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Theorem qqhval 28393
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 2403 . . . 4  |-  ( f  =  R  ->  ZZ  =  ZZ )
2 fveq2 5848 . . . . . . 7  |-  ( f  =  R  ->  ( ZRHom `  f )  =  ( ZRHom `  R
) )
3 qqhval.3 . . . . . . 7  |-  L  =  ( ZRHom `  R
)
42, 3syl6eqr 2461 . . . . . 6  |-  ( f  =  R  ->  ( ZRHom `  f )  =  L )
54cnveqd 4998 . . . . 5  |-  ( f  =  R  ->  `' ( ZRHom `  f )  =  `' L )
6 fveq2 5848 . . . . 5  |-  ( f  =  R  ->  (Unit `  f )  =  (Unit `  R ) )
75, 6imaeq12d 5157 . . . 4  |-  ( f  =  R  ->  ( `' ( ZRHom `  f ) " (Unit `  f ) )  =  ( `' L "
(Unit `  R )
) )
8 fveq2 5848 . . . . . . 7  |-  ( f  =  R  ->  (/r `  f )  =  (/r `  R ) )
9 qqhval.1 . . . . . . 7  |-  ./  =  (/r
`  R )
108, 9syl6eqr 2461 . . . . . 6  |-  ( f  =  R  ->  (/r `  f )  =  ./  )
114fveq1d 5850 . . . . . 6  |-  ( f  =  R  ->  (
( ZRHom `  f
) `  x )  =  ( L `  x ) )
124fveq1d 5850 . . . . . 6  |-  ( f  =  R  ->  (
( ZRHom `  f
) `  y )  =  ( L `  y ) )
1310, 11, 12oveq123d 6298 . . . . 5  |-  ( f  =  R  ->  (
( ( ZRHom `  f ) `  x
) (/r `  f ) ( ( ZRHom `  f
) `  y )
)  =  ( ( L `  x ) 
./  ( L `  y ) ) )
1413opeq2d 4165 . . . 4  |-  ( f  =  R  ->  <. (
x  /  y ) ,  ( ( ( ZRHom `  f ) `  x ) (/r `  f
) ( ( ZRHom `  f ) `  y
) ) >.  =  <. ( x  /  y ) ,  ( ( L `
 x )  ./  ( L `  y ) ) >. )
151, 7, 14mpt2eq123dv 6339 . . 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 5050 . 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 28392 . 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 10913 . . . 4  |-  ZZ  e.  _V
19 fvex 5858 . . . . . . 7  |-  ( ZRHom `  R )  e.  _V
203, 19eqeltri 2486 . . . . . 6  |-  L  e. 
_V
2120cnvex 6730 . . . . 5  |-  `' L  e.  _V
22 imaexg 6720 . . . . 5  |-  ( `' L  e.  _V  ->  ( `' L " (Unit `  R ) )  e. 
_V )
2321, 22ax-mp 5 . . . 4  |-  ( `' L " (Unit `  R ) )  e. 
_V
2418, 23mpt2ex 6860 . . 3  |-  ( x  e.  ZZ ,  y  e.  ( `' L " (Unit `  R )
)  |->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)  e.  _V
2524rnex 6717 . 2  |-  ran  (
x  e.  ZZ , 
y  e.  ( `' L " (Unit `  R ) )  |->  <.
( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  e.  _V
2616, 17, 25fvmpt 5931 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 1405    e. wcel 1842   _Vcvv 3058   <.cop 3977   `'ccnv 4821   ran crn 4823   "cima 4825   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279    / cdiv 10246   ZZcz 10904   1rcur 17471  Unitcui 17606  /rcdvr 17649   ZRHomczrh 18835  QQHomcqqh 28391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-neg 9843  df-z 10905  df-qqh 28392
This theorem is referenced by:  qqhval2  28401
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