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

Theorem qqhrhm 27795
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 23670 . 2  |-  QQ  =  ( Base `  Q )
31qrng1 23673 . 2  |-  1  =  ( 1r `  Q )
4 eqid 2467 . 2  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 qex 11206 . . 3  |-  QQ  e.  _V
6 cnfldmul 18296 . . . 4  |-  x.  =  ( .r ` fld )
71, 6ressmulr 14625 . . 3  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
85, 7ax-mp 5 . 2  |-  x.  =  ( .r `  Q )
9 eqid 2467 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
101qdrng 23671 . . 3  |-  Q  e.  DivRing
11 drngrng 17274 . . 3  |-  ( Q  e.  DivRing  ->  Q  e.  Ring )
1210, 11mp1i 12 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  Q  e.  Ring )
13 isfld 17276 . . . . 5  |-  ( R  e. Field 
<->  ( R  e.  DivRing  /\  R  e.  CRing ) )
1413simplbi 460 . . . 4  |-  ( R  e. Field  ->  R  e.  DivRing )
1514adantr 465 . . 3  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  R  e.  DivRing )
16 drngrng 17274 . . 3  |-  ( R  e.  DivRing  ->  R  e.  Ring )
1715, 16syl 16 . 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 27791 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R ) `  1
)  =  ( 1r
`  R ) )
2214, 21sylan 471 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  ( (QQHom `  R ) `  1
)  =  ( 1r
`  R ) )
23 eqid 2467 . . . 4  |-  (Unit `  R )  =  (Unit `  R )
24 eqid 2467 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
2513simprbi 464 . . . . 5  |-  ( R  e. Field  ->  R  e.  CRing )
2625ad2antrr 725 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  R  e.  CRing )
2720zrhrhm 18418 . . . . . . 7  |-  ( R  e.  Ring  ->  L  e.  (ring RingHom  R ) )
28 zringbas 18364 . . . . . . . 8  |-  ZZ  =  ( Base ` ring )
2928, 18rhmf 17247 . . . . . . 7  |-  ( L  e.  (ring RingHom  R )  ->  L : ZZ --> B )
3017, 27, 293syl 20 . . . . . 6  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  L : ZZ
--> B )
3130adantr 465 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  L : ZZ --> B )
32 qnumcl 14149 . . . . . 6  |-  ( x  e.  QQ  ->  (numer `  x )  e.  ZZ )
3332ad2antrl 727 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  x )  e.  ZZ )
3431, 33ffvelrnd 6033 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (numer `  x ) )  e.  B )
3514ad2antrr 725 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  R  e.  DivRing )
36 simplr 754 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(chr `  R )  =  0 )
3735, 36jca 532 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( R  e.  DivRing  /\  (chr `  R )  =  0 ) )
38 qdencl 14150 . . . . . . 7  |-  ( x  e.  QQ  ->  (denom `  x )  e.  NN )
3938ad2antrl 727 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  NN )
4039nnzd 10977 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  ZZ )
4139nnne0d 10592 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  =/=  0 )
42 eqid 2467 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
4318, 20, 42elzrhunit 27785 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  x )  e.  ZZ  /\  (denom `  x )  =/=  0
) )  ->  ( L `  (denom `  x
) )  e.  (Unit `  R ) )
4437, 40, 41, 43syl12anc 1226 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (denom `  x ) )  e.  (Unit `  R )
)
45 qnumcl 14149 . . . . . 6  |-  ( y  e.  QQ  ->  (numer `  y )  e.  ZZ )
4645ad2antll 728 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  y )  e.  ZZ )
4731, 46ffvelrnd 6033 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (numer `  y ) )  e.  B )
48 qdencl 14150 . . . . . . 7  |-  ( y  e.  QQ  ->  (denom `  y )  e.  NN )
4948ad2antll 728 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  NN )
5049nnzd 10977 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  ZZ )
5149nnne0d 10592 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  =/=  0 )
5218, 20, 42elzrhunit 27785 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  y )  e.  ZZ  /\  (denom `  y )  =/=  0
) )  ->  ( L `  (denom `  y
) )  e.  (Unit `  R ) )
5337, 50, 51, 52syl12anc 1226 . . . 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 27606 . . 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 14155 . . . . . . 7  |-  ( x  e.  QQ  ->  x  =  ( (numer `  x )  /  (denom `  x ) ) )
5655fveq2d 5876 . . . . . 6  |-  ( x  e.  QQ  ->  (
(QQHom `  R ) `  x )  =  ( (QQHom `  R ) `  ( (numer `  x
)  /  (denom `  x ) ) ) )
5756ad2antrl 727 . . . . 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 27793 . . . . . 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 1230 . . . . 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 2508 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  x )  =  ( ( L `
 (numer `  x
) )  ./  ( L `  (denom `  x
) ) ) )
61 qeqnumdivden 14155 . . . . . . 7  |-  ( y  e.  QQ  ->  y  =  ( (numer `  y )  /  (denom `  y ) ) )
6261fveq2d 5876 . . . . . 6  |-  ( y  e.  QQ  ->  (
(QQHom `  R ) `  y )  =  ( (QQHom `  R ) `  ( (numer `  y
)  /  (denom `  y ) ) ) )
6362ad2antll 728 . . . . 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 27793 . . . . . 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 1230 . . . . 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 2508 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  y )  =  ( ( L `
 (numer `  y
) )  ./  ( L `  (denom `  y
) ) ) )
6760, 66oveq12d 6313 . . 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 727 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  x  =  ( (numer `  x )  /  (denom `  x ) ) )
6961ad2antll 728 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
y  =  ( (numer `  y )  /  (denom `  y ) ) )
7068, 69oveq12d 6313 . . . . . 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 10979 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  x )  e.  CC )
7240zcnd 10979 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  CC )
7346zcnd 10979 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  y )  e.  CC )
7450zcnd 10979 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  CC )
7571, 72, 73, 74, 41, 51divmuldivd 10373 . . . . . 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 2508 . . . . 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 5876 . . . 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 10984 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  x
)  x.  (numer `  y ) )  e.  ZZ )
7940, 50zmulcld 10984 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  e.  ZZ )
8072, 74, 41, 51mulne0d 10213 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  =/=  0 )
8118, 19, 20qqhvq 27793 . . . . 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 1230 . . . 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 16 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  R  e.  Ring )
8483, 27syl 16 . . . . . 6  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  L  e.  (ring RingHom  R ) )
85 zringmulr 18367 . . . . . . 7  |-  x.  =  ( .r ` ring )
8628, 85, 9rhmmul 17248 . . . . . 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 1228 . . . . 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 17248 . . . . . 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 1228 . . . . 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 6313 . . . 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 2512 . . 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 2519 . 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 18295 . . . 4  |-  +  =  ( +g  ` fld )
941, 93ressplusg 14614 . . 3  |-  ( QQ  e.  _V  ->  +  =  ( +g  `  Q
) )
955, 94ax-mp 5 . 2  |-  +  =  ( +g  `  Q )
9618, 19, 20qqhf 27792 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )
9714, 96sylan 471 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )
9833, 50zmulcld 10984 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  x
)  x.  (denom `  y ) )  e.  ZZ )
9931, 98ffvelrnd 6033 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(numer `  x )  x.  (denom `  y )
) )  e.  B
)
10046, 40zmulcld 10984 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  y
)  x.  (denom `  x ) )  e.  ZZ )
10131, 100ffvelrnd 6033 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(numer `  y )  x.  (denom `  x )
) )  e.  B
)
10223, 9unitmulcl 17185 . . . . . 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 1228 . . . . 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 2555 . . . 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 27605 . . . 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 1230 . . 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 6313 . . . . . 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 10376 . . . . . 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 2508 . . . . 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 5876 . . . 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 10982 . . . . 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 27793 . . . . 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 1230 . . . 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 17246 . . . . . 6  |-  ( L  e.  (ring RingHom  R )  ->  L  e.  (ring  GrpHom  R ) )
11584, 114syl 16 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  L  e.  (ring  GrpHom  R ) )
116 zringplusg 18365 . . . . . . 7  |-  +  =  ( +g  ` ring )
11728, 116, 24ghmlin 16144 . . . . . 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 6310 . . . . 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 1228 . . . 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 2512 . . 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 27636 . . . . . 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 1246 . . . . 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 2512 . . . 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 27636 . . . . . . 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 1246 . . . . . 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 9629 . . . . . . . 8  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  =  ( (denom `  y
)  x.  (denom `  x ) ) )
127126fveq2d 5876 . . . . . . 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 6311 . . . . . 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 2518 . . . . 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 2508 . . . 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 6313 . . 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 2518 . 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 17250 1  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q RingHom  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118   -->wf 5590   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    / cdiv 10218   NNcn 10548   ZZcz 10876   QQcq 11194  numercnumer 14142  denomcdenom 14143   Basecbs 14507   ↾s cress 14508   +g cplusg 14572   .rcmulr 14573   0gc0g 14712    GrpHom cghm 16136   1rcur 17025   Ringcrg 17070   CRingccrg 17071  Unitcui 17160  /rcdvr 17203   RingHom crh 17233   DivRingcdr 17267  Fieldcfield 17268  ℂfldccnfld 18290  ℤringzring 18358   ZRHomczrh 18406  chrcchr 18408  QQHomcqqh 27778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-numer 14144  df-denom 14145  df-gz 14324  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-ghm 16137  df-od 16426  df-cmn 16673  df-mgp 17014  df-ur 17026  df-ring 17072  df-cring 17073  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-rnghom 17236  df-drng 17269  df-field 17270  df-subrg 17298  df-cnfld 18291  df-zring 18359  df-zrh 18410  df-chr 18412  df-qqh 27779
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator