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

Theorem qqhghm 26429
Description: The QQHom homomorphism is a group homomorphism if the target structure is a division ring. (Contributed by Thierry Arnoux, 9-Nov-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
qqhghm  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )

Proof of Theorem qqhghm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qqhrhm.1 . . 3  |-  Q  =  (flds  QQ )
21qrngbas 22880 . 2  |-  QQ  =  ( Base `  Q )
3 qqhval2.0 . 2  |-  B  =  ( Base `  R
)
4 qex 10977 . . 3  |-  QQ  e.  _V
5 cnfldadd 17835 . . . 4  |-  +  =  ( +g  ` fld )
61, 5ressplusg 14292 . . 3  |-  ( QQ  e.  _V  ->  +  =  ( +g  `  Q
) )
74, 6ax-mp 5 . 2  |-  +  =  ( +g  `  Q )
8 eqid 2443 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
91qdrng 22881 . . 3  |-  Q  e.  DivRing
10 drnggrp 16852 . . 3  |-  ( Q  e.  DivRing  ->  Q  e.  Grp )
119, 10mp1i 12 . 2  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  Q  e.  Grp )
12 drnggrp 16852 . . 3  |-  ( R  e.  DivRing  ->  R  e.  Grp )
1312adantr 465 . 2  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  R  e.  Grp )
14 qqhval2.1 . . 3  |-  ./  =  (/r
`  R )
15 qqhval2.2 . . 3  |-  L  =  ( ZRHom `  R
)
163, 14, 15qqhf 26427 . 2  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )
17 drngrng 16851 . . . . 5  |-  ( R  e.  DivRing  ->  R  e.  Ring )
1817ad2antrr 725 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  R  e.  Ring )
1917adantr 465 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  R  e.  Ring )
2015zrhrhm 17955 . . . . . . 7  |-  ( R  e.  Ring  ->  L  e.  (ring RingHom  R ) )
21 zringbas 17901 . . . . . . . 8  |-  ZZ  =  ( Base ` ring )
2221, 3rhmf 16828 . . . . . . 7  |-  ( L  e.  (ring RingHom  R )  ->  L : ZZ --> B )
2319, 20, 223syl 20 . . . . . 6  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  L : ZZ
--> B )
2423adantr 465 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  L : ZZ
--> B )
25 qnumcl 13830 . . . . . . 7  |-  ( x  e.  QQ  ->  (numer `  x )  e.  ZZ )
2625ad2antrl 727 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (numer `  x
)  e.  ZZ )
27 qdencl 13831 . . . . . . . 8  |-  ( y  e.  QQ  ->  (denom `  y )  e.  NN )
2827ad2antll 728 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  y
)  e.  NN )
2928nnzd 10758 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  y
)  e.  ZZ )
3026, 29zmulcld 10765 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (numer `  x )  x.  (denom `  y ) )  e.  ZZ )
3124, 30ffvelrnd 5856 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( L `  ( (numer `  x
)  x.  (denom `  y ) ) )  e.  B )
32 qnumcl 13830 . . . . . . 7  |-  ( y  e.  QQ  ->  (numer `  y )  e.  ZZ )
3332ad2antll 728 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (numer `  y
)  e.  ZZ )
34 qdencl 13831 . . . . . . . 8  |-  ( x  e.  QQ  ->  (denom `  x )  e.  NN )
3534ad2antrl 727 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  x
)  e.  NN )
3635nnzd 10758 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  x
)  e.  ZZ )
3733, 36zmulcld 10765 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (numer `  y )  x.  (denom `  x ) )  e.  ZZ )
3824, 37ffvelrnd 5856 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( L `  ( (numer `  y
)  x.  (denom `  x ) ) )  e.  B )
3918, 20syl 16 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  L  e.  (ring RingHom  R ) )
40 zringmulr 17904 . . . . . . 7  |-  x.  =  ( .r ` ring )
41 eqid 2443 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
4221, 40, 41rhmmul 16829 . . . . . 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 ) ) ) )
4339, 36, 29, 42syl3anc 1218 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( L `  ( (denom `  x
)  x.  (denom `  y ) ) )  =  ( ( L `
 (denom `  x
) ) ( .r
`  R ) ( L `  (denom `  y ) ) ) )
44 simpl 457 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( R  e.  DivRing  /\  (chr `  R
)  =  0 ) )
4535nnne0d 10378 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  x
)  =/=  0 )
46 eqid 2443 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
473, 15, 46elzrhunit 26420 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  x )  e.  ZZ  /\  (denom `  x )  =/=  0
) )  ->  ( L `  (denom `  x
) )  e.  (Unit `  R ) )
