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Theorem qqhrhm 28131
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 23930 . 2  |-  QQ  =  ( Base `  Q )
31qrng1 23933 . 2  |-  1  =  ( 1r `  Q )
4 eqid 2457 . 2  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 qex 11219 . . 3  |-  QQ  e.  _V
6 cnfldmul 18553 . . . 4  |-  x.  =  ( .r ` fld )
71, 6ressmulr 14769 . . 3  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
85, 7ax-mp 5 . 2  |-  x.  =  ( .r `  Q )
9 eqid 2457 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
101qdrng 23931 . . 3  |-  Q  e.  DivRing
11 drngring 17530 . . 3  |-  ( Q  e.  DivRing  ->  Q  e.  Ring )
1210, 11mp1i 12 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  Q  e.  Ring )
13 isfld 17532 . . . . 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 drngring 17530 . . 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 28127 . . 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 2457 . . . 4  |-  (Unit `  R )  =  (Unit `  R )
24 eqid 2457 . . . 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 18676 . . . . . . 7  |-  ( R  e.  Ring  ->  L  e.  (ring RingHom  R ) )
28 zringbas 18621 . . . . . . . 8  |-  ZZ  =  ( Base ` ring )
2928, 18rhmf 17502 . . . . . . 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 14285 . . . . . 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 755 . . . . . 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 14286 . . . . . . 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 10989 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  ZZ )
4139nnne0d 10601 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  =/=  0 )
42 eqid 2457 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
4318, 20, 42elzrhunit 28121 . . . . 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 14285 . . . . . 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 14286 . . . . . . 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 10989 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  ZZ )
5149nnne0d 10601 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  =/=  0 )
5218, 20, 42elzrhunit 28121 . . . . 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 27942 . . 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 14291 . . . . . . 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 28129 . . . . . 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 2498 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  x )  =  ( ( L `
 (numer `  x
) )  ./  ( L `  (denom `  x
) ) ) )
61 qeqnumdivden 14291 . . . . . . 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 28129 . . . . . 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 2498 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  y )  =  ( ( L `
 (numer `  y
) )  ./  ( L `  (denom `  y
) ) ) )
6760, 66oveq12d 6314 . . 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 6314 . . . . . 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 10991 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  x )  e.  CC )
7240zcnd 10991 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  CC )
7346zcnd 10991 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  y )  e.  CC )
7450zcnd 10991 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  CC )
7571, 72, 73, 74, 41, 51divmuldivd 10382 . . . . . 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 2498 . . . . 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 10996 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  x
)  x.  (numer `  y ) )  e.  ZZ )
7940, 50zmulcld 10996 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  e.  ZZ )
8072, 74, 41, 51mulne0d 10222 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  =/=  0 )
8118, 19, 20qqhvq 28129 . . . . 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 18624 . . . . . . 7  |-  x.  =  ( .r ` ring )
8628, 85, 9rhmmul 17503 . . . . . 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 17503 . . . . . 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 6314 . . . 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 2502 . . 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 2509 . 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 18552 . . . 4  |-  +  =  ( +g  ` fld )
941, 93ressplusg 14758 . . 3  |-  ( QQ  e.  _V  ->  +  =  ( +g  `  Q
) )
955, 94ax-mp 5 . 2  |-  +  =  ( +g  `  Q )
9618, 19, 20qqhf 28128 . . 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 10996 . . . . 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 10996 . . . . 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 17440 . . . . . 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 2545 . . . 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 27941 . . . 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 6314 . . . . . 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 10385 . . . . . 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 2498 . . . . 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 10994 . . . . 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 28129 . . . . 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 17501 . . . . . 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 18622 . . . . . . 7  |-  +  =  ( +g  ` ring )
11728, 116, 24ghmlin 16399 . . . . . 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 6311 . . . . 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 2502 . . 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 27972 . . . . . 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 2502 . . . 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 27972 . . . . . . 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 9634 . . . . . . . 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 6312 . . . . . 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 2508 . . . . 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 2498 . . . 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 6314 . . 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 2508 . 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 17505 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 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    / cdiv 10227   NNcn 10556   ZZcz 10885   QQcq 11207  numercnumer 14278  denomcdenom 14279   Basecbs 14644   ↾s cress 14645   +g cplusg 14712   .rcmulr 14713   0gc0g 14857    GrpHom cghm 16391   1rcur 17280   Ringcrg 17325   CRingccrg 17326  Unitcui 17415  /rcdvr 17458   RingHom crh 17488   DivRingcdr 17523  Fieldcfield 17524  ℂfldccnfld 18547  ℤringzring 18615   ZRHomczrh 18664  chrcchr 18666  QQHomcqqh 28114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-numer 14280  df-denom 14281  df-gz 14460  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-ghm 16392  df-od 16680  df-cmn 16927  df-mgp 17269  df-ur 17281  df-ring 17327  df-cring 17328  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-dvr 17459  df-rnghom 17491  df-drng 17525  df-field 17526  df-subrg 17554  df-cnfld 18548  df-zring 18616  df-zrh 18668  df-chr 18670  df-qqh 28115
This theorem is referenced by: (None)
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