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Theorem qqhucn 28847
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 . . . 4  |-  ( ph  ->  R  e.  DivRing )
2 qqhucn.4 . . . 4  |-  ( ph  ->  (chr `  R )  =  0 )
3 qqhucn.b . . . . 5  |-  B  =  ( Base `  R
)
4 eqid 2462 . . . . 5  |-  (/r `  R
)  =  (/r `  R
)
5 eqid 2462 . . . . 5  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
63, 4, 5qqhf 28841 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )
71, 2, 6syl2anc 671 . . 3  |-  ( ph  ->  (QQHom `  R ) : QQ --> B )
8 simpr 467 . . . . 5  |-  ( (
ph  /\  e  e.  RR+ )  ->  e  e.  RR+ )
9 qqhucn.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  R  e. NrmRing )
10 nrgngp 21720 . . . . . . . . . . . . . . 15  |-  ( R  e. NrmRing  ->  R  e. NrmGrp )
119, 10syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  R  e. NrmGrp )
1211ad2antrr 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  R  e. NrmGrp )
137ffvelrnda 6050 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  QQ )  ->  ( (QQHom `  R ) `  p
)  e.  B )
1413adantr 471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  p )  e.  B
)
157adantr 471 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  QQ )  ->  (QQHom `  R ) : QQ --> B )
1615ffvelrnda 6050 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  q )  e.  B
)
17 eqid 2462 . . . . . . . . . . . . . 14  |-  ( norm `  R )  =  (
norm `  R )
18 eqid 2462 . . . . . . . . . . . . . 14  |-  ( -g `  R )  =  (
-g `  R )
19 eqid 2462 . . . . . . . . . . . . . 14  |-  ( dist `  R )  =  (
dist `  R )
2017, 3, 18, 19ngpdsr 21673 . . . . . . . . . . . . 13  |-  ( ( 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
) ) ) )
2112, 14, 16, 20syl3anc 1276 . . . . . . . . . . . 12  |-  ( ( ( 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
) ) ) )
22 simpr 467 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  q  e.  QQ )
23 simplr 767 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  p  e.  QQ )
24 qsubdrg 19075 . . . . . . . . . . . . . . . . . . 19  |-  ( QQ  e.  (SubRing ` fld )  /\  (flds  QQ )  e.  DivRing )
2524simpli 464 . . . . . . . . . . . . . . . . . 18  |-  QQ  e.  (SubRing ` fld )
26 subrgsubg 18069 . . . . . . . . . . . . . . . . . 18  |-  ( QQ  e.  (SubRing ` fld )  ->  QQ  e.  (SubGrp ` fld ) )
2725, 26ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  QQ  e.  (SubGrp ` fld )
28 cnfldsub 19051 . . . . . . . . . . . . . . . . . 18  |-  -  =  ( -g ` fld )
29 qqhucn.q . . . . . . . . . . . . . . . . . 18  |-  Q  =  (flds  QQ )
30 eqid 2462 . . . . . . . . . . . . . . . . . 18  |-  ( -g `  Q )  =  (
-g `  Q )
3128, 29, 30subgsub 16884 . . . . . . . . . . . . . . . . 17  |-  ( ( QQ  e.  (SubGrp ` fld )  /\  q  e.  QQ  /\  p  e.  QQ )  ->  ( q  -  p )  =  ( q ( -g `  Q
) p ) )
3227, 31mp3an1 1360 . . . . . . . . . . . . . . . 16  |-  ( ( q  e.  QQ  /\  p  e.  QQ )  ->  ( q  -  p
)  =  ( q ( -g `  Q
) p ) )
3322, 23, 32syl2anc 671 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
q  -  p )  =  ( q (
-g `  Q )
p ) )
3433fveq2d 5896 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  ( q  -  p
) )  =  ( (QQHom `  R ) `  ( q ( -g `  Q ) p ) ) )
353, 4, 5, 29qqhghm 28843 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )
361, 2, 35syl2anc 671 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  (QQHom `  R )  e.  ( Q  GrpHom  R ) )
3736ad2antrr 737 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (QQHom `  R )  e.  ( Q  GrpHom  R ) )
3829qrngbas 24513 . . . . . . . . . . . . . . . 16  |-  QQ  =  ( Base `  Q )
3938, 30, 18ghmsub 16946 . . . . . . . . . . . . . . 15  |-  ( ( (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
) ) )
4037, 22, 23, 39syl3anc 1276 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
(QQHom `  R ) `  ( q ( -g `  Q ) p ) )  =  ( ( (QQHom `  R ) `  q ) ( -g `  R ) ( (QQHom `  R ) `  p
) ) )
4134, 40eqtr2d 2497 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  q )
( -g `  R ) ( (QQHom `  R
) `  p )
)  =  ( (QQHom `  R ) `  (
q  -  p ) ) )
4241fveq2d 5896 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( norm `  R ) `  ( ( (QQHom `  R ) `  q
) ( -g `  R
) ( (QQHom `  R ) `  p
) ) )  =  ( ( norm `  R
) `  ( (QQHom `  R ) `  (
q  -  p ) ) ) )
439, 1elind 3630 . . . . . . . . . . . . . 14  |-  ( ph  ->  R  e.  (NrmRing  i^i  DivRing ) )
4443ad2antrr 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  R  e.  (NrmRing  i^i  DivRing ) )
