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Theorem qqhval2 27627
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 3122 . . . 4  |-  ( R  e.  DivRing  ->  R  e.  _V )
21adantr 465 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  R  e.  _V )
3 qqhval2.1 . . . 4  |-  ./  =  (/r
`  R )
4 eqid 2467 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 qqhval2.2 . . . 4  |-  L  =  ( ZRHom `  R
)
63, 4, 5qqhval 27619 . . 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 2468 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ZZ  =  ZZ )
9 qqhval2.0 . . . . 5  |-  B  =  ( Base `  R
)
10 eqid 2467 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
119, 5, 10zrhunitpreima 27623 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( `' L " (Unit `  R
) )  =  ( ZZ  \  { 0 } ) )
12 mpt2eq12 6341 . . . 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 661 . . 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 5230 . 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 1683 . . . 4  |-  F/ e ( R  e.  DivRing  /\  (chr `  R )  =  0 )
16 nfab1 2631 . . . 4  |-  F/_ e { e  |  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >. }
17 nfcv 2629 . . . 4  |-  F/_ e { <. q ,  s
>.  |  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) ) 
./  ( L `  (denom `  q ) ) ) ) }
18 simpr 461 . . . . . . . . . 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 11189 . . . . . . . . . . . 12  |-  ZZ  C_  QQ
20 simplrl 759 . . . . . . . . . . . 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 3502 . . . . . . . . . . 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 760 . . . . . . . . . . . . 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 3488 . . . . . . . . . . . 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 3502 . . . . . . . . . . 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 3489 . . . . . . . . . . . 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 4041 . . . . . . . . . . . . 13  |-  ( y  e.  { 0 }  <-> 
y  =  0 )
2726necon3bbii 2728 . . . . . . . . . . . 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 11203 . . . . . . . . . . 11  |-  ( ( x  e.  QQ  /\  y  e.  QQ  /\  y  =/=  0 )  ->  (
x  /  y )  e.  QQ )
3021, 24, 28, 29syl3anc 1228 . . . . . . . . . 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 757 . . . . . . . . . . 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 758 . . . . . . . . . . 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 27626 . . . . . . . . . . . 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 2475 . . . . . . . . . . 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 1235 . . . . . . . . . 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 6309 . . . . . . . . . . 11  |-  ( x  /  y )  e. 
_V
37 ovex 6309 . . . . . . . . . . 11  |-  ( ( L `  x ) 
./  ( L `  y ) )  e. 
_V
38 opeq12 4215 . . . . . . . . . . . . 13  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  ->  <. q ,  s >.  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)
3938eqeq2d 2481 . . . . . . . . . . . 12  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( e  =  <. q ,  s >.  <->  e  =  <. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
) )
40 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
q  =  ( x  /  y ) )
4140eleq1d 2536 . . . . . . . . . . . . 13  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( q  e.  QQ  <->  ( x  /  y )  e.  QQ ) )
42 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
s  =  ( ( L `  x ) 
./  ( L `  y ) ) )
4340fveq2d 5870 . . . . . . . . . . . . . . . 16  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
(numer `  q )  =  (numer `  ( x  /  y ) ) )
4443fveq2d 5870 . . . . . . . . . . . . . . 15  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( L `  (numer `  q ) )  =  ( L `  (numer `  ( x  /  y
) ) ) )
4540fveq2d 5870 . . . . . . . . . . . . . . . 16  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
(denom `  q )  =  (denom `  ( x  /  y ) ) )
4645fveq2d 5870 . . . . . . . . . . . . . . 15  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( L `  (denom `  q ) )  =  ( L `  (denom `  ( x  /  y
) ) ) )
4744, 46oveq12d 6302 . . . . . . . . . . . . . 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 2489 . . . . . . . . . . . . 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 710 . . . . . . . . . . . 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 710 . . . . . . . . . . 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 3206 . . . . . . . . . 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 1226 . . . . . . . . 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 2962 . . . . . . 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 1951 . . . . . . 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 763 . . . . . . . . . 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 14132 . . . . . . . . . 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 14133 . . . . . . . . . . . 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 10965 . . . . . . . . . 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 10568 . . . . . . . . . . 11  |-  ( (denom `  q )  e.  NN  ->  (denom `  q )  =/=  0 )
64 elsni 4052 . . . . . . . . . . . 12  |-  ( (denom `  q )  e.  {
0 }  ->  (denom `  q )  =  0 )
6564necon3ai 2695 . . . . . . . . . . 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 3487 . . . . . . . . 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 755 . . . . . . . . . 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 14138 . . . . . . . . . . . 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 764 . . . . . . . . . . 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 4221 . . . . . . . . . 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 2508 . . . . . . . . 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 6291 . . . . . . . . . . . 12  |-  ( x  =  (numer `  q
)  ->  ( x  /  y )  =  ( (numer `  q
)  /  y ) )
75 fveq2 5866 . . . . . . . . . . . . 13  |-  ( x  =  (numer `  q
)  ->  ( L `  x )  =  ( L `  (numer `  q ) ) )
7675oveq1d 6299 . . . . . . . . . . . 12  |-  ( x  =  (numer `  q
)  ->  ( ( L `  x )  ./  ( L `  y
) )  =  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) )
7774, 76opeq12d 4221 . . . . . . . . . . 11  |-  ( x  =  (numer `  q
)  ->  <. ( x  /  y ) ,  ( ( L `  x )  ./  ( L `  y )
) >.  =  <. (
(numer `  q )  /  y ) ,  ( ( L `  (numer `  q ) ) 
./  ( L `  y ) ) >.
)
7877eqeq2d 2481 . . . . . . . . . 10  |-  ( x  =  (numer `  q
)  ->  ( e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.  <->  e  =  <. ( (numer `  q )  /  y
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) >. ) )
79 oveq2 6292 . . . . . . . . . . . 12  |-  ( y  =  (denom `  q
)  ->  ( (numer `  q )  /  y
)  =  ( (numer `  q )  /  (denom `  q ) ) )
80 fveq2 5866 . . . . . . . . . . . . 13  |-  ( y  =  (denom `  q
)  ->  ( L `  y )  =  ( L `  (denom `  q ) ) )
8180oveq2d 6300 . . . . . . . . . . . 12  |-  ( y  =  (denom `  q
)  ->  ( ( L `  (numer `  q
) )  ./  ( L `  y )
)  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) )
8279, 81opeq12d 4221 . . . . . . . . . . 11  |-  ( y  =  (denom `  q
)  ->  <. ( (numer `  q )  /  y
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) >.  =  <. ( (numer `  q )  /  (denom `  q )
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) )
>. )
8382eqeq2d 2481 . . . . . . . . . 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 3225 . . . . . . . . 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 1228 . . . . . . . 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 1699 . . . . . . 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 830 . . . . 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 2454 . . . . 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 4755 . . . . 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 3522 . . 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 2467 . . . 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 6396 . . 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 4507 . . 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 2533 . 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 2512 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 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   E.wrex 2815   _Vcvv 3113    \ cdif 3473   {csn 4027   <.cop 4033   {copab 4504    |-> cmpt 4505   `'ccnv 4998   ran crn 5000   "cima 5002   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   0cc0 9492    / cdiv 10206   NNcn 10536   ZZcz 10864   QQcq 11182  numercnumer 14125  denomcdenom 14126   Basecbs 14490   0gc0g 14695   1rcur 16955  Unitcui 17089  /rcdvr 17132   DivRingcdr 17196   ZRHomczrh 18332  chrcchr 18334  QQHomcqqh 27617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-fz 11673  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-dvds 13848  df-gcd 14004  df-numer 14127  df-denom 14128  df-gz 14307  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-0g 14697  df-mnd 15732  df-mhm 15786  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-ghm 16070  df-od 16359  df-cmn 16606  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-rnghom 17165  df-drng 17198  df-subrg 17227  df-cnfld 18220  df-zring 18285  df-zrh 18336  df-chr 18338  df-qqh 27618
This theorem is referenced by:  qqhvval  27628  qqhf  27631
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