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Theorem qqhval2 28651
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 3087 . . . 4  |-  ( R  e.  DivRing  ->  R  e.  _V )
21adantr 466 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  R  e.  _V )
3 qqhval2.1 . . . 4  |-  ./  =  (/r
`  R )
4 eqid 2420 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 qqhval2.2 . . . 4  |-  L  =  ( ZRHom `  R
)
63, 4, 5qqhval 28643 . . 3  |-  ( R  e.  _V  ->  (QQHom `  R )  =  ran  ( x  e.  ZZ ,  y  e.  ( `' L " (Unit `  R ) )  |->  <.
( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
) )
72, 6syl 17 . 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 2421 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ZZ  =  ZZ )
9 qqhval2.0 . . . . 5  |-  B  =  ( Base `  R
)
10 eqid 2420 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
119, 5, 10zrhunitpreima 28647 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( `' L " (Unit `  R
) )  =  ( ZZ  \  { 0 } ) )
12 mpt2eq12 6356 . . . 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 665 . . 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 5073 . 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 1751 . . . 4  |-  F/ e ( R  e.  DivRing  /\  (chr `  R )  =  0 )
16 nfab1 2584 . . . 4  |-  F/_ e { e  |  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >. }
17 nfcv 2582 . . . 4  |-  F/_ e { <. q ,  s
>.  |  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) ) 
./  ( L `  (denom `  q ) ) ) ) }
18 simpr 462 . . . . . . . . . 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 11260 . . . . . . . . . . . 12  |-  ZZ  C_  QQ
20 simplrl 768 . . . . . . . . . . . 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 3459 . . . . . . . . . . 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 769 . . . . . . . . . . . . 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 3445 . . . . . . . . . . . 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 3459 . . . . . . . . . . 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 3446 . . . . . . . . . . . 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 4007 . . . . . . . . . . . . 13  |-  ( y  e.  { 0 }  <-> 
y  =  0 )
2726necon3bbii 2683 . . . . . . . . . . . 12  |-  ( -.  y  e.  { 0 }  <->  y  =/=  0
)
2825, 27sylib 199 . . . . . . . . . . 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 11274 . . . . . . . . . . 11  |-  ( ( x  e.  QQ  /\  y  e.  QQ  /\  y  =/=  0 )  ->  (
x  /  y )  e.  QQ )
3021, 24, 28, 29syl3anc 1264 . . . . . . . . . 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 766 . . . . . . . . . . 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 767 . . . . . . . . . . 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 28650 . . . . . . . . . . . 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 2428 . . . . . . . . . . 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 1271 . . . . . . . . . 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 6324 . . . . . . . . . . 11  |-  ( x  /  y )  e. 
_V
37 ovex 6324 . . . . . . . . . . 11  |-  ( ( L `  x ) 
./  ( L `  y ) )  e. 
_V
38 opeq12 4183 . . . . . . . . . . . . 13  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  ->  <. q ,  s >.  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)
3938eqeq2d 2434 . . . . . . . . . . . 12  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( e  =  <. q ,  s >.  <->  e  =  <. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
) )
40 simpl 458 . . . . . . . . . . . . . 14  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
q  =  ( x  /  y ) )
4140eleq1d 2489 . . . . . . . . . . . . 13  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( q  e.  QQ  <->  ( x  /  y )  e.  QQ ) )
42 simpr 462 . . . . . . . . . . . . . 14  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
s  =  ( ( L `  x ) 
./  ( L `  y ) ) )
4340fveq2d 5876 . . . . . . . . . . . . . . . 16  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
(numer `  q )  =  (numer `  ( x  /  y ) ) )
4443fveq2d 5876 . . . . . . . . . . . . . . 15  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( L `  (numer `  q ) )  =  ( L `  (numer `  ( x  /  y
) ) ) )
4540fveq2d 5876 . . . . . . . . . . . . . . . 16  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
(denom `  q )  =  (denom `  ( x  /  y ) ) )
4645fveq2d 5876 . . . . . . . . . . . . . . 15  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( L `  (denom `  q ) )  =  ( L `  (denom `  ( x  /  y
) ) ) )
4744, 46oveq12d 6314 . . . . . . . . . . . . . 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 2442 . . . . . . . . . . . . 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 715 . . . . . . . . . . . 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 715 . . . . . . . . . . 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 3171 . . . . . . . . . 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 1262 . . . . . . . . 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 435 . . . . . . . 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 2922 . . . . . . 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 430 . . . . . 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 1825 . . . . . . 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 772 . . . . . . . . . 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 14649 . . . . . . . . . 10  |-  ( q  e.  QQ  ->  (numer `  q )  e.  ZZ )