45 qqhucn.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  Z  e. NrmMod )
4645ad2antrr 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  Z  e. NrmMod )
472ad2antrr 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (chr `  R )  =  0 )
48 qsubcl 11317 . . . . . . . . . . . . . 14  |-  ( ( q  e.  QQ  /\  p  e.  QQ )  ->  ( q  -  p
)  e.  QQ )
4922, 23, 48syl2anc 671 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
q  -  p )  e.  QQ )
50 qqhucn.z . . . . . . . . . . . . . 14  |-  Z  =  ( ZMod `  R
)
5117, 50qqhnm 28845 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) ) )
5244, 46, 47, 49, 51syl31anc 1279 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( norm `  R ) `  ( (QQHom `  R
) `  ( q  -  p ) ) )  =  ( abs `  (
q  -  p ) ) )
5321, 42, 523eqtrd 2500 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
( (QQHom `  R
) `  p )
( dist `  R )
( (QQHom `  R
) `  q )
)  =  ( abs `  ( q  -  p
) ) )
5414, 16ovresd 6469 . . . . . . . . . . 11  |-  ( ( ( 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
) ) )
55 qsscn 11309 . . . . . . . . . . . . . 14  |-  QQ  C_  CC
5655, 23sseldi 3442 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  p  e.  CC )
5755, 22sseldi 3442 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  q  e.  CC )
58 eqid 2462 . . . . . . . . . . . . . 14  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
5958cnmetdval 21846 . . . . . . . . . . . . 13  |-  ( ( p  e.  CC  /\  q  e.  CC )  ->  ( p ( abs 
o.  -  ) q
)  =  ( abs `  ( p  -  q
) ) )
6056, 57, 59syl2anc 671 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
p ( abs  o.  -  ) q )  =  ( abs `  (
p  -  q ) ) )
6123, 22ovresd 6469 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
p ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( p ( abs  o.  -  ) q ) )
6257, 56abssubd 13570 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  ( abs `  ( q  -  p ) )  =  ( abs `  (
p  -  q ) ) )
6360, 61, 623eqtr4d 2506 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  QQ )  /\  q  e.  QQ )  ->  (
p ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) ) q )  =  ( abs `  ( q  -  p ) ) )
6453, 54, 633eqtr4rd 2507 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )
6564breq1d 4428 . . . . . . . . 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
) )
6665biimpd 212 . . . . . . . 8  |-  ( ( ( 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 ) )
6766ralrimiva 2814 . . . . . . 7  |-  ( (
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 )
)
6867ralrimiva 2814 . . . . . 6  |-  ( 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 ) )
6968adantr 471 . . . . 5  |-  ( (
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 )
)
70 breq2 4422 . . . . . . . 8  |-  ( d  =  e  ->  (
( p ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) q )  <  d  <->  ( p
( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) q )  <  e
) )
7170imbi1d 323 . . . . . . 7  |-  ( 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 )
) )
72712ralbidv 2844 . . . . . 6  |-  ( 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 )
) )
7372rspcev 3162 . . . . 5  |-  ( ( 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 )
)
748, 69, 73syl2anc 671 . . . 4  |-  ( (
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 )
)
7574ralrimiva 2814 . . 3  |-  ( 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 )
)
76 eqid 2462 . . . 4  |-  (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) )  =  (metUnif `  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) )
77 qqhucn.v . . . 4  |-  V  =  (metUnif `  ( ( dist `  R )  |`  ( B  X.  B
) ) )
78 0z 10982 . . . . . 6  |-  0  e.  ZZ
79 zq 11304 . . . . . 6  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
80 ne0i 3749 . . . . . 6  |-  ( 0  e.  QQ  ->  QQ  =/=  (/) )
8178, 79, 80mp2b 10 . . . . 5  |-  QQ  =/=  (/)
8281a1i 11 . . . 4  |-  ( ph  ->  QQ  =/=  (/) )
83 drngring 18037 . . . . 5  |-  ( R  e.  DivRing  ->  R  e.  Ring )
84 eqid 2462 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
853, 84ringidcl 17856 . . . . 5  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  B )
86 ne0i 3749 . . . . 5  |-  ( ( 1r `  R )  e.  B  ->  B  =/=  (/) )
871, 83, 85, 864syl 19 . . . 4  |-  ( ph  ->  B  =/=  (/) )
88 cnfldxms 21852 . . . . . . . 8  |-fld  e.  *MetSp
89 qex 11310 . . . . . . . 8  |-  QQ  e.  _V
90 ressxms 21595 . . . . . . . 8  |-  ( (fld  e. 
