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

Theorem qqhrhm 28786
Description: The QQHom homomorphism is a ring homomorphism if the target structure is a field. If the target structure is a division ring, it is a group homomorphism, but not a ring homomorphism, because it does not preserve the ring multiplication operation. (Contributed by Thierry Arnoux, 29-Oct-2017.)
Hypotheses
Ref Expression
qqhval2.0  |-  B  =  ( Base `  R
)
qqhval2.1  |-  ./  =  (/r
`  R )
qqhval2.2  |-  L  =  ( ZRHom `  R
)
qqhrhm.1  |-  Q  =  (flds  QQ )
Assertion
Ref Expression
qqhrhm  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q RingHom  R ) )

Proof of Theorem qqhrhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qqhrhm.1 . . 3  |-  Q  =  (flds  QQ )
21qrngbas 24450 . 2  |-  QQ  =  ( Base `  Q )
31qrng1 24453 . 2  |-  1  =  ( 1r `  Q )
4 eqid 2450 . 2  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 qex 11273 . . 3  |-  QQ  e.  _V
6 cnfldmul 18969 . . . 4  |-  x.  =  ( .r ` fld )
71, 6ressmulr 15243 . . 3  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
85, 7ax-mp 5 . 2  |-  x.  =  ( .r `  Q )
9 eqid 2450 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
101qdrng 24451 . . 3  |-  Q  e.  DivRing
11 drngring 17975 . . 3  |-  ( Q  e.  DivRing  ->  Q  e.  Ring )
1210, 11mp1i 13 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  Q  e.  Ring )
13 isfld 17977 . . . . 5  |-  ( R  e. Field 
<->  ( R  e.  DivRing  /\  R  e.  CRing ) )
1413simplbi 462 . . . 4  |-  ( R  e. Field  ->  R  e.  DivRing )
1514adantr 467 . . 3  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  R  e.  DivRing )
16 drngring 17975 . . 3  |-  ( R  e.  DivRing  ->  R  e.  Ring )
1715, 16syl 17 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  R  e.  Ring )
18 qqhval2.0 . . . 4  |-  B  =  ( Base `  R
)
19 qqhval2.1 . . . 4  |-  ./  =  (/r
`  R )
20 qqhval2.2 . . . 4  |-  L  =  ( ZRHom `  R
)
2118, 19, 20qqh1 28782 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R ) `  1
)  =  ( 1r
`  R ) )
2214, 21sylan 474 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R ) `  1
)  =  ( 1r
`  R ) )
23 eqid 2450 . . . 4  |-  (Unit `  R )  =  (Unit `  R )
24 eqid 2450 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
2513simprbi 466 . . . . 5  |-  ( R  e. Field  ->  R  e.  CRing )
2625ad2antrr 731 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  R  e.  CRing )
2720zrhrhm 19076 . . . . . . 7  |-  ( R  e.  Ring  ->  L  e.  (ring RingHom  R ) )
28 zringbas 19038 . . . . . . . 8  |-  ZZ  =  ( Base ` ring )
2928, 18rhmf 17947 . . . . . . 7  |-  ( L  e.  (ring RingHom  R )  ->  L : ZZ --> B )
3017, 27, 293syl 18 . . . . . 6  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  L : ZZ
--> B )
3130adantr 467 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  L : ZZ --> B )
32 qnumcl 14682 . . . . . 6  |-  ( x  e.  QQ  ->  (numer `  x )  e.  ZZ )
3332ad2antrl 733 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  x )  e.  ZZ )
3431, 33ffvelrnd 6021 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (numer `  x ) )  e.  B )
3514ad2antrr 731 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  R  e.  DivRing )
36 simplr 761 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(chr `  R )  =  0 )