4844, 36, 45, 47syl12anc 1216 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( L `  (denom `  x )
)  e.  (Unit `  R ) )
4928nnne0d 10378 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  y
)  =/=  0 )
503, 15, 46elzrhunit 26420 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  y )  e.  ZZ  /\  (denom `  y )  =/=  0
) )  ->  ( L `  (denom `  y
) )  e.  (Unit `  R ) )
5144, 29, 49, 50syl12anc 1216 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( L `  (denom `  y )
)  e.  (Unit `  R ) )
52 eqid 2443 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
5352, 41unitmulcl 16768 . . . . . 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 ) )
5418, 48, 51, 53syl3anc 1218 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( ( L `  (denom `  x
) ) ( .r
`  R ) ( L `  (denom `  y ) ) )  e.  (Unit `  R
) )
5543, 54eqeltrd 2517 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( L `  ( (denom `  x
)  x.  (denom `  y ) ) )  e.  (Unit `  R
) )
563, 52, 8, 14dvrdir 26270 . . . 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 )
) ) ) ) )
5718, 31, 38, 55, 56syl13anc 1220 . . 3  |-  ( ( ( R  e.  DivRing  /\  (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 )
) ) ) ) )
58 qeqnumdivden 13836 . . . . . . . 8  |-  ( x  e.  QQ  ->  x  =  ( (numer `  x )  /  (denom `  x ) ) )
5958ad2antrl 727 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  x  =  ( (numer `  x )  /  (denom `  x )
) )
60 qeqnumdivden 13836 . . . . . . . 8  |-  ( y  e.  QQ  ->  y  =  ( (numer `  y )  /  (denom `  y ) ) )
6160ad2antll 728 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  y  =  ( (numer `  y )  /  (denom `  y )
) )
6259, 61oveq12d 6121 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( x  +  y )  =  ( ( (numer `  x )  /  (denom `  x ) )  +  ( (numer `  y
)  /  (denom `  y ) ) ) )
6326zcnd 10760 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (numer `  x
)  e.  CC )
6436zcnd 10760 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  x
)  e.  CC )
6533zcnd 10760 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (numer `  y
)  e.  CC )
6629zcnd 10760 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  y
)  e.  CC )
6763, 64, 65, 66, 45, 49divadddivd 10163 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (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 ) ) ) )
6862, 67eqtrd 2475 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (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 ) ) ) )
6968fveq2d 5707 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (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 ) ) ) ) )
7030, 37zaddcld 10763 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (
(numer `  x )  x.  (denom `  y )
)  +  ( (numer `  y )  x.  (denom `  x ) ) )  e.  ZZ )
7136, 29zmulcld 10765 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (denom `  x )  x.  (denom `  y ) )  e.  ZZ )
7264, 66, 45, 49mulne0d 10000 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (denom `  x )  x.  (denom `  y ) )  =/=  0 )
733, 14, 15qqhvq 26428 . . . . 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 ) ) ) ) )
7444, 70, 71, 72, 73syl13anc 1220 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (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 ) ) ) ) )
75 rhmghm 16827 . . . . . 6  |-  ( L  e.  (ring RingHom  R )  ->  L  e.  (ring  GrpHom  R ) )
7639, 75syl 16 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  L  e.  (ring  GrpHom  R ) )
77 zringplusg 17902 . . . . . . 7  |-  +  =  ( +g  ` ring )
7821, 77, 8ghmlin 15764 . . . . . 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 )
) ) ) )
7978oveq1d 6118 . . . . 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 ) ) ) ) )
8076, 30, 37, 79syl3anc 1218 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (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 ) ) ) ) )
8169, 74, 803eqtrd 2479 . . 3  |-  ( ( ( R  e.  DivRing  /\  (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 ) ) ) ) )
8258fveq2d 5707 . . . . . 6  |-  ( x  e.  QQ  ->  (
(QQHom `  R ) `  x )  =  ( (QQHom `  R ) `  ( (numer `  x
)  /  (denom `  x ) ) ) )
8382ad2antrl 727 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (QQHom `  R ) `  x
)  =  ( (QQHom `  R ) `  (
(numer `  x )  /  (denom `  x )
) ) )
843, 14, 15qqhvq 26428 . . . . . 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