*MetSp  /\  QQ  e.  _V )  ->  (flds  QQ )  e.  *MetSp )
9188, 89, 90mp2an 683 . . . . . . 7  |-  (flds  QQ )  e.  *MetSp
9229, 91eqeltri 2536 . . . . . 6  |-  Q  e. 
*MetSp
93 cnfldds 19035 . . . . . . . . 9  |-  ( abs 
o.  -  )  =  ( dist ` fld )
9429, 93ressds 15366 . . . . . . . 8  |-  ( QQ  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  Q
) )
9589, 94ax-mp 5 . . . . . . 7  |-  ( abs 
o.  -  )  =  ( dist `  Q )
9638, 95xmsxmet2 21529 . . . . . 6  |-  ( Q  e.  *MetSp  ->  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  e.  ( *Met `  QQ ) )
9792, 96mp1i 13 . . . . 5  |-  ( ph  ->  ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) )  e.  ( *Met `  QQ ) )
98 xmetpsmet 21418 . . . . 5  |-  ( ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) )  e.  ( *Met `  QQ )  ->  ( ( abs 
o.  -  )  |`  ( QQ  X.  QQ ) )  e.  (PsMet `  QQ ) )
9997, 98syl 17 . . . 4  |-  ( ph  ->  ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) )  e.  (PsMet `  QQ ) )
100 ngpxms 21670 . . . . . 6  |-  ( R  e. NrmGrp  ->  R  e.  *MetSp )
1013, 19xmsxmet2 21529 . . . . . 6  |-  ( R  e.  *MetSp  ->  (
( dist `  R )  |`  ( B  X.  B
) )  e.  ( *Met `  B
) )
1029, 10, 100, 1014syl 19 . . . . 5  |-  ( ph  ->  ( ( dist `  R
)  |`  ( B  X.  B ) )  e.  ( *Met `  B ) )
103 xmetpsmet 21418 . . . . 5  |-  ( ( ( dist `  R
)  |`  ( B  X.  B ) )  e.  ( *Met `  B )  ->  (
( dist `  R )  |`  ( B  X.  B
) )  e.  (PsMet `  B ) )
104102, 103syl 17 . . . 4  |-  ( ph  ->  ( ( dist `  R
)  |`  ( B  X.  B ) )  e.  (PsMet `  B )
)
10576, 77, 82, 87, 99, 104metucn 21641 . . 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 ) ) ) )
1067, 75, 105mpbir2and 938 . 2  |-  ( ph  ->  (QQHom `  R )  e.  ( (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) Cnu V ) )
107 qqhucn.u . . . . . 6  |-  U  =  (UnifSt `  Q )
10829fveq2i 5895 . . . . . 6  |-  (UnifSt `  Q )  =  (UnifSt `  (flds  QQ ) )
109 ressuss 21333 . . . . . . 7  |-  ( QQ  e.  _V  ->  (UnifSt `  (flds  QQ ) )  =  ( (UnifSt ` fld )t  ( QQ  X.  QQ ) ) )
11089, 109ax-mp 5 . . . . . 6  |-  (UnifSt `  (flds  QQ ) )  =  ( (UnifSt ` fld )t  ( QQ  X.  QQ ) )
111107, 108, 1103eqtri 2488 . . . . 5  |-  U  =  ( (UnifSt ` fld )t  ( QQ  X.  QQ ) )
112 eqid 2462 . . . . . . 7  |-  (UnifSt ` fld )  =  (UnifSt ` fld )
113112cnflduss 22378 . . . . . 6  |-  (UnifSt ` fld )  =  (metUnif `  ( abs  o. 
-  ) )
114113oveq1i 6330 . . . . 5  |-  ( (UnifSt ` fld )t  ( QQ  X.  QQ ) )  =  ( (metUnif `  ( abs  o. 