3735, 36jca 535 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( R  e.  DivRing  /\  (chr `  R )  =  0 ) )
38 qdencl 14683 . . . . . . 7  |-  ( x  e.  QQ  ->  (denom `  x )  e.  NN )
3938ad2antrl 733 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  NN )
4039nnzd 11036 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  ZZ )
4139nnne0d 10651 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  =/=  0 )
42 eqid 2450 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
4318, 20, 42elzrhunit 28776 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  x )  e.  ZZ  /\  (denom `  x )  =/=  0
) )  ->  ( L `  (denom `  x
) )  e.  (Unit `  R ) )
4437, 40, 41, 43syl12anc 1265 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (denom `  x ) )  e.  (Unit `  R )
)
45 qnumcl 14682 . . . . . 6  |-  ( y  e.  QQ  ->  (numer `  y )  e.  ZZ )
4645ad2antll 734 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  y )  e.  ZZ )
4731, 46ffvelrnd 6021 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (numer `  y ) )  e.  B )
48 qdencl 14683 . . . . . . 7  |-  ( y  e.  QQ  ->  (denom `  y )  e.  NN )
4948ad2antll 734 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  NN )
5049nnzd 11036 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  ZZ )
5149nnne0d 10651 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  =/=  0 )
5218, 20, 42elzrhunit 28776 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  y )  e.  ZZ  /\  (denom `  y )  =/=  0
) )  ->  ( L `  (denom `  y
) )  e.  (Unit `  R ) )
5337, 50, 51, 52syl12anc 1265 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (denom `  y ) )  e.  (Unit `  R )
)
5418, 23, 24, 19, 9, 26, 34, 44, 47, 53rdivmuldivd 28547 . . 3  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( ( L `
 (numer `  x
) )  ./  ( L `  (denom `  x
) ) ) ( .r `  R ) ( ( L `  (numer `  y ) ) 
./  ( L `  (denom `  y ) ) ) )  =  ( ( ( L `  (numer `  x ) ) ( .r `  R
) ( L `  (numer `  y ) ) )  ./  ( ( L `  (denom `  x
) ) ( .r
`  R ) ( L `  (denom `  y ) ) ) ) )
55 qeqnumdivden 14688 . . . . . . 7  |-  ( x  e.  QQ  ->  x  =  ( (numer `  x )  /  (denom `  x ) ) )
5655fveq2d 5867 . . . . . 6  |-  ( x  e.  QQ  ->  (
(QQHom `  R ) `  x )  =  ( (QQHom `  R ) `  ( (numer `  x
)  /  (denom `  x ) ) ) )
5756ad2antrl 733 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  x )  =  ( (QQHom `  R ) `  (
(numer `  x )  /  (denom `  x )
) ) )
5818, 19, 20qqhvq 28784 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(numer `  x )  e.  ZZ  /\  (denom `  x )  e.  ZZ  /\  (denom `  x )  =/=  0 ) )  -> 
( (QQHom `  R
) `  ( (numer `  x )  /  (denom `  x ) ) )  =  ( ( L `
 (numer `  x
) )  ./  ( L `  (denom `  x
) ) ) )
5937, 33, 40, 41, 58syl13anc 1269 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  ( (numer `  x )  /  (denom `  x ) ) )  =  ( ( L `
 (numer `  x
) )  ./  ( L `  (denom `  x
) ) ) )
6057, 59eqtrd 2484 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  x )  =  ( ( L `
 (numer `  x
) )  ./  ( L `  (denom `  x
) ) ) )
61 qeqnumdivden 14688 . . . . . . 7  |-  ( y  e.  QQ  ->  y  =  ( (numer `  y )  /  (denom `  y ) ) )
6261fveq2d 5867 . . . . . 6  |-  ( y  e.  QQ  ->  (
(QQHom `  R ) `  y )  =  ( (QQHom `  R ) `  ( (numer `  y