) ) ) )
8544, 26, 36, 45, 84syl13anc 1220 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (QQHom `  R ) `  (
(numer `  x )  /  (denom `  x )
) )  =  ( ( L `  (numer `  x ) )  ./  ( L `  (denom `  x ) ) ) )
8652, 21, 14, 40rhmdvd 26301 . . . . . 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 )
) ) ) )
8739, 26, 36, 29, 48, 51, 86syl132anc 1236 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (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 ) ) ) ) )
8883, 85, 873eqtrd 2479 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (QQHom `  R ) `  x
)  =  ( ( L `  ( (numer `  x )  x.  (denom `  y ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) )
8960fveq2d 5707 . . . . . 6  |-  ( y  e.  QQ  ->  (
(QQHom `  R ) `  y )  =  ( (QQHom `  R ) `  ( (numer `  y
)  /  (denom `  y ) ) ) )
9089ad2antll 728 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (QQHom `  R ) `  y
)  =  ( (QQHom `  R ) `  (
(numer `  y )  /  (denom `  y )
) ) )
9152, 21, 14, 40rhmdvd 26301 . . . . . . 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 )
) ) ) )
9239, 33, 29, 36, 51, 48, 91syl132anc 1236 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (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 ) ) ) ) )
933, 14, 15qqhvq 26428 . . . . . . 7  |-  ( ( ( 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
) ) ) )
9444, 33, 29, 49, 93syl13anc 1220 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (QQHom `  R ) `  (
(numer `  y )  /  (denom `  y )
) )  =  ( ( L `  (numer `  y ) )  ./  ( L `  (denom `  y ) ) ) )
9564, 66mulcomd 9419 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (denom `  x )  x.  (denom `  y ) )  =  ( (denom `  y
)  x.  (denom `  x ) ) )
9695fveq2d 5707 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( L `  ( (denom `  x
)  x.  (denom `  y ) ) )  =  ( L `  ( (denom `  y )  x.  (denom `  x )
) ) )
9796oveq2d 6119 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (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 ) ) ) ) )
9892, 94, 973eqtr4d 2485 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (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 ) ) ) ) )
9990, 98eqtrd 2475 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (QQHom `  R ) `  y
)  =  ( ( L `  ( (numer `  y )  x.  (denom `  x ) ) ) 
./  ( L `  ( (denom `  x )  x.  (denom `  y )
) ) ) )
10088, 99oveq12d 6121 . . 3  |-  ( ( ( R  e.  DivRing  /\  (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 )
) ) ) ) )
10157, 81, 1003eqtr4d 2485 . 2  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (QQHom `  R ) `  (
x  +  y ) )  =  ( ( (QQHom `  R ) `  x ) ( +g  `  R ) ( (QQHom `  R ) `  y
) ) )
1022, 3, 7, 8, 11, 13, 16, 101isghmd 15768 1  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  e.  ( Q 
GrpHom  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   _Vcvv 2984   -->wf 5426   ` cfv 5430  (class class class)co 6103   0cc0 9294    + caddc 9297    x. cmul 9299    / cdiv 10005   NNcn 10334   ZZcz 10658   QQcq 10965  numercnumer 13823  denomcdenom 13824   Basecbs 14186   ↾s cress 14187   +g cplusg 14250   .rcmulr 14251   0gc0g 14390   Grpcgrp 15422    GrpHom cghm 15756   Ringcrg 16657  Unitcui 16743  /rcdvr 16786   RingHom crh 16816   DivRingcdr 16844  ℂfldccnfld 17830  ℤringzring 17895   ZRHomczrh 17943  chrcchr 17945  QQHomcqqh 26413
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373  ax-mulf 9374
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-tpos 6757  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-q 10966  df-rp 11004  df-fz 11450  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-dvds 13548  df-gcd 13703  df-numer 13825  df-denom 13826  df-gz 14003  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-tset 14269  df-ple 14270  df-ds 14272  df-unif 14273  df-0g 14392  df-mnd 15427  df-mhm 15476  df-grp 15557  df-minusg 15558  df-sbg 15559  df-mulg 15560  df-subg 15690  df-ghm 15757  df-od 16044  df-cmn 16291  df-mgp 16604  df-ur 16616  df-rng 16659  df-cring 16660  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-dvr 16787  df-rnghom 16818  df-drng 16846  df-subrg 16875  df-cnfld 17831  df-zring 17896  df-zrh 17947  df-chr 17949  df-qqh 26414
This theorem is referenced by:  qqhcn  26432  qqhucn  26433
  Copyright terms: Public domain W3C validator