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Theorem qqhval2 26265
Description: Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 26-Oct-2017.)
Hypotheses
Ref Expression
qqhval2.0  |-  B  =  ( Base `  R
)
qqhval2.1  |-  ./  =  (/r
`  R )
qqhval2.2  |-  L  =  ( ZRHom `  R
)
Assertion
Ref Expression
qqhval2  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  =  ( q  e.  QQ  |->  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) )
Distinct variable groups:    ./ , q    B, q    L, q    R, q

Proof of Theorem qqhval2
Dummy variables  x  y  e  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2971 . . . 4  |-  ( R  e.  DivRing  ->  R  e.  _V )
21adantr 462 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  R  e.  _V )
3 qqhval2.1 . . . 4  |-  ./  =  (/r
`  R )
4 eqid 2433 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 qqhval2.2 . . . 4  |-  L  =  ( ZRHom `  R
)
63, 4, 5qqhval 26257 . . 3  |-  ( R  e.  _V  ->  (QQHom `  R )  =  ran  ( x  e.  ZZ ,  y  e.  ( `' L " (Unit `  R ) )  |->  <.
( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
) )
72, 6syl 16 . 2  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  =  ran  (
x  e.  ZZ , 
y  e.  ( `' L " (Unit `  R ) )  |->  <.
( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
) )
8 eqidd 2434 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ZZ  =  ZZ )
9 qqhval2.0 . . . . 5  |-  B  =  ( Base `  R
)
10 eqid 2433 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
119, 5, 10zrhunitpreima 26261 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( `' L " (Unit `  R
) )  =  ( ZZ  \  { 0 } ) )
12 mpt2eq12 6135 . . . 4  |-  ( ( ZZ  =  ZZ  /\  ( `' L " (Unit `  R ) )  =  ( ZZ  \  {
0 } ) )  ->  ( x  e.  ZZ ,  y  e.  ( `' L "
(Unit `  R )
)  |->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)  =  ( x  e.  ZZ ,  y  e.  ( ZZ  \  { 0 } ) 
|->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
) )
138, 11, 12syl2anc 654 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( x  e.  ZZ ,  y  e.  ( `' L "
(Unit `  R )
)  |->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)  =  ( x  e.  ZZ ,  y  e.  ( ZZ  \  { 0 } ) 
|->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
) )
1413rneqd 5054 . 2  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ran  ( x  e.  ZZ ,  y  e.  ( `' L " (Unit `  R )
)  |->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)  =  ran  (
x  e.  ZZ , 
y  e.  ( ZZ 
\  { 0 } )  |->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
) )
15 nfv 1672 . . . 4  |-  F/ e ( R  e.  DivRing  /\  (chr `  R )  =  0 )
16 nfab1 2571 . . . 4  |-  F/_ e { e  |  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >. }
17 nfcv 2569 . . . 4  |-  F/_ e { <. q ,  s
>.  |  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) ) 
./  ( L `  (denom `  q ) ) ) ) }
18 simpr 458 . . . . . . . . . 10  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  e  =  <. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)
19 zssq 10948 . . . . . . . . . . . 12  |-  ZZ  C_  QQ
20 simplrl 752 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  x  e.  ZZ )
2119, 20sseldi 3342 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  x  e.  QQ )
22 simplrr 753 . . . . . . . . . . . . 13  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  y  e.  ( ZZ  \  { 0 } ) )
2322eldifad 3328 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  y  e.  ZZ )
2419, 23sseldi 3342 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  y  e.  QQ )
2522eldifbd 3329 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  -.  y  e.  { 0 } )
26 elsn 3879 . . . . . . . . . . . . 13  |-  ( y  e.  { 0 }  <-> 
y  =  0 )
2726necon3bbii 2629 . . . . . . . . . . . 12  |-  ( -.  y  e.  { 0 }  <->  y  =/=  0
)
2825, 27sylib 196 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  y  =/=  0 )
29 qdivcl 10962 . . . . . . . . . . 11  |-  ( ( x  e.  QQ  /\  y  e.  QQ  /\  y  =/=  0 )  ->  (
x  /  y )  e.  QQ )
3021, 24, 28, 29syl3anc 1211 . . . . . . . . . 10  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  ( x  /  y )  e.  QQ )
31 simplll 750 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  R  e.  DivRing )
32 simpllr 751 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  (chr `  R
)  =  0 )
339, 3, 5qqhval2lem 26264 . . . . . . . . . . . 12  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ  /\  y  =/=  0 ) )  -> 
( ( L `  (numer `  ( x  / 
y ) ) ) 
./  ( L `  (denom `  ( x  / 
y ) ) ) )  =  ( ( L `  x ) 
./  ( L `  y ) ) )
3433eqcomd 2438 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  ZZ  /\  y  e.  ZZ  /\  y  =/=  0 ) )  -> 
( ( L `  x )  ./  ( L `  y )
)  =  ( ( L `  (numer `  ( x  /  y
) ) )  ./  ( L `  (denom `  ( x  /  y
) ) ) ) )
3531, 32, 20, 23, 28, 34syl23anc 1218 . . . . . . . . . 10  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  ( ( L `  x )  ./  ( L `  y
) )  =  ( ( L `  (numer `  ( x  /  y
) ) )  ./  ( L `  (denom `  ( x  /  y
) ) ) ) )
36 ovex 6105 . . . . . . . . . . 11  |-  ( x  /  y )  e. 
