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Theorem qqhghm 26337
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 22811 . 2  |-  QQ  =  ( Base `  Q )
3 qqhval2.0 . 2  |-  B  =  ( Base `  R
)
4 qex 10961 . . 3  |-  QQ  e.  _V
5 cnfldadd 17723 . . . 4  |-  +  =  ( +g  ` fld )
61, 5ressplusg 14276 . . 3  |-  ( QQ  e.  _V  ->  +  =  ( +g  `  Q
) )
74, 6ax-mp 5 . 2  |-  +  =  ( +g  `  Q )
8 eqid 2441 . 2  |-  ( +g  `  R )  =  ( +g  `  R )
91qdrng 22812 . . 3  |-  Q  e.  DivRing
10 drnggrp 16820 . . 3  |-  ( Q  e.  DivRing  ->  Q  e.  Grp )
119, 10mp1i 12 . 2  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  Q  e.  Grp )
12 drnggrp 16820 . . 3  |-  ( R  e.  DivRing  ->  R  e.  Grp )
1312adantr 462 . 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 26335 . 2  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
) : QQ --> B )
17 drngrng 16819 . . . . 5  |-  ( R  e.  DivRing  ->  R  e.  Ring )
1817ad2antrr 720 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  R  e.  Ring )
1917adantr 462 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  R  e.  Ring )
2015zrhrhm 17843 . . . . . . 7  |-  ( R  e.  Ring  ->  L  e.  (ring RingHom  R ) )
21 zringbas 17789 . . . . . . . 8  |-  ZZ  =  ( Base ` ring )
2221, 3rhmf 16804 . . . . . . 7  |-  ( L  e.  (ring RingHom  R )  ->  L : ZZ --> B )
2319, 20, 223syl 20 . . . . . 6  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  L : ZZ
--> B )
2423adantr 462 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  L : ZZ
--> B )
25 qnumcl 13814 . . . . . . 7  |-  ( x  e.  QQ  ->  (numer `  x )  e.  ZZ )
2625ad2antrl 722 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (numer `  x
)  e.  ZZ )
27 qdencl 13815 . . . . . . . 8  |-  ( y  e.  QQ  ->  (denom `  y )  e.  NN )
2827ad2antll 723 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  y
)  e.  NN )
2928nnzd 10742 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  y
)  e.  ZZ )
3026, 29zmulcld 10749 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (numer `  x )  x.  (denom `  y ) )  e.  ZZ )
3124, 30ffvelrnd 5841 . . . 4  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( L `  ( (numer `  x
)  x.  (denom `  y ) ) )  e.  B )
32 qnumcl 13814 . . . . . . 7  |-  ( y  e.  QQ  ->  (numer `  y )  e.  ZZ )
3332ad2antll 723 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (numer `  y
)  e.  ZZ )
34 qdencl 13815 . . . . . . . 8  |-  ( x  e.  QQ  ->  (denom `  x )  e.  NN )
3534ad2antrl 722 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  x
)  e.  NN )
3635nnzd 10742 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  x
)  e.  ZZ )
3733, 36zmulcld 10749 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (numer `  y )  x.  (denom `  x ) )  e.  ZZ )
3824, 37ffvelrnd 5841 . . . 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 17792 . . . . . . 7  |-  x.  =  ( .r ` ring )
41 eqid 2441 . . . . . . 7  |-  ( .r
`  R )  =  ( .r `  R
)
4221, 40, 41rhmmul 16805 . . . . . 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 1213 . . . . 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 454 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( R  e.  DivRing  /\  (chr `  R
)  =  0 ) )
4535nnne0d 10362 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  x
)  =/=  0 )
46 eqid 2441 . . . . . . . 8  |-  ( 0g
`  R )  =  ( 0g `  R
)
473, 15, 46elzrhunit 26328 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  x )  e.  ZZ  /\  (denom `  x )  =/=  0
) )  ->  ( L `  (denom `  x
) )  e.  (Unit `  R ) )
4844, 36, 45, 47syl12anc 1211 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( L `  (denom `  x )
)  e.  (Unit `  R ) )
4928nnne0d 10362 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  y
)  =/=  0 )
503, 15, 46elzrhunit 26328 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
(denom `  y )  e.  ZZ  /\  (denom `  y )  =/=  0
) )  ->  ( L `  (denom `  y
) )  e.  (Unit `  R ) )
5144, 29, 49, 50syl12anc 1211 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( L `  (denom `  y )
)  e.  (Unit `  R ) )
52 eqid 2441 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
5352, 41unitmulcl 16746 . . . . . 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 1213 . . . . 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 2515 . . . 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 26177 . . . 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 1215 . . 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 13820 . . . . . . . 8  |-  ( x  e.  QQ  ->  x  =  ( (numer `  x )  /  (denom `  x ) ) )
5958ad2antrl 722 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  x  =  ( (numer `  x )  /  (denom `  x )