)  /  (denom `  y ) ) ) )
6362ad2antll 734 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  y )  =  ( (QQHom `  R ) `  (
(numer `  y )  /  (denom `  y )
) ) )
6418, 19, 20qqhvq 28784 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(numer `  y )  e.  ZZ  /\  (denom `  y )  e.  ZZ  /\  (denom `  y )  =/=  0 ) )  -> 
( (QQHom `  R
) `  ( (numer `  y )  /  (denom `  y ) ) )  =  ( ( L `
 (numer `  y
) )  ./  ( L `  (denom `  y
) ) ) )
6537, 46, 50, 51, 64syl13anc 1269 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  ( (numer `  y )  /  (denom `  y ) ) )  =  ( ( L `
 (numer `  y
) )  ./  ( L `  (denom `  y
) ) ) )
6663, 65eqtrd 2484 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  y )  =  ( ( L `
 (numer `  y
) )  ./  ( L `  (denom `  y
) ) ) )
6760, 66oveq12d 6306 . . 3  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( (QQHom `  R ) `  x
) ( .r `  R ) ( (QQHom `  R ) `  y
) )  =  ( ( ( L `  (numer `  x ) ) 
./  ( L `  (denom `  x ) ) ) ( .r `  R ) ( ( L `  (numer `  y ) )  ./  ( L `  (denom `  y ) ) ) ) )
6855ad2antrl 733 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  x  =  ( (numer `  x )  /  (denom `  x ) ) )
6961ad2antll 734 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
y  =  ( (numer `  y )  /  (denom `  y ) ) )
7068, 69oveq12d 6306 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( x  x.  y
)  =  ( ( (numer `  x )  /  (denom `  x )
)  x.  ( (numer `  y )  /  (denom `  y ) ) ) )
7133zcnd 11038 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  x )  e.  CC )
7240zcnd 11038 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  CC )
7346zcnd 11038 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  y )  e.  CC )
7450zcnd 11038 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  CC )
7571, 72, 73, 74, 41, 51divmuldivd 10421 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( (numer `  x )  /  (denom `  x ) )  x.  ( (numer `  y
)  /  (denom `  y ) ) )  =  ( ( (numer `  x )  x.  (numer `  y ) )  / 
( (denom `  x
)  x.  (denom `  y ) ) ) )
7670, 75eqtrd 2484 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( x  x.  y
)  =  ( ( (numer `  x )  x.  (numer `  y )
)  /  ( (denom `  x )  x.  (denom `  y ) ) ) )
7776fveq2d 5867 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  ( x  x.  y ) )  =  ( (QQHom `  R
) `  ( (
(numer `  x )  x.  (numer `  y )
)  /  ( (denom `  x )  x.  (denom `  y ) ) ) ) )
7833, 46zmulcld 11043 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  x
)  x.  (numer `  y ) )  e.  ZZ )
7940, 50zmulcld 11043 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  e.  ZZ )
8072, 74, 41, 51mulne0d 10261 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  =/=  0 )
8118, 19, 20qqhvq 28784 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
( (numer `  x
)  x.  (numer `  y ) )  e.  ZZ  /\  ( (denom `  x )  x.  (denom `  y ) )  e.  ZZ  /\  ( (denom `  x )  x.  (denom `  y ) )  =/=  0 ) )  -> 
( (QQHom `  R
) `  ( (
(numer `  x )  x.  (numer `  y )
)  /  ( (denom `  x )  x.  (denom `  y ) ) ) )  =  ( ( L `  ( (numer `  x )  x.  (numer `  y ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) )
8237, 78, 79, 80, 81syl13anc 1269 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  ( (
(numer `  x )  x.  (numer `  y )
)  /  ( (denom `  x )  x.  (denom `  y ) ) ) )  =  ( ( L `  ( (numer `  x )  x.  (numer `  y ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) )
8335, 16syl 17 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  R  e.  Ring )
8483, 27syl 17 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  L  e.  (ring RingHom  R ) )
85 zringmulr 19041 . . . . . . 7  |-  x.  =  ( .r ` ring )
8628, 85, 9rhmmul 17948 . . . . . 6  |-  ( ( L  e.  (ring RingHom  R )  /\  (numer `  x )  e.  ZZ  /\  (numer `  y )  e.  ZZ )  ->  ( L `  ( (numer `  x )  x.  (numer `  y )
) )  =  ( ( L `  (numer `  x ) ) ( .r `  R ) ( L `  (numer `  y ) ) ) )
8784, 33, 46, 86syl3anc 1267 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(numer `  x )  x.  (numer `  y )
) )  =  ( ( L `  (numer `  x ) ) ( .r `  R ) ( L `  (numer `  y ) ) ) )
8828, 85, 9rhmmul 17948 . . . . . 6  |-  ( ( L  e.  (ring RingHom  R )  /\  (denom `  x )  e.  ZZ  /\  (denom `  y )  e.  ZZ )  ->  ( L `  ( (denom `  x )  x.  (denom `  y )
) )  =  ( ( L `  (denom `  x ) ) ( .r `  R ) ( L `  (denom `  y ) ) ) )
8984, 40, 50, 88syl3anc 1267 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(denom `  x )  x.  (denom `  y )
) )  =  ( ( L `  (denom `  x ) ) ( .r `  R ) ( L `  (denom `  y ) ) ) )
9087, 89oveq12d 6306 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( L `  ( (numer `  x )  x.  (numer `  y )
) )  ./  ( L `  ( (denom `  x )  x.  (denom `  y ) ) ) )  =  ( ( ( L `  (numer `  x ) ) ( .r `  R ) ( L `  (numer `  y ) ) ) 
./  ( ( L `
 (denom `  x
) ) ( .r
`  R ) ( L `  (denom `  y ) ) ) ) )
9177, 82, 903eqtrd 2488 . . 3  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  ( x  x.  y ) )  =  ( ( ( L `
 (numer `  x
) ) ( .r
`  R ) ( L `  (numer `  y ) ) ) 
./  ( ( L `
 (denom `  x
) ) ( .r
`  R ) ( L `  (denom `  y ) ) ) ) )
9254, 67, 913eqtr4rd 2495 . 2  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  ( x  x.  y ) )  =  ( ( (QQHom `  R ) `  x
) ( .r `  R ) ( (QQHom `  R ) `  y
) ) )
93 cnfldadd 18968 . . . 4  |-  +  =  ( +g  ` fld )
941, 93ressplusg 15232 . . 3  |-  ( QQ  e.  _V  ->  +  =  ( +g  `  Q
) )
955, 94ax-mp 5 . 2  |-  +  =  ( +g  `  Q )
9618, 19, 20qqhf 28783 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )
9714, 96sylan 474 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )
9833, 50zmulcld 11043 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  x
)  x.  (denom `  y ) )  e.  ZZ )
9931, 98ffvelrnd 6021 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(numer `  x )  x.  (denom `  y )
) )  e.  B
)
10046, 40zmulcld 11043 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  y
)  x.  (denom `  x ) )  e.  ZZ )
10131, 100ffvelrnd 6021 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(numer `  y )  x.  (denom `  x )
) )  e.  B
)
10223, 9unitmulcl 17885 . . . . . 6  |-  ( ( R  e.  Ring  /\  ( L `  (denom `  x
) )  e.  (Unit `  R )  /\  ( L `  (denom `  y
) )  e.  (Unit `  R ) )  -> 
( ( L `  (denom `  x ) ) ( .r `  R
) ( L `  (denom `  y ) ) )  e.  (Unit `  R ) )
10383, 44, 53, 102syl3anc 1267 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( L `  (denom `  x ) ) ( .r `  R
) ( L `  (denom `  y ) ) )  e.  (Unit `  R ) )