-  ) )t  ( QQ 
X.  QQ ) )
115 cnxmet 21848 . . . . . . 7  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
116 xmetpsmet 21418 . . . . . . 7  |-  ( ( abs  o.  -  )  e.  ( *Met `  CC )  ->  ( abs 
o.  -  )  e.  (PsMet `  CC ) )
117115, 116ax-mp 5 . . . . . 6  |-  ( abs 
o.  -  )  e.  (PsMet `  CC )
118 restmetu 21640 . . . . . 6  |-  ( ( QQ  =/=  (/)  /\  ( abs  o.  -  )  e.  (PsMet `  CC )  /\  QQ  C_  CC )  ->  ( (metUnif `  ( abs  o.  -  ) )t  ( QQ  X.  QQ ) )  =  (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) )
11981, 117, 55, 118mp3an 1373 . . . . 5  |-  ( (metUnif `  ( abs  o.  -  ) )t  ( QQ  X.  QQ ) )  =  (metUnif `  ( ( abs  o.  -  )  |`  ( QQ 
X.  QQ ) ) )
120111, 114, 1193eqtri 2488 . . . 4  |-  U  =  (metUnif `  ( ( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) )
121120a1i 11 . . 3  |-  ( ph  ->  U  =  (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) )
122121oveq1d 6335 . 2  |-  ( ph  ->  ( U Cnu V )  =  ( (metUnif `  (
( abs  o.  -  )  |`  ( QQ  X.  QQ ) ) ) Cnu V ) )
123106, 122eleqtrrd 2543 1  |-  ( ph  ->  (QQHom `  R )  e.  ( U Cnu V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   _Vcvv 3057    i^i cin 3415    C_ wss 3416   (/)c0 3743   class class class wbr 4418    X. cxp 4854    |` cres 4858    o. ccom 4860   -->wf 5601   ` cfv 5605  (class class class)co 6320   CCcc 9568   0cc0 9570    < clt 9706    - cmin 9891   ZZcz 10971   QQcq 11298   RR+crp 11336   abscabs 13352   Basecbs 15176   ↾s cress 15177   distcds 15254   ↾t crest 15374   -gcsg 16726  SubGrpcsubg 16866    GrpHom cghm 16935   1rcur 17790   Ringcrg 17835  /rcdvr 17965   DivRingcdr 18030  SubRingcsubrg 18059  PsMetcpsmet 19009   *Metcxmt 19010  metUnifcmetu 19016  ℂfldccnfld 19025   ZRHomczrh 19126   ZModczlm 19127  chrcchr 19128  UnifStcuss 21323   Cnucucn 21345   *MetSpcxme 21387   normcnm 21646  NrmGrpcngp 21647  NrmRingcnrg 21649  NrmModcnlm 21650  QQHomcqqh 28827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-inf2 8177  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648  ax-addf 9649  ax-mulf 9650
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-tpos 7004  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-sup 7987  df-inf 7988  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-7 10706  df-8 10707  df-9 10708  df-10 10709  df-n0 10904  df-z 10972  df-dec 11086  df-uz 11194  df-q 11299  df-rp 11337  df-xneg 11443  df-xadd 11444  df-xmul 11445  df-ico 11675  df-fz 11820  df-fzo 11953  df-fl 12066  df-mod 12135  df-seq 12252  df-exp 12311  df-cj 13217  df-re 13218  df-im 13219  df-sqrt 13353  df-abs 13354  df-dvds 14361  df-gcd 14524  df-numer 14739  df-denom 14740  df-gz 14929  df-struct 15178  df-ndx 15179  df-slot 15180  df-base 15181  df-sets 15182  df-ress 15183  df-plusg 15258  df-mulr 15259  df-starv 15260  df-sca 15261  df-vsca 15262  df-tset 15264  df-ple 15265  df-ds 15267  df-unif 15268  df-rest 15376  df-topn 15377  df-0g 15395  df-topgen 15397  df-mgm 16543  df-sgrp 16582  df-mnd 16592  df-mhm 16637  df-grp 16728  df-minusg 16729  df-sbg 16730  df-mulg 16731  df-subg 16869  df-ghm 16936  df-od 17227  df-cmn 17487  df-abl 17488  df-mgp 17779  df-ur 17791  df-ring 17837  df-cring 17838  df-oppr 17906  df-dvdsr 17924  df-unit 17925  df-invr 17955  df-dvr 17966  df-rnghom 17998  df-drng 18032  df-subrg 18061  df-abv 18100  df-lmod 18148  df-nzr 18537  df-psmet 19017  df-xmet 19018  df-met 19019  df-bl 19020  df-mopn 19021  df-fbas 19022  df-fg 19023  df-metu 19024  df-cnfld 19026  df-zring 19095  df-zrh 19130  df-zlm 19131  df-chr 19132  df-top 19976  df-bases 19977  df-topon 19978  df-topsp 19979  df-fil 20916  df-ust 21270  df-uss 21326  df-ucn 21346  df-xms 21390  df-ms 21391  df-nm 21652  df-ngp 21653  df-nrg 21655  df-nlm 21656  df-qqh 28828
This theorem is referenced by:  rrhcn  28852
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