Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  qqhval Structured version   Visualization version   Unicode version

Theorem qqhval 28778
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 2452 . . . 4  |-  ( f  =  R  ->  ZZ  =  ZZ )
2 fveq2 5865 . . . . . . 7  |-  ( f  =  R  ->  ( ZRHom `  f )  =  ( ZRHom `  R
) )
3 qqhval.3 . . . . . . 7  |-  L  =  ( ZRHom `  R
)
42, 3syl6eqr 2503 . . . . . 6  |-  ( f  =  R  ->  ( ZRHom `  f )  =  L )
54cnveqd 5010 . . . . 5  |-  ( f  =  R  ->  `' ( ZRHom `  f )  =  `' L )
6 fveq2 5865 . . . . 5  |-  ( f  =  R  ->  (Unit `  f )  =  (Unit `  R ) )
75, 6imaeq12d 5169 . . . 4  |-  ( f  =  R  ->  ( `' ( ZRHom `  f ) " (Unit `  f ) )  =  ( `' L "
(Unit `  R )
) )
8 fveq2 5865 . . . . . . 7  |-  ( f  =  R  ->  (/r `  f )  =  (/r `  R ) )
9 qqhval.1 . . . . . . 7  |-  ./  =  (/r
`  R )
108, 9syl6eqr 2503 . . . . . 6  |-  ( f  =  R  ->  (/r `  f )  =  ./  )
114fveq1d 5867 . . . . . 6  |-  ( f  =  R  ->  (
( ZRHom `  f
) `  x )  =  ( L `  x ) )
124fveq1d 5867 . . . . . 6  |-  ( f  =  R  ->  (
( ZRHom `  f
) `  y )  =  ( L `  y ) )
1310, 11, 12oveq123d 6311 . . . . 5  |-  ( f  =  R  ->  (
( ( ZRHom `  f ) `  x
) (/r `  f ) ( ( ZRHom `  f
) `  y )
)  =  ( ( L `  x ) 
./  ( L `  y ) ) )
1413opeq2d 4173 . . . 4  |-  ( f  =  R  ->  <. (
x  /  y ) ,  ( ( ( ZRHom `  f ) `  x ) (/r `  f
) ( ( ZRHom `  f ) `  y
) ) >.  =  <. ( x  /  y ) ,  ( ( L `
 x )  ./  ( L `  y ) ) >. )
151, 7, 14mpt2eq123dv 6353 . . 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 28777 . 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 10946 . . . 4  |-  ZZ  e.  _V
19 fvex 5875 . . . . . . 7  |-  ( ZRHom `  R )  e.  _V
203, 19eqeltri 2525 . . . . . 6  |-  L  e. 
_V
2120cnvex 6740 . . . . 5  |-  `' L  e.  _V
22 imaexg 6730 . . . . 5  |-  ( `' L  e.  _V  ->  ( `' L " (Unit `  R ) )  e. 
_V )
2321, 22ax-mp 5 . . . 4  |-  ( `' L " (Unit `  R ) )  e. 
_V
2418, 23mpt2ex 6870 . . 3  |-  ( x  e.  ZZ ,  y  e.  ( `' L " (Unit `  R )
)  |->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)  e.  _V
2524rnex 6727 . 2  |-  ran  (
x  e.  ZZ , 
y  e.  ( `' L " (Unit `  R ) )  |->  <.
( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  e.  _V
2616, 17, 25fvmpt 5948 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 1444    e. wcel 1887   _Vcvv 3045   <.cop 3974   `'ccnv 4833   ran crn 4835   "cima 4837   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292    / cdiv 10269   ZZcz 10937   1rcur 17735  Unitcui 17867  /rcdvr 17910   ZRHomczrh 19071  QQHomcqqh 28776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-neg 9863  df-z 10938  df-qqh 28777
This theorem is referenced by:  qqhval2  28786
  Copyright terms: Public domain W3C validator