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Theorem qqhval2 26433
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 3002 . . . 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 2443 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 qqhval2.2 . . . 4  |-  L  =  ( ZRHom `  R
)
63, 4, 5qqhval 26425 . . 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 2444 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ZZ  =  ZZ )
9 qqhval2.0 . . . . 5  |-  B  =  ( Base `  R
)
10 eqid 2443 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
119, 5, 10zrhunitpreima 26429 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( `' L " (Unit `  R
) )  =  ( ZZ  \  { 0 } ) )
12 mpt2eq12 6167 . . . 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 5088 . 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 1673 . . . 4  |-  F/ e ( R  e.  DivRing  /\  (chr `  R )  =  0 )
16 nfab1 2591 . . . 4  |-  F/_ e { e  |  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >. }
17 nfcv 2589 . . . 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 10981 . . . . . . . . . . . 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 3375 . . . . . . . . . . 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 3361 . . . . . . . . . . . 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 3375 . . . . . . . . . . 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 3362 . . . . . . . . . . . 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 3912 . . . . . . . . . . . . 13  |-  ( y  e.  { 0 }  <-> 
y  =  0 )
2726necon3bbii 2633 . . . . . . . . . . . 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 10995 . . . . . . . . . . 11  |-  ( ( x  e.  QQ  /\  y  e.  QQ  /\  y  =/=  0 )  ->  (
x  /  y )  e.  QQ )
3021, 24, 28, 29syl3anc 1218 . . . . . . . . . 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 26432 . . . . . . . . . . . 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 2448 . . . . . . . . . . 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 1225 . . . . . . . . . 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 6137 . . . . . . . . . . 11  |-  ( x  /  y )  e. 
_V
37 ovex 6137 . . . . . . . . . . 11  |-  ( ( L `  x ) 
./  ( L `  y ) )  e. 
_V
38 opeq12 4082 . . . . . . . . . . . . 13  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  ->  <. q ,  s >.  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)
3938eqeq2d 2454 . . . . . . . . . . . 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 2509 . . . . . . . . . . . . 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 5716 . . . . . . . . . . . . . . . 16  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
(numer `  q )  =  (numer `  ( x  /  y ) ) )
4443fveq2d 5716 . . . . . . . . . . . . . . 15  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( L `  (numer `  q ) )  =  ( L `  (numer `  ( x  /  y
) ) ) )
4540fveq2d 5716 . . . . . . . . . . . . . . . 16  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
(denom `  q )  =  (denom `  ( x  /  y ) ) )
4645fveq2d 5716 . . . . . . . . . . . . . . 15  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( L `  (denom `  q ) )  =  ( L `  (denom `  ( x  /  y
) ) ) )
4744, 46oveq12d 6130 . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . 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 3086 . . . . . . . . . 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 1216 . . . . . . . . 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 2869 . . . . . . 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 1926 . . . . . . 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 13839 . . . . . . . . . 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 13840 . . . . . . . . . . . 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 10767 . . . . . . . . . 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 10375 . . . . . . . . . . 11  |-  ( (denom `  q )  e.  NN  ->  (denom `  q )  =/=  0 )
64 elsni 3923 . . . . . . . . . . . 12  |-  ( (denom `  q )  e.  {
0 }  ->  (denom `  q )  =  0 )
6564necon3ai 2675 . . . . . . . . . . 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 3360 . . . . . . . . 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 13845 . . . . . . . . . . . 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 4088 . . . . . . . . . 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 2475 . . . . . . . . 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 6119 . . . . . . . . . . . 12  |-  ( x  =  (numer `  q
)  ->  ( x  /  y )  =  ( (numer `  q
)  /  y ) )
75 fveq2 5712 . . . . . . . . . . . . 13  |-  ( x  =  (numer `  q
)  ->  ( L `  x )  =  ( L `  (numer `  q ) ) )
7675oveq1d 6127 . . . . . . . . . . . 12  |-  ( x  =  (numer `  q
)  ->  ( ( L `  x )  ./  ( L `  y
) )  =  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) )
7774, 76opeq12d 4088 . . . . . . . . . . 11  |-  ( x  =  (numer `  q
)  ->  <. ( x  /  y ) ,  ( ( L `  x )  ./  ( L `  y )
) >.  =  <. (
(numer `  q )  /  y ) ,  ( ( L `  (numer `  q ) ) 
./  ( L `  y ) ) >.
)
7877eqeq2d 2454 . . . . . . . . . 10  |-  ( x  =  (numer `  q
)  ->  ( e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.  <->  e  =  <. ( (numer `  q )  /  y
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) >. ) )
79 oveq2 6120 . . . . . . . . . . . 12  |-  ( y  =  (denom `  q
)  ->  ( (numer `  q )  /  y
)  =  ( (numer `  q )  /  (denom `  q ) ) )
80 fveq2 5712 . . . . . . . . . . . . 13  |-  ( y  =  (denom `  q
)  ->  ( L `  y )  =  ( L `  (denom `  q ) ) )
8180oveq2d 6128 . . . . . . . . . . . 12  |-  ( y  =  (denom `  q
)  ->  ( ( L `  (numer `  q
) )  ./  ( L `  y )
)  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) )
8279, 81opeq12d 4088 . . . . . . . . . . 11  |-  ( y  =  (denom `  q
)  ->  <. ( (numer `  q )  /  y
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) >.  =  <. ( (numer `  q )  /  (denom `  q )
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) )
>. )
8382eqeq2d 2454 . . . . . . . . . 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 3102 . . . . . . . . 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 1218 . . . . . . . 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 1689 . . . . . . 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 828 . . . . 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 2431 . . . . 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 4618 . . . . 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 3395 . . 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 2443 . . . 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 6221 . . 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 4373 . . 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 2500 . 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 2479 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 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    =/= wne 2620   E.wrex 2737   _Vcvv 2993    \ cdif 3346   {csn 3898   <.cop 3904   {copab 4370    e. cmpt 4371   `'ccnv 4860   ran crn 4862   "cima 4864   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   0cc0 9303    / cdiv 10014   NNcn 10343   ZZcz 10667   QQcq 10974  numercnumer 13832  denomcdenom 13833   Basecbs 14195   0gc0g 14399   1rcur 16625  Unitcui 16753  /rcdvr 16796   DivRingcdr 16854   ZRHomczrh 17953  chrcchr 17955  QQHomcqqh 26423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-fz 11459  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-dvds 13557  df-gcd 13712  df-numer 13834  df-denom 13835  df-gz 14012  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-0g 14401  df-mnd 15436  df-mhm 15485  df-grp 15566  df-minusg 15567  df-sbg 15568  df-mulg 15569  df-subg 15699  df-ghm 15766  df-od 16053  df-cmn 16300  df-mgp 16614  df-ur 16626  df-rng 16669  df-cring 16670  df-oppr 16737  df-dvdsr 16755  df-unit 16756  df-invr 16786  df-dvr 16797  df-rnghom 16828  df-drng 16856  df-subrg 16885  df-cnfld 17841  df-zring 17906  df-zrh 17957  df-chr 17959  df-qqh 26424
This theorem is referenced by:  qqhvval  26434  qqhf  26437
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