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Theorem qqhrhm 28788
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 24443 . 2  |-  QQ  =  ( Base `  Q )
31qrng1 24446 . 2  |-  1  =  ( 1r `  Q )
4 eqid 2422 . 2  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 qex 11276 . . 3  |-  QQ  e.  _V
6 cnfldmul 18963 . . . 4  |-  x.  =  ( .r ` fld )
71, 6ressmulr 15237 . . 3  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
85, 7ax-mp 5 . 2  |-  x.  =  ( .r `  Q )
9 eqid 2422 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
101qdrng 24444 . . 3  |-  Q  e.  DivRing
11 drngring 17969 . . 3  |-  ( Q  e.  DivRing  ->  Q  e.  Ring )
1210, 11mp1i 13 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  Q  e.  Ring )
13 isfld 17971 . . . . 5  |-  ( R  e. Field 
<->  ( R  e.  DivRing  /\  R  e.  CRing ) )
1413simplbi 461 . . . 4  |-  ( R  e. Field  ->  R  e.  DivRing )
1514adantr 466 . . 3  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  R  e.  DivRing )
16 drngring 17969 . . 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 28784 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R ) `  1
)  =  ( 1r
`  R ) )
2214, 21sylan 473 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R ) `  1
)  =  ( 1r
`  R ) )
23 eqid 2422 . . . 4  |-  (Unit `  R )  =  (Unit `  R )
24 eqid 2422 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
2513simprbi 465 . . . . 5  |-  ( R  e. Field  ->  R  e.  CRing )
2625ad2antrr 730 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  R  e.  CRing )
2720zrhrhm 19069 . . . . . . 7  |-  ( R  e.  Ring  ->  L  e.  (ring RingHom  R ) )
28 zringbas 19031 . . . . . . . 8  |-  ZZ  =  ( Base ` ring )
2928, 18rhmf 17941 . . . . . . 7  |-  ( L  e.  (ring RingHom  R )  ->  L : ZZ --> B )
3017, 27, 293syl 18 . . . . . 6  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  L : ZZ
--> B )
3130adantr 466 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  L : ZZ --> B )
32 qnumcl 14676 . . . . . 6  |-  ( x  e.  QQ  ->  (numer `  x )  e.  ZZ )
3332ad2antrl 732 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  x )  e.  ZZ )
3431, 33ffvelrnd 6034 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (numer `  x ) )  e.  B )
3514ad2antrr 730 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  R  e.  DivRing )
36 simplr 760 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(chr `  R )  =  0 )
3735, 36jca 534 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( R  e.  DivRing  /\  (chr `  R )  =  0 ) )
38 qdencl 14677 . . . . . . 7  |-  ( x  e.  QQ  ->  (denom `  x )  e.  NN )
3938ad2antrl 732 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  NN )
4039nnzd 11039 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  ZZ )
4139nnne0d 10654 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  =/=  0 )
42 eqid 2422 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
4318, 20, 42elzrhunit 28778 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  x )  e.  ZZ  /\  (denom `  x )  =/=  0
) )  ->  ( L `  (denom `  x
) )  e.  (Unit `  R ) )
4437, 40, 41, 43syl12anc 1262 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (denom `  x ) )  e.  (Unit `  R )
)
45 qnumcl 14676 . . . . . 6  |-  ( y  e.  QQ  ->  (numer `  y )  e.  ZZ )
4645ad2antll 733 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  y )  e.  ZZ )
4731, 46ffvelrnd 6034 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (numer `  y ) )  e.  B )
48 qdencl 14677 . . . . . . 7  |-  ( y  e.  QQ  ->  (denom `  y )  e.  NN )
4948ad2antll 733 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  NN )
5049nnzd 11039 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  ZZ )
5149nnne0d 10654 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  =/=  0 )
5218, 20, 42elzrhunit 28778 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  y )  e.  ZZ  /\  (denom `  y )  =/=  0
) )  ->  ( L `  (denom `  y
) )  e.  (Unit `  R ) )
5337, 50, 51, 52syl12anc 1262 . . . 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 28549 . . 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 14682 . . . . . . 7  |-  ( x  e.  QQ  ->  x  =  ( (numer `  x )  /  (denom `  x ) ) )
5655fveq2d 5881 . . . . . 6  |-  ( x  e.  QQ  ->  (
(QQHom `  R ) `  x )  =  ( (QQHom `  R ) `  ( (numer `  x
)  /  (denom `  x ) ) ) )
5756ad2antrl 732 . . . . 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 28786 . . . . . 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 1266 . . . . 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 2463 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  x )  =  ( ( L `
 (numer `  x
) )  ./  ( L `  (denom `  x
) ) ) )
61 qeqnumdivden 14682 . . . . . . 7  |-  ( y  e.  QQ  ->  y  =  ( (numer `  y )  /  (denom `  y ) ) )
6261fveq2d 5881 . . . . . 6  |-  ( y  e.  QQ  ->  (
(QQHom `  R ) `  y )  =  ( (QQHom `  R ) `  ( (numer `  y
)  /  (denom `  y ) ) ) )
6362ad2antll 733 . . . . 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 28786 . . . . . 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 1266 . . . . 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 2463 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  y )  =  ( ( L `
 (numer `  y
) )  ./  ( L `  (denom `  y
) ) ) )
6760, 66oveq12d 6319 . . 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 732 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  x  =  ( (numer `  x )  /  (denom `  x ) ) )
6961ad2antll 733 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
y  =  ( (numer `  y )  /  (denom `  y ) ) )
7068, 69oveq12d 6319 . . . . . 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 11041 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  x )  e.  CC )
7240zcnd 11041 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  CC )
7346zcnd 11041 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  y )  e.  CC )