5957, 58syl 17 . . . . . . . . 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 14650 . . . . . . . . . . . 12  |-  ( q  e.  QQ  ->  (denom `  q )  e.  NN )
6157, 60syl 17 . . . . . . . . . . 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 11028 . . . . . . . . . 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 10631 . . . . . . . . . . 11  |-  ( (denom `  q )  e.  NN  ->  (denom `  q )  =/=  0 )
64 elsni 4018 . . . . . . . . . . . 12  |-  ( (denom `  q )  e.  {
0 }  ->  (denom `  q )  =  0 )
6564necon3ai 2650 . . . . . . . . . . 11  |-  ( (denom `  q )  =/=  0  ->  -.  (denom `  q
)  e.  { 0 } )
6661, 63, 653syl 18 . . . . . . . . . 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 3444 . . . . . . . . 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 762 . . . . . . . . . 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 14655 . . . . . . . . . . . 12  |-  ( q  e.  QQ  ->  q  =  ( (numer `  q )  /  (denom `  q ) ) )
7057, 69syl 17 . . . . . . . . . . 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 773 . . . . . . . . . . 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 4189 . . . . . . . . . 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 2461 . . . . . . . . 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 6303 . . . . . . . . . . . 12  |-  ( x  =  (numer `  q
)  ->  ( x  /  y )  =  ( (numer `  q
)  /  y ) )
75 fveq2 5872 . . . . . . . . . . . . 13  |-  ( x  =  (numer `  q
)  ->  ( L `  x )  =  ( L `  (numer `  q ) ) )
7675oveq1d 6311 . . . . . . . . . . . 12  |-  ( x  =  (numer `  q
)  ->  ( ( L `  x )  ./  ( L `  y
) )  =  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) )
7774, 76opeq12d 4189 . . . . . . . . . . 11  |-  ( x  =  (numer `  q
)  ->  <. ( x  /  y ) ,  ( ( L `  x )  ./  ( L `  y )
) >.  =  <. (
(numer `  q )  /  y ) ,  ( ( L `  (numer `  q ) ) 
./  ( L `  y ) ) >.
)
7877eqeq2d 2434 . . . . . . . . . 10  |-  ( x  =  (numer `  q
)  ->  ( e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.  <->  e  =  <. ( (numer `  q )  /  y
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) >. ) )
79 oveq2 6304 . . . . . . . . . . . 12  |-  ( y  =  (denom `  q
)  ->  ( (numer `  q )  /  y
)  =  ( (numer `  q )  /  (denom `  q ) ) )
80 fveq2 5872 . . . . . . . . . . . . 13  |-  ( y  =  (denom `  q
)  ->  ( L `  y )  =  ( L `  (denom `  q ) ) )
8180oveq2d 6312 . . . . . . . . . . . 12  |-  ( y  =  (denom `  q
)  ->  ( ( L `  (numer `  q
) )  ./  ( L `  y )
)  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) )
8279, 81opeq12d 4189 . . . . . . . . . . 11  |-  ( y  =  (denom `  q
)  ->  <. ( (numer `  q )  /  y
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) >.  =  <. ( (numer `  q )  /  (denom `  q )
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) )
>. )
8382eqeq2d 2434 . . . . . . . . . 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 3190 . . . . . . . . 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 1264 . . . . . . . 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 1767 . . . . . . 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 216 . . . . . 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 840 . . . . 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 2407 . . . . 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 4720 . . . . 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 291 . . . 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 3479 . . 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 2420 . . . 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 6411 . . 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 4477 . . 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 2486 . 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 2465 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 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1867   {cab 2405    =/= wne 2616   E.wrex 2774   _Vcvv 3078    \ cdif 3430   {csn 3993   <.cop 3999   {copab 4474    |-> cmpt 4475   `'ccnv 4844   ran crn 4846   "cima 4848   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   0cc0 9528    / cdiv 10258   NNcn 10598   ZZcz 10926   QQcq 11253  numercnumer 14642  denomcdenom 14643   Basecbs 15073   0gc0g 15290   1rcur 17663  Unitcui 17795  /rcdvr 17838   DivRingcdr 17903   ZRHomczrh 18995  chrcchr 18997  QQHomcqqh 28641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-tpos 6972  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-inf 7954  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-fz 11772  df-fl 12014  df-mod 12083  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-dvds 14273  df-gcd 14432  df-numer 14644  df-denom 14645  df-gz 14826  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-starv 15157  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-0g 15292  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-mhm 16526  df-grp 16617  df-minusg 16618  df-sbg 16619  df-mulg 16620  df-subg 16758  df-ghm 16825  df-od 17113  df-cmn 17360  df-mgp 17652  df-ur 17664  df-ring 17710  df-cring 17711  df-oppr 17779  df-dvdsr 17797  df-unit 17798  df-invr 17828  df-dvr 17839  df-rnghom 17871  df-drng 17905  df-subrg 17934  df-cnfld 18899  df-zring 18967  df-zrh 18999  df-chr 19001  df-qqh 28642
This theorem is referenced by:  qqhvval  28652  qqhf  28655
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