10489, 103eqeltrd 2528 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(denom `  x )  x.  (denom `  y )
) )  e.  (Unit `  R ) )
10518, 23, 24, 19dvrdir 28546 . . . 4  |-  ( ( R  e.  Ring  /\  (
( L `  (
(numer `  x )  x.  (denom `  y )
) )  e.  B  /\  ( L `  (
(numer `  y )  x.  (denom `  x )
) )  e.  B  /\  ( L `  (
(denom `  x )  x.  (denom `  y )
) )  e.  (Unit `  R ) ) )  ->  ( ( ( L `  ( (numer `  x )  x.  (denom `  y ) ) ) ( +g  `  R
) ( L `  ( (numer `  y )  x.  (denom `  x )
) ) )  ./  ( L `  ( (denom `  x )  x.  (denom `  y ) ) ) )  =  ( ( ( L `  (
(numer `  x )  x.  (denom `  y )
) )  ./  ( L `  ( (denom `  x )  x.  (denom `  y ) ) ) ) ( +g  `  R
) ( ( L `
 ( (numer `  y )  x.  (denom `  x ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) ) )
10683, 99, 101, 104, 105syl13anc 1269 . . 3  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( ( L `
 ( (numer `  x )  x.  (denom `  y ) ) ) ( +g  `  R
) ( L `  ( (numer `  y )  x.  (denom `  x )
) ) )  ./  ( L `  ( (denom `  x )  x.  (denom `  y ) ) ) )  =  ( ( ( L `  (
(numer `  x )  x.  (denom `  y )
) )  ./  ( L `  ( (denom `  x )  x.  (denom `  y ) ) ) ) ( +g  `  R
) ( ( L `
 ( (numer `  y )  x.  (denom `  x ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) ) )
10768, 69oveq12d 6306 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( x  +  y )  =  ( ( (numer `  x )  /  (denom `  x )
)  +  ( (numer `  y )  /  (denom `  y ) ) ) )
10871, 72, 73, 74, 41, 51divadddivd 10424 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( (numer `  x )  /  (denom `  x ) )  +  ( (numer `  y
)  /  (denom `  y ) ) )  =  ( ( ( (numer `  x )  x.  (denom `  y )
)  +  ( (numer `  y )  x.  (denom `  x ) ) )  /  ( (denom `  x )  x.  (denom `  y ) ) ) )
109107, 108eqtrd 2484 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( x  +  y )  =  ( ( ( (numer `  x
)  x.  (denom `  y ) )  +  ( (numer `  y
)  x.  (denom `  x ) ) )  /  ( (denom `  x )  x.  (denom `  y ) ) ) )
110109fveq2d 5867 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  ( x  +  y ) )  =  ( (QQHom `  R ) `  (
( ( (numer `  x )  x.  (denom `  y ) )  +  ( (numer `  y
)  x.  (denom `  x ) ) )  /  ( (denom `  x )  x.  (denom `  y ) ) ) ) )
11198, 100zaddcld 11041 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( (numer `  x )  x.  (denom `  y ) )  +  ( (numer `  y
)  x.  (denom `  x ) ) )  e.  ZZ )
11218, 19, 20qqhvq 28784 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
( ( (numer `  x )  x.  (denom `  y ) )  +  ( (numer `  y
)  x.  (denom `  x ) ) )  e.  ZZ  /\  (
(denom `  x )  x.  (denom `  y )
)  e.  ZZ  /\  ( (denom `  x )  x.  (denom `  y )
)  =/=  0 ) )  ->  ( (QQHom `  R ) `  (
( ( (numer `  x )  x.  (denom `  y ) )  +  ( (numer `  y
)  x.  (denom `  x ) ) )  /  ( (denom `  x )  x.  (denom `  y ) ) ) )  =  ( ( L `  ( ( (numer `  x )  x.  (denom `  y )
)  +  ( (numer `  y )  x.  (denom `  x ) ) ) )  ./  ( L `  ( (denom `  x
)  x.  (denom `  y ) ) ) ) )
11337, 111, 79, 80, 112syl13anc 1269 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  ( (
( (numer `  x
)  x.  (denom `  y ) )  +  ( (numer `  y
)  x.  (denom `  x ) ) )  /  ( (denom `  x )  x.  (denom `  y ) ) ) )  =  ( ( L `  ( ( (numer `  x )  x.  (denom `  y )
)  +  ( (numer `  y )  x.  (denom `  x ) ) ) )  ./  ( L `  ( (denom `  x