) )
60 qeqnumdivden 13820 . . . . . . . 8  |-  ( y  e.  QQ  ->  y  =  ( (numer `  y )  /  (denom `  y ) ) )
6160ad2antll 723 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  y  =  ( (numer `  y )  /  (denom `  y )
) )
6259, 61oveq12d 6108 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( x  +  y )  =  ( ( (numer `  x )  /  (denom `  x ) )  +  ( (numer `  y
)  /  (denom `  y ) ) ) )
6326zcnd 10744 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (numer `  x
)  e.  CC )
6436zcnd 10744 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  x
)  e.  CC )
6533zcnd 10744 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (numer `  y
)  e.  CC )
6629zcnd 10744 . . . . . . 7  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  (denom `  y
)  e.  CC )
6763, 64, 65, 66, 45, 49divadddivd 10147 . . . . . 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 2473 . . . . 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 5692 . . . 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 10747 . . . . 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 10749 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (denom `  x )  x.  (denom `  y ) )  e.  ZZ )
7264, 66, 45, 49mulne0d 9984 . . . . 5  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (denom `  x )  x.  (denom `  y ) )  =/=  0 )
733, 14, 15qqhvq 26336 . . . . 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 1215 . . . 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 16803 . . . . . 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 17790 . . . . . . 7  |-  +  =  ( +g  ` ring )
7821, 77, 8ghmlin 15745 . . . . . 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 6105 . . . . 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 1213 . . . 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 2477 . . 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 5692 . . . . . 6  |-  ( x  e.  QQ  ->  (
(QQHom `  R ) `  x )  =  ( (QQHom `  R ) `  ( (numer `  x
)  /  (denom `  x ) ) ) )
8382ad2antrl 722 . . . . 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 26336 . . . . . 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 1215 . . . . 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 26208 . . . . . 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 1231 . . . . 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 2477 . . . 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 5692 . . . . . 6  |-  ( y  e.  QQ  ->  (
(QQHom `  R ) `  y )  =  ( (QQHom `  R ) `  ( (numer `  y
)  /  (denom `  y ) ) ) )
9089ad2antll 723 . . . . 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 26208 . . . . . . 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 1231 . . . . . 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 26336 . . . . . . 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 1215 . . . . . 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 9403 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  QQ  /\  y  e.  QQ )
)  ->  ( (denom `  x )  x.  (denom `  y ) )  =  ( (denom `  y
)  x.  (denom `  x ) ) )
9695fveq2d 5692 . . . . . . 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 6106 . . . . . 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 2483 . . . . 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 2473 . . . 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 6108 . . 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 2483 . 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 15749 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278    + caddc 9281    x. cmul 9283    / cdiv 9989   NNcn 10318   ZZcz 10642   QQcq 10949  numercnumer 13807  denomcdenom 13808   Basecbs 14170   ↾s cress 14171   +g cplusg 14234   .rcmulr 14235   0gc0g 14374   Grpcgrp 15406    GrpHom cghm 15737   Ringcrg 16635  Unitcui 16721  /rcdvr 16764   RingHom crh 16794   DivRingcdr 16812  ℂfldccnfld 17718  ℤringzring 17783   ZRHomczrh 17831  chrcchr 17833  QQHomcqqh 26321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-fz 11434  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-numer 13809  df-denom 13810  df-gz 13987  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-0g 14376  df-mnd 15411  df-mhm 15460  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-ghm 15738  df-od 16025  df-cmn 16272  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-rnghom 16796  df-drng 16814  df-subrg 16843  df-cnfld 17719  df-zring 17784  df-zrh 17835  df-chr 17837  df-qqh 26322
This theorem is referenced by:  qqhcn  26340  qqhucn  26341
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