7450zcnd 11041 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  CC )
7571, 72, 73, 74, 41, 51divmuldivd 10424 . . . . . 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 2463 . . . . 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 5881 . . . 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 11046 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  x
)  x.  (numer `  y ) )  e.  ZZ )
7940, 50zmulcld 11046 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  e.  ZZ )
8072, 74, 41, 51mulne0d 10264 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  =/=  0 )
8118, 19, 20qqhvq 28786 . . . . 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 1266 . . . 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 19034 . . . . . . 7  |-  x.  =  ( .r ` ring )
8628, 85, 9rhmmul 17942 . . . . . 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 1264 . . . . 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 17942 . . . . . 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 1264 . . . . 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 6319 . . . 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 2467 . . 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 2474 . 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 18962 . . . 4  |-  +  =  ( +g  ` fld )
941, 93ressplusg 15226 . . 3  |-  ( QQ  e.  _V  ->  +  =  ( +g  `  Q
) )
955, 94ax-mp 5 . 2  |-  +  =  ( +g  `  Q )
9618, 19, 20qqhf 28785 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )
9714, 96sylan 473 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )
9833, 50zmulcld 11046 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  x
)  x.  (denom `  y ) )  e.  ZZ )
9931, 98ffvelrnd 6034 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(numer `  x )  x.  (denom `  y )
) )  e.  B
)
10046, 40zmulcld 11046 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  y
)  x.  (denom `  x ) )  e.  ZZ )
10131, 100ffvelrnd 6034 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(numer `  y )  x.  (denom `  x )
) )  e.  B
)
10223, 9unitmulcl 17879 . . . . . 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 1264 . . . . 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 2510 . . . 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 28548 . . . 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 1266 . . 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 6319 . . . . . 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 10427 . . . . . 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 2463 . . . . 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 5881 . . . 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 11044 . . . . 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 28786 . . . . 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 1266 . . . 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 17940 . . . . . 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 19032 . . . . . . 7  |-  +  =  ( +g  ` ring )
11728, 116, 24ghmlin 16875 . . . . . 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 6316 . . . . 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 1264 . . . 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 2467 . . 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 28579 . . . . . 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 1282 . . . . 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 2467 . . . 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 28579 . . . . . . 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 1282 . . . . . 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 9664 . . . . . . . 8  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  =  ( (denom `  y
)  x.  (denom `  x ) ) )
127126fveq2d 5881 . . . . . . 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 6317 . . . . . 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 2473 . . . . 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 2463 . . . 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 6319 . . 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 2473 . 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 17944 1  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q RingHom  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   _Vcvv 3081   -->wf 5593   ` cfv 5597  (class class class)co 6301   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    / cdiv 10269   NNcn 10609   ZZcz 10937   QQcq 11264  numercnumer 14669  denomcdenom 14670   Basecbs 15108   ↾s cress 15109   +g cplusg 15177   .rcmulr 15178   0gc0g 15325    GrpHom cghm 16867   1rcur 17722   Ringcrg 17767   CRingccrg 17768  Unitcui 17854  /rcdvr 17897   RingHom crh 17927   DivRingcdr 17962  Fieldcfield 17963  ℂfldccnfld 18957  ℤringzring 19025   ZRHomczrh 19057  chrcchr 19059  QQHomcqqh 28771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-tpos 6977  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-oadd 7190  df-er 7367  df-map 7478  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-sup 7958  df-inf 7959  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-fz 11785  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-dvds 14293  df-gcd 14456  df-numer 14671  df-denom 14672  df-gz 14861  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-0g 15327  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-mhm 16569  df-grp 16660  df-minusg 16661  df-sbg 16662  df-mulg 16663  df-subg 16801  df-ghm 16868  df-od 17159  df-cmn 17419  df-mgp 17711  df-ur 17723  df-ring 17769  df-cring 17770  df-oppr 17838  df-dvdsr 17856  df-unit 17857  df-invr 17887  df-dvr 17898  df-rnghom 17930  df-drng 17964  df-field 17965  df-subrg 17993  df-cnfld 18958  df-zring 19026  df-zrh 19061  df-chr 19063  df-qqh 28772
This theorem is referenced by: (None)
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