)  x.  (denom `  y ) ) ) ) )
114 rhmghm 17946 . . . . . 6  |-  ( L  e.  (ring RingHom  R )  ->  L  e.  (ring  GrpHom  R ) )
11584, 114syl 17 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  L  e.  (ring  GrpHom  R ) )
116 zringplusg 19039 . . . . . . 7  |-  +  =  ( +g  ` ring )
11728, 116, 24ghmlin 16881 . . . . . 6  |-  ( ( L  e.  (ring  GrpHom  R )  /\  ( (numer `  x )  x.  (denom `  y ) )  e.  ZZ  /\  ( (numer `  y )  x.  (denom `  x ) )  e.  ZZ )  ->  ( L `  ( (
(numer `  x )  x.  (denom `  y )
)  +  ( (numer `  y )  x.  (denom `  x ) ) ) )  =  ( ( L `  ( (numer `  x )  x.  (denom `  y ) ) ) ( +g  `  R
) ( L `  ( (numer `  y )  x.  (denom `  x )
) ) ) )
118117oveq1d 6303 . . . . 5  |-  ( ( L  e.  (ring  GrpHom  R )  /\  ( (numer `  x )  x.  (denom `  y ) )  e.  ZZ  /\  ( (numer `  y )  x.  (denom `  x ) )  e.  ZZ )  ->  (
( L `  (
( (numer `  x
)  x.  (denom `  y ) )  +  ( (numer `  y
)  x.  (denom `  x ) ) ) )  ./  ( L `  ( (denom `  x
)  x.  (denom `  y ) ) ) )  =  ( ( ( L `  (
(numer `  x )  x.  (denom `  y )
) ) ( +g  `  R ) ( L `
 ( (numer `  y )  x.  (denom `  x ) ) ) )  ./  ( L `  ( (denom `  x
)  x.  (denom `  y ) ) ) ) )
119115, 98, 100, 118syl3anc 1267 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( L `  ( ( (numer `  x )  x.  (denom `  y ) )  +  ( (numer `  y
)  x.  (denom `  x ) ) ) )  ./  ( L `  ( (denom `  x
)  x.  (denom `  y ) ) ) )  =  ( ( ( L `  (
(numer `  x )  x.  (denom `  y )
) ) ( +g  `  R ) ( L `
 ( (numer `  y )  x.  (denom `  x ) ) ) )  ./  ( L `  ( (denom `  x
)  x.  (denom `  y ) ) ) ) )
120110, 113, 1193eqtrd 2488 . . 3  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  ( x  +  y ) )  =  ( ( ( L `  ( (numer `  x )  x.  (denom `  y ) ) ) ( +g  `  R
) ( L `  ( (numer `  y )  x.  (denom `  x )
) ) )  ./  ( L `  ( (denom `  x )  x.  (denom `  y ) ) ) ) )
12123, 28, 19, 85rhmdvd 28577 . . . . . 6  |-  ( ( L  e.  (ring RingHom  R )  /\  ( (numer `  x )  e.  ZZ  /\  (denom `  x )  e.  ZZ  /\  (denom `  y )  e.  ZZ )  /\  (
( L `  (denom `  x ) )  e.  (Unit `  R )  /\  ( L `  (denom `  y ) )  e.  (Unit `  R )
) )  ->  (
( L `  (numer `  x ) )  ./  ( L `  (denom `  x ) ) )  =  ( ( L `
 ( (numer `  x )  x.  (denom `  y ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) )
12284, 33, 40, 50, 44, 53, 121syl132anc 1285 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( L `  (numer `  x ) ) 
./  ( L `  (denom `  x ) ) )  =  ( ( L `  ( (numer `  x )  x.  (denom `  y ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) )
12357, 59, 1223eqtrd 2488 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  x )  =  ( ( L `
 ( (numer `  x )  x.  (denom `  y ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) )
12423, 28, 19, 85rhmdvd 28577 . . . . . . 7  |-  ( ( L  e.  (ring RingHom  R )  /\  ( (numer `  y )  e.  ZZ  /\  (denom `  y )  e.  ZZ  /\  (denom `  x )  e.  ZZ )  /\  (
( L `  (denom `  y ) )  e.  (Unit `  R )  /\  ( L `  (denom `  x ) )  e.  (Unit `  R )
) )  ->  (
( L `  (numer `  y ) )  ./  ( L `  (denom `  y ) ) )  =  ( ( L `
 ( (numer `  y )  x.  (denom `  x ) ) ) 
./  ( L `  ( (denom `  y )  x.  (denom `  x )
) ) ) )
12584, 46, 50, 40, 53, 44, 124syl132anc 1285 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( L `  (numer `  y ) ) 