_V
37 ovex 6105 . . . . . . . . . . 11  |-  ( ( L `  x ) 
./  ( L `  y ) )  e. 
_V
38 opeq12 4049 . . . . . . . . . . . . 13  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  ->  <. q ,  s >.  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)
3938eqeq2d 2444 . . . . . . . . . . . 12  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( e  =  <. q ,  s >.  <->  e  =  <. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
) )
40 simpl 454 . . . . . . . . . . . . . 14  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
q  =  ( x  /  y ) )
4140eleq1d 2499 . . . . . . . . . . . . 13  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( q  e.  QQ  <->  ( x  /  y )  e.  QQ ) )
42 simpr 458 . . . . . . . . . . . . . 14  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
s  =  ( ( L `  x ) 
./  ( L `  y ) ) )
4340fveq2d 5683 . . . . . . . . . . . . . . . 16  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
(numer `  q )  =  (numer `  ( x  /  y ) ) )
4443fveq2d 5683 . . . . . . . . . . . . . . 15  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( L `  (numer `  q ) )  =  ( L `  (numer `  ( x  /  y
) ) ) )
4540fveq2d 5683 . . . . . . . . . . . . . . . 16  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
(denom `  q )  =  (denom `  ( x  /  y ) ) )
4645fveq2d 5683 . . . . . . . . . . . . . . 15  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( L `  (denom `  q ) )  =  ( L `  (denom `  ( x  /  y
) ) ) )
4744, 46oveq12d 6098 . . . . . . . . . . . . . 14  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( ( L `  (numer `  q ) ) 
./  ( L `  (denom `  q ) ) )  =  ( ( L `  (numer `  ( x  /  y
) ) )  ./  ( L `  (denom `  ( x  /  y
) ) ) ) )
4842, 47eqeq12d 2447 . . . . . . . . . . . . 13  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( s  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) )  <-> 
( ( L `  x )  ./  ( L `  y )
)  =  ( ( L `  (numer `  ( x  /  y
) ) )  ./  ( L `  (denom `  ( x  /  y
) ) ) ) ) )
4941, 48anbi12d 703 . . . . . . . . . . . 12  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) ) 
./  ( L `  (denom `  q ) ) ) )  <->  ( (
x  /  y )  e.  QQ  /\  (
( L `  x
)  ./  ( L `  y ) )  =  ( ( L `  (numer `  ( x  / 
y ) ) ) 
./  ( L `  (denom `  ( x  / 
y ) ) ) ) ) ) )
5039, 49anbi12d 703 . . . . . . . . . . 11  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( ( e  = 
<. q ,  s >.  /\  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) )  <->  ( e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.  /\  ( ( x  / 
y )  e.  QQ  /\  ( ( L `  x )  ./  ( L `  y )
)  =  ( ( L `  (numer `  ( x  /  y
) ) )  ./  ( L `  (denom `  ( x  /  y
) ) ) ) ) ) ) )
5136, 37, 50spc2ev 3054 . . . . . . . . . 10  |-  ( ( e  =  <. (
x  /  y ) ,  ( ( L `
 x )  ./  ( L `  y ) ) >.  /\  (
( x  /  y
)  e.  QQ  /\  ( ( L `  x )  ./  ( L `  y )
)  =  ( ( L `  (numer `  ( x  /  y
) ) )  ./  ( L `  (denom `  ( x  /  y
) ) ) ) ) )  ->  E. q E. s ( e  = 
<. q ,  s >.  /\  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) ) )
5218, 30, 35, 51syl12anc 1209 . . . . . . . . 9  |-  ( ( ( ( R  e.  DivRing 
/\  (chr `  R
)  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ( ZZ  \  {
0 } ) ) )  /\  e  = 
<. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
)  ->  E. q E. s ( e  = 
<. q ,  s >.  /\  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) ) )
