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Theorem qqhrhm 26421
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 22871 . 2  |-  QQ  =  ( Base `  Q )
31qrng1 22874 . 2  |-  1  =  ( 1r `  Q )
4 eqid 2443 . 2  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 qex 10968 . . 3  |-  QQ  e.  _V
6 cnfldmul 17827 . . . 4  |-  x.  =  ( .r ` fld )
71, 6ressmulr 14294 . . 3  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
85, 7ax-mp 5 . 2  |-  x.  =  ( .r `  Q )
9 eqid 2443 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
101qdrng 22872 . . 3  |-  Q  e.  DivRing
11 drngrng 16842 . . 3  |-  ( Q  e.  DivRing  ->  Q  e.  Ring )
1210, 11mp1i 12 . 2  |-  ( ( R  e. Field  /\  (chr `  R )  =  0 )  ->  Q  e.  Ring )
13 isfld 16844 . . . . 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 16842 . . 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 26417 . . 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 2443 . . . 4  |-  (Unit `  R )  =  (Unit `  R )
24 eqid 2443 . . . 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 17946 . . . . . . 7  |-  ( R  e.  Ring  ->  L  e.  (ring RingHom  R ) )
28 zringbas 17892 . . . . . . . 8  |-  ZZ  =  ( Base ` ring )
2928, 18rhmf 16819 . . . . . . 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 13821 . . . . . 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 5847 . . . 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 13822 . . . . . . 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 10749 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  ZZ )
4139nnne0d 10369 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  =/=  0 )
42 eqid 2443 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
4318, 20, 42elzrhunit 26411 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  x )  e.  ZZ  /\  (denom `  x )  =/=  0
) )  ->  ( L `  (denom `  x
) )  e.  (Unit `  R ) )
4437, 40, 41, 43syl12anc 1216 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (denom `  x ) )  e.  (Unit `  R )
)
45 qnumcl 13821 . . . . . 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 5847 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (numer `  y ) )  e.  B )
48 qdencl 13822 . . . . . . 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 10749 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  ZZ )
5149nnne0d 10369 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  =/=  0 )
5218, 20, 42elzrhunit 26411 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  y )  e.  ZZ  /\  (denom `  y )  =/=  0
) )  ->  ( L `  (denom `  y
) )  e.  (Unit `  R ) )
5337, 50, 51, 52syl12anc 1216 . . . 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 26262 . . 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 13827 . . . . . . 7  |-  ( x  e.  QQ  ->  x  =  ( (numer `  x )  /  (denom `  x ) ) )
5655fveq2d 5698 . . . . . 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 26419 . . . . . 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 1220 . . . . 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 2475 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  x )  =  ( ( L `
 (numer `  x
) )  ./  ( L `  (denom `  x
) ) ) )
61 qeqnumdivden 13827 . . . . . . 7  |-  ( y  e.  QQ  ->  y  =  ( (numer `  y )  /  (denom `  y ) ) )
6261fveq2d 5698 . . . . . 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 26419 . . . . . 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 1220 . . . . 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 2475 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (QQHom `  R
) `  y )  =  ( ( L `
 (numer `  y
) )  ./  ( L `  (denom `  y
) ) ) )
6760, 66oveq12d 6112 . . 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 6112 . . . . . 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 10751 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  x )  e.  CC )
7240zcnd 10751 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  x )  e.  CC )
7346zcnd 10751 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(numer `  y )  e.  CC )
7450zcnd 10751 . . . . . . 7  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
(denom `  y )  e.  CC )
7571, 72, 73, 74, 41, 51divmuldivd 10151 . . . . . 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 2475 . . . . 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 5698 . . . 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 10756 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  x
)  x.  (numer `  y ) )  e.  ZZ )
7940, 50zmulcld 10756 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  e.  ZZ )
8072, 74, 41, 51mulne0d 9991 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  =/=  0 )
8118, 19, 20qqhvq 26419 . . . . 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 1220 . . . 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 17895 . . . . . . 7  |-  x.  =  ( .r ` ring )