./  ( L `  (denom `  y ) ) )  =  ( ( L `  ( (numer `  y )  x.  (denom `  x ) ) ) 
./  ( L `  ( (denom `  y )  x.  (denom `  x )
) ) ) )
12672, 74mulcomd 9661 . . . . . . . 8  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  =  ( (denom `  y
)  x.  (denom `  x ) ) )
127126fveq2d 5867 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(denom `  x )  x.  (denom `  y )
) )  =  ( L `  ( (denom `  y )  x.  (denom `  x ) ) ) )
128127oveq2d 6304 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( L `  ( (numer `  y )  x.  (denom `  x )
) )  ./  ( L `  ( (denom `  x )  x.  (denom `  y ) ) ) )  =  ( ( L `  ( (numer `  y )  x.  (denom `  x ) ) ) 
./  ( L `  ( (denom `  y )  x.  (denom `  x )
) ) ) )
129125, 65, 1283eqtr4d 2494 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  ( (numer `  y )  /  (denom `  y ) ) )  =  ( ( L `
 ( (numer `  y )  x.  (denom `  x ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) )
13063, 129eqtrd 2484 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  y )  =  ( ( L `
 ( (numer `  y )  x.  (denom `  x ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) )
131123, 130oveq12d 6306 . . 3  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( ( (QQHom `  R ) `  x
) ( +g  `  R
) ( (QQHom `  R ) `  y
) )  =  ( ( ( L `  ( (numer `  x )  x.  (denom `  y )
) )  ./  ( L `  ( (denom `  x )  x.  (denom `  y ) ) ) ) ( +g  `  R
) ( ( L `
 ( (numer `  y )  x.  (denom `  x ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) ) )
132106, 120, 1313eqtr4d 2494 . 2  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  ( x  +  y ) )  =  ( ( (QQHom `  R ) `  x
) ( +g  `  R
) ( (QQHom `  R ) `  y
) ) )
1332, 3, 4, 8, 9, 12, 17, 22, 92, 18, 95, 24, 97, 132isrhmd 17950 1  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q RingHom  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   _Vcvv 3044   -->wf 5577   ` cfv 5581  (class class class)co 6288   0cc0 9536   1c1 9537    + caddc 9539    x. cmul 9541    / cdiv 10266   NNcn 10606   ZZcz 10934   QQcq 11261  numercnumer 14675  denomcdenom 14676   Basecbs 15114   ↾s cress 15115   +g cplusg 15183   .rcmulr 15184   0gc0g 15331    GrpHom cghm 16873   1rcur 17728   Ringcrg 17773   CRingccrg 17774  Unitcui 17860  /rcdvr 17903   RingHom crh 17933   DivRingcdr 17968  Fieldcfield 17969  ℂfldccnfld 18963  ℤringzring 19032   ZRHomczrh 19064  chrcchr 19066  QQHomcqqh 28769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-tpos 6970  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-fz 11782  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-dvds 14299  df-gcd 14462  df-numer 14677  df-denom 14678  df-gz 14867  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-0g 15333  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-grp 16666  df-minusg 16667  df-sbg 16668  df-mulg 16669  df-subg 16807  df-ghm 16874  df-od 17165  df-cmn 17425  df-mgp 17717  df-ur 17729  df-ring 17775  df-cring 17776  df-oppr 17844  df-dvdsr 17862  df-unit 17863  df-invr 17893  df-dvr 17904  df-rnghom 17936  df-drng 17970  df-field 17971  df-subrg 17999  df-cnfld 18964  df-zring 19033  df-zrh 19068  df-chr 19070  df-qqh 28770
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator