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Theorem qqhval 26568
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 2455 . . . 4  |-  ( f  =  R  ->  ZZ  =  ZZ )
2 fveq2 5802 . . . . . . 7  |-  ( f  =  R  ->  ( ZRHom `  f )  =  ( ZRHom `  R
) )
3 qqhval.3 . . . . . . 7  |-  L  =  ( ZRHom `  R
)
42, 3syl6eqr 2513 . . . . . 6  |-  ( f  =  R  ->  ( ZRHom `  f )  =  L )
54cnveqd 5126 . . . . 5  |-  ( f  =  R  ->  `' ( ZRHom `  f )  =  `' L )
6 fveq2 5802 . . . . 5  |-  ( f  =  R  ->  (Unit `  f )  =  (Unit `  R ) )
75, 6imaeq12d 5281 . . . 4  |-  ( f  =  R  ->  ( `' ( ZRHom `  f ) " (Unit `  f ) )  =  ( `' L "
(Unit `  R )
) )
8 fveq2 5802 . . . . . . 7  |-  ( f  =  R  ->  (/r `  f )  =  (/r `  R ) )
9 qqhval.1 . . . . . . 7  |-  ./  =  (/r
`  R )
108, 9syl6eqr 2513 . . . . . 6  |-  ( f  =  R  ->  (/r `  f )  =  ./  )
114fveq1d 5804 . . . . . 6  |-  ( f  =  R  ->  (
( ZRHom `  f
) `  x )  =  ( L `  x ) )
124fveq1d 5804 . . . . . 6  |-  ( f  =  R  ->  (
( ZRHom `  f
) `  y )  =  ( L `  y ) )
1310, 11, 12oveq123d 6224 . . . . 5  |-  ( f  =  R  ->  (
( ( ZRHom `  f ) `  x
) (/r `  f ) ( ( ZRHom `  f
) `  y )
)  =  ( ( L `  x ) 
./  ( L `  y ) ) )
1413opeq2d 4177 . . . 4  |-  ( f  =  R  ->  <. (
x  /  y ) ,  ( ( ( ZRHom `  f ) `  x ) (/r `  f
) ( ( ZRHom `  f ) `  y
) ) >.  =  <. ( x  /  y ) ,  ( ( L `
 x )  ./  ( L `  y ) ) >. )
151, 7, 14mpt2eq123dv 6260 . . 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 5178 . 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 26567 . 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 10769 . . . 4  |-  ZZ  e.  _V
19 fvex 5812 . . . . . . 7  |-  ( ZRHom `  R )  e.  _V
203, 19eqeltri 2538 . . . . . 6  |-  L  e. 
_V
2120cnvex 6638 . . . . 5  |-  `' L  e.  _V
22 imaexg 6628 . . . . 5  |-  ( `' L  e.  _V  ->  ( `' L " (Unit `  R ) )  e. 
_V )
2321, 22ax-mp 5 . . . 4  |-  ( `' L " (Unit `  R ) )  e. 
_V
2418, 23mpt2ex 6763 . . 3  |-  ( x  e.  ZZ ,  y  e.  ( `' L " (Unit `  R )
)  |->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)  e.  _V
2524rnex 6625 . 2  |-  ran  (
x  e.  ZZ , 
y  e.  ( `' L " (Unit `  R ) )  |->  <.
( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  e.  _V
2616, 17, 25fvmpt 5886 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 1370    e. wcel 1758   _Vcvv 3078   <.cop 3994   `'ccnv 4950   ran crn 4952   "cima 4954   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205    / cdiv 10107   ZZcz 10760   1rcur 16728  Unitcui 16857  /rcdvr 16900   ZRHomczrh 18059  QQHomcqqh 26566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-neg 9712  df-z 10761  df-qqh 26567
This theorem is referenced by:  qqhval2  26576
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