8628, 85, 9rhmmul 16820 . . . . . 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 1218 . . . . 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 16820 . . . . . 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 1218 . . . . 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 6112 . . . 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 2479 . . 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 2486 . 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 17826 . . . 4  |-  +  =  ( +g  ` fld )
941, 93ressplusg 14283 . . 3  |-  ( QQ  e.  _V  ->  +  =  ( +g  `  Q
) )
955, 94ax-mp 5 . 2  |-  +  =  ( +g  `  Q )
9618, 19, 20qqhf 26418 . . 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 10756 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  x
)  x.  (denom `  y ) )  e.  ZZ )
9931, 98ffvelrnd 5847 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(numer `  x )  x.  (denom `  y )
) )  e.  B
)
10046, 40zmulcld 10756 . . . . 5  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (numer `  y
)  x.  (denom `  x ) )  e.  ZZ )
10131, 100ffvelrnd 5847 . . . 4  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( L `  (
(numer `  y )  x.  (denom `  x )
) )  e.  B
)
10223, 9unitmulcl 16759 . . . . . 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 1218 . . . . 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 2517 . . . 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 26261 . . . 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 1220 . . 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 6112 . . . . . 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 10154 . . . . . 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 2475 . . . . 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 5698 . . . 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 10754 . . . . 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 26419 . . . . 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 1220 . . . 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 16818 . . . . . 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 17893 . . . . . . 7  |-  +  =  ( +g  ` ring )
11728, 116, 24ghmlin 15755 . . . . . 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 6109 . . . . 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 1218 . . . 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 2479 . . 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 26292 . . . . . 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 1236 . . . . 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 2479 . . . 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 26292 . . . . . . 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 1236 . . . . . 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 9410 . . . . . . . 8  |-  ( ( ( R  e. Field  /\  (chr `  R )  =  0 )  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  -> 
( (denom `  x
)  x.  (denom `  y ) )  =  ( (denom `  y
)  x.  (denom `  x ) ) )
127126fveq2d 5698 . . . . . . 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 6110 . . . . . 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 2485 . . . . 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 2475 . . . 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 6112 . . 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 2485 . 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 16822 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 965    = wceq 1369    e. wcel 1756    =/= wne 2609   _Vcvv 2975   -->wf 5417   ` cfv 5421  (class class class)co 6094   0cc0 9285   1c1 9286    + caddc 9288    x. cmul 9290    / cdiv 9996   NNcn 10325   ZZcz 10649   QQcq 10956  numercnumer 13814  denomcdenom 13815   Basecbs 14177   ↾s cress 14178   +g cplusg 14241   .rcmulr 14242   0gc0g 14381    GrpHom cghm 15747   1rcur 16606   Ringcrg 16648   CRingccrg 16649  Unitcui 16734  /rcdvr 16777   RingHom crh 16807   DivRingcdr 16835  Fieldcfield 16836  ℂfldccnfld 17821  ℤringzring 17886   ZRHomczrh 17934  chrcchr 17936  QQHomcqqh 26404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364  ax-mulf 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-tpos 6748  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-q 10957  df-rp 10995  df-fz 11441  df-fl 11645  df-mod 11712  df-seq 11810  df-exp 11869  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-dvds 13539  df-gcd 13694  df-numer 13816  df-denom 13817  df-gz 13994  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-starv 14256  df-tset 14260  df-ple 14261  df-ds 14263  df-unif 14264  df-0g 14383  df-mnd 15418  df-mhm 15467  df-grp 15548  df-minusg 15549  df-sbg 15550  df-mulg 15551  df-subg 15681  df-ghm 15748  df-od 16035  df-cmn 16282  df-mgp 16595  df-ur 16607  df-rng 16650  df-cring 16651  df-oppr 16718  df-dvdsr 16736  df-unit 16737  df-invr 16767  df-dvr 16778  df-rnghom 16809  df-drng 16837  df-field 16838  df-subrg 16866  df-cnfld 17822  df-zring 17887  df-zrh 17938  df-chr 17940  df-qqh 26405
This theorem is referenced by: (None)
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