nlm 20178  df-qqh 26401
This theorem is referenced by:  rrhcn  26425
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