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Theorem qqhval 26355
Description: Value of the canonical homormorphism from the rational number to a field. (Contributed by Thierry Arnoux, 22-Oct-2017.)
Hypotheses
Ref Expression
qqhval.1  |-  ./  =  (/r
`  R )
qqhval.2  |-  .1.  =  ( 1r `  R )
qqhval.3  |-  L  =  ( ZRHom `  R
)
Assertion
Ref Expression
qqhval  |-  ( R  e.  _V  ->  (QQHom `  R )  =  ran  ( x  e.  ZZ ,  y  e.  ( `' L " (Unit `  R ) )  |->  <.
( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
) )
Distinct variable groups:    x, y, R    y, L
Allowed substitution hints:    ./ ( x, y)    .1. ( x, y)    L( x)

Proof of Theorem qqhval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 eqidd 2439 . . . 4  |-  ( f  =  R  ->  ZZ  =  ZZ )
2 fveq2 5686 . . . . . . 7  |-  ( f  =  R  ->  ( ZRHom `  f )  =  ( ZRHom `  R
) )
3 qqhval.3 . . . . . . 7  |-  L  =  ( ZRHom `  R
)
42, 3syl6eqr 2488 . . . . . 6  |-  ( f  =  R  ->  ( ZRHom `  f )  =  L )
54cnveqd 5010 . . . . 5  |-  ( f  =  R  ->  `' ( ZRHom `  f )  =  `' L )
6 fveq2 5686 . . . . 5  |-  ( f  =  R  ->  (Unit `  f )  =  (Unit `  R ) )
75, 6imaeq12d 5165 . . . 4  |-  ( f  =  R  ->  ( `' ( ZRHom `  f ) " (Unit `  f ) )  =  ( `' L "
(Unit `  R )
) )
8 fveq2 5686 . . . . . . 7  |-  ( f  =  R  ->  (/r `  f )  =  (/r `  R ) )
9 qqhval.1 . . . . . . 7  |-  ./  =  (/r
`  R )
108, 9syl6eqr 2488 . . . . . 6  |-  ( f  =  R  ->  (/r `  f )  =  ./  )
114fveq1d 5688 . . . . . 6  |-  ( f  =  R  ->  (
( ZRHom `  f
) `  x )  =  ( L `  x ) )
124fveq1d 5688 . . . . . 6  |-  ( f  =  R  ->  (
( ZRHom `  f
) `  y )  =  ( L `  y ) )
1310, 11, 12oveq123d 6107 . . . . 5  |-  ( f  =  R  ->  (
( ( ZRHom `  f ) `  x
) (/r `  f ) ( ( ZRHom `  f
) `  y )
)  =  ( ( L `  x ) 
./  ( L `  y ) ) )
1413opeq2d 4061 . . . 4  |-  ( f  =  R  ->  <. (
x  /  y ) ,  ( ( ( ZRHom `  f ) `  x ) (/r `  f
) ( ( ZRHom `  f ) `  y
) ) >.  =  <. ( x  /  y ) ,  ( ( L `
 x )  ./  ( L `  y ) ) >. )
151, 7, 14mpt2eq123dv 6143 . . 3  |-  ( f  =  R  ->  (
x  e.  ZZ , 
y  e.  ( `' ( ZRHom `  f
) " (Unit `  f ) )  |->  <.
( x  /  y
) ,  ( ( ( ZRHom `  f
) `  x )
(/r `  f ) ( ( ZRHom `  f
) `  y )
) >. )  =  ( x  e.  ZZ , 
y  e.  ( `' L " (Unit `  R ) )  |->  <.
( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
) )
1615rneqd 5062 . 2  |-  ( f  =  R  ->  ran  ( x  e.  ZZ ,  y  e.  ( `' ( ZRHom `  f ) " (Unit `  f ) )  |->  <.
( x  /  y
) ,  ( ( ( ZRHom `  f
) `  x )
(/r `  f ) ( ( ZRHom `  f
) `  y )
) >. )  =  ran  ( x  e.  ZZ ,  y  e.  ( `' L " (Unit `  R ) )  |->  <.
( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
) )
17 df-qqh 26354 . 2  |- QQHom  =  ( f  e.  _V  |->  ran  ( x  e.  ZZ ,  y  e.  ( `' ( ZRHom `  f ) " (Unit `  f ) )  |->  <.
( x  /  y
) ,  ( ( ( ZRHom `  f
) `  x )
(/r `  f ) ( ( ZRHom `  f
) `  y )
) >. ) )
18 zex 10647 . . . 4  |-  ZZ  e.  _V
19 fvex 5696 . . . . . . 7  |-  ( ZRHom `  R )  e.  _V
203, 19eqeltri 2508 . . . . . 6  |-  L  e. 
_V
2120cnvex 6520 . . . . 5  |-  `' L  e.  _V
22 imaexg 6510 . . . . 5  |-  ( `' L  e.  _V  ->  ( `' L " (Unit `  R ) )  e. 
_V )
2321, 22ax-mp 5 . . . 4  |-  ( `' L " (Unit `  R ) )  e. 
_V
2418, 23mpt2ex 6645 . . 3  |-  ( x  e.  ZZ ,  y  e.  ( `' L " (Unit `  R )
)  |->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)  e.  _V
2524rnex 6507 . 2  |-  ran  (
x  e.  ZZ , 
y  e.  ( `' L " (Unit `  R ) )  |->  <.
( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  e.  _V
2616, 17, 25fvmpt 5769 1  |-  ( R  e.  _V  ->  (QQHom `  R )  =  ran  ( x  e.  ZZ ,  y  e.  ( `' L " (Unit `  R ) )  |->  <.
( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2967   <.cop 3878   `'ccnv 4834   ran crn 4836   "cima 4838   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088    / cdiv 9985   ZZcz 10638   1rcur 16591  Unitcui 16719  /rcdvr 16762   ZRHomczrh 17906  QQHomcqqh 26353
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-neg 9590  df-z 10639  df-qqh 26354
This theorem is referenced by:  qqhval2  26363
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