5352ex 434 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
x  e.  ZZ  /\  y  e.  ( ZZ  \  { 0 } ) ) )  ->  (
e  =  <. (
x  /  y ) ,  ( ( L `
 x )  ./  ( L `  y ) ) >.  ->  E. q E. s ( e  = 
<. q ,  s >.  /\  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) ) ) )
5453rexlimdvva 2838 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.  ->  E. q E. s
( e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) ) )
5554imp 429 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)  ->  E. q E. s ( e  = 
<. q ,  s >.  /\  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) ) )
56 19.42vv 1924 . . . . . . 7  |-  ( E. q E. s ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  <->  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  E. q E. s ( e  = 
<. q ,  s >.  /\  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) ) ) )
57 simprrl 756 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  q  e.  QQ )
58 qnumcl 13801 . . . . . . . . . 10  |-  ( q  e.  QQ  ->  (numer `  q )  e.  ZZ )
5957, 58syl 16 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  (numer `  q )  e.  ZZ )
60 qdencl 13802 . . . . . . . . . . . 12  |-  ( q  e.  QQ  ->  (denom `  q )  e.  NN )
6157, 60syl 16 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  (denom `  q )  e.  NN )
6261nnzd 10734 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  (denom `  q )  e.  ZZ )
63 nnne0 10342 . . . . . . . . . . 11  |-  ( (denom `  q )  e.  NN  ->  (denom `  q )  =/=  0 )
64 elsni 3890 . . . . . . . . . . . 12  |-  ( (denom `  q )  e.  {
0 }  ->  (denom `  q )  =  0 )
6564necon3ai 2641 . . . . . . . . . . 11  |-  ( (denom `  q )  =/=  0  ->  -.  (denom `  q
)  e.  { 0 } )
6661, 63, 653syl 20 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  -.  (denom `  q )  e. 
{ 0 } )
6762, 66eldifd 3327 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  (denom `  q )  e.  ( ZZ  \  { 0 } ) )
68 simprl 748 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  e  =  <. q ,  s
>. )
69 qeqnumdivden 13807 . . . . . . . . . . . 12  |-  ( q  e.  QQ  ->  q  =  ( (numer `  q )  /  (denom `  q ) ) )
7057, 69syl 16 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  q  =  ( (numer `  q )  /  (denom `  q ) ) )
71 simprrr 757 . . . . . . . . . . 11  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  s  =  ( ( L `
 (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) )
7270, 71opeq12d 4055 . . . . . . . . . 10  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  <. q ,  s >.  =  <. ( (numer `  q )  /  (denom `  q )
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) )
>. )
7368, 72eqtrd 2465 . . . . . . . . 9  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  e  =  <. ( (numer `  q )  /  (denom `  q ) ) ,  ( ( L `  (numer `  q ) ) 
./  ( L `  (denom `  q ) ) ) >. )
74 oveq1 6087 . . . . . . . . . . . 12  |-  ( x  =  (numer `  q
)  ->  ( x  /  y )  =  ( (numer `  q
)  /  y ) )
75 fveq2 5679 . . . . . . . . . . . . 13  |-  ( x  =  (numer `  q
)  ->  ( L `  x )  =  ( L `  (numer `  q ) ) )
7675oveq1d 6095 . . . . . . . . . . . 12  |-  ( x  =  (numer `  q
)  ->  ( ( L `  x )  ./  ( L `  y
) )  =  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) )
7774, 76opeq12d 4055 . . . . . . . . . . 11  |-  ( x  =  (numer `  q
)  ->  <. ( x  /  y ) ,  ( ( L `  x )  ./  ( L `  y )
) >.  =  <. (
(numer `  q )  /  y ) ,  ( ( L `  (numer `  q ) ) 
./  ( L `  y ) ) >.
)
7877eqeq2d 2444 . . . . . . . . . 10  |-  ( x  =  (numer `  q
)  ->  ( e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.  <->  e  =  <. ( (numer `  q )  /  y
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) >. ) )
79 oveq2 6088 . . . . . . . . . . . 12  |-  ( y  =  (denom `  q
)  ->  ( (numer `  q )  /  y
)  =  ( (numer `  q )  /  (denom `  q ) ) )
80 fveq2 5679 . . . . . . . . . . . . 13  |-  ( y  =  (denom `  q
)  ->  ( L `  y )  =  ( L `  (denom `  q ) ) )
8180oveq2d 6096 . . . . . . . . . . . 12  |-  ( y  =  (denom `  q
)  ->  ( ( L `  (numer `  q
) )  ./  ( L `  y )
)  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) )
8279, 81opeq12d 4055 . . . . . . . . . . 11  |-  ( y  =  (denom `  q
)  ->  <. ( (numer `  q )  /  y
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) >.  =  <. ( (numer `  q )  /  (denom `  q )
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) )
>. )
8382eqeq2d 2444 . . . . . . . . . 10  |-  ( y  =  (denom `  q
)  ->  ( e  =  <. ( (numer `  q )  /  y
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) >.  <->  e  =  <. ( (numer `  q )  /  (denom `  q )
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) )
>. ) )
8478, 83rspc2ev 3070 . . . . . . . . 9  |-  ( ( (numer `  q )  e.  ZZ  /\  (denom `  q )  e.  ( ZZ  \  { 0 } )  /\  e  =  <. ( (numer `  q )  /  (denom `  q ) ) ,  ( ( L `  (numer `  q ) ) 
./  ( L `  (denom `  q ) ) ) >. )  ->  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)
8559, 67, 73, 84syl3anc 1211 . . . . . . . 8  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)
8685exlimivv 1688 . . . . . . 7  |-  ( E. q E. s ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  (
e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )  ->  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)
8756, 86sylbir 213 . . . . . 6  |-  ( ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  /\  E. q E. s ( e  =  <. q ,  s
>.  /\  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) ) 
./  ( L `  (denom `  q ) ) ) ) ) )  ->  E. x  e.  ZZ  E. y  e.  ( ZZ 
\  { 0 } ) e  =  <. ( x  /  y ) ,  ( ( L `
 x )  ./  ( L `  y ) ) >. )
8855, 87impbida 821 . . . . 5  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.  <->  E. q E. s ( e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) ) )
89 abid 2421 . . . . 5  |-  ( e  e.  { e  |  E. x  e.  ZZ  E. y  e.  ( ZZ 
\  { 0 } ) e  =  <. ( x  /  y ) ,  ( ( L `
 x )  ./  ( L `  y ) ) >. }  <->  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)
90 elopab 4585 . . . . 5  |-  ( e  e.  { <. q ,  s >.  |  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) }  <->  E. q E. s
( e  =  <. q ,  s >.  /\  (
q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) ) )
9188, 89, 903bitr4g 288 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( e  e.  { e  |  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >. } 
<->  e  e.  { <. q ,  s >.  |  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) } ) )
9215, 16, 17, 91eqrd 3362 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  { e  |  E. x  e.  ZZ  E. y  e.  ( ZZ 
\  { 0 } ) e  =  <. ( x  /  y ) ,  ( ( L `
 x )  ./  ( L `  y ) ) >. }  =  { <. q ,  s >.  |  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) ) 
./  ( L `  (denom `  q ) ) ) ) } )
93 eqid 2433 . . . 4  |-  ( x  e.  ZZ ,  y  e.  ( ZZ  \  { 0 } ) 
|->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)  =  ( x  e.  ZZ ,  y  e.  ( ZZ  \  { 0 } ) 
|->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)
9493rnmpt2 6189 . . 3  |-  ran  (
x  e.  ZZ , 
y  e.  ( ZZ 
\  { 0 } )  |->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)  =  { e  |  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >. }
95 df-mpt 4340 . . 3  |-  ( q  e.  QQ  |->  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) )  =  { <. q ,  s >.  |  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q
) )  ./  ( L `  (denom `  q
) ) ) ) }
9692, 94, 953eqtr4g 2490 . 2  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ran  ( x  e.  ZZ ,  y  e.  ( ZZ  \  { 0 } ) 
|->  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)  =  ( q  e.  QQ  |->  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) )
977, 14, 963eqtrd 2469 1  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  =  ( q  e.  QQ  |->  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362   E.wex 1589    e. wcel 1755   {cab 2419    =/= wne 2596   E.wrex 2706   _Vcvv 2962    \ cdif 3313   {csn 3865   <.cop 3871   {copab 4337    e. cmpt 4338   `'ccnv 4826   ran crn 4828   "cima 4830   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   0cc0 9270    / cdiv 9981   NNcn 10310   ZZcz 10634   QQcq 10941  numercnumer 13794  denomcdenom 13795   Basecbs 14157   0gc0g 14361   1rcur 16579  Unitcui 16665  /rcdvr 16708   DivRingcdr 16756   ZRHomczrh 17773  chrcchr 17775  QQHomcqqh 26255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-tpos 6734  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-fz 11425  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-dvds 13519  df-gcd 13674  df-numer 13796  df-denom 13797  df-gz 13974  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-0g 14363  df-mnd 15398  df-mhm 15447  df-grp 15525  df-minusg 15526  df-sbg 15527  df-mulg 15528  df-subg 15658  df-ghm 15725  df-od 16012  df-cmn 16259  df-mgp 16566  df-rng 16580  df-cring 16581  df-ur 16582  df-oppr 16649  df-dvdsr 16667  df-unit 16668  df-invr 16698  df-dvr 16709  df-rnghom 16740  df-drng 16758  df-subrg 16787  df-cnfld 17663  df-zring 17726  df-zrh 17777  df-chr 17779  df-qqh 26256
This theorem is referenced by:  qqhvval  26266  qqhf  26269
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