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Theorem qqhval2 24319
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 2924 . . . 4  |-  ( R  e.  DivRing  ->  R  e.  _V )
21adantr 452 . . 3  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  R  e.  _V )
3 qqhval2.1 . . . 4  |-  ./  =  (/r
`  R )
4 eqid 2404 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
5 qqhval2.2 . . . 4  |-  L  =  ( ZRHom `  R
)
63, 4, 5qqhval 24311 . . 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 2405 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ZZ  =  ZZ )
9 qqhval2.0 . . . . 5  |-  B  =  ( Base `  R
)
10 eqid 2404 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
119, 5, 10zrhunitpreima 24315 . . . 4  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  ( `' L " (Unit `  R
) )  =  ( ZZ  \  { 0 } ) )
12 mpt2eq12 6093 . . . 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 643 . . 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 5056 . 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 1626 . . . 4  |-  F/ e ( R  e.  DivRing  /\  (chr `  R )  =  0 )
16 nfab1 2542 . . . 4  |-  F/_ e { e  |  E. x  e.  ZZ  E. y  e.  ( ZZ  \  {
0 } ) e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >. }
17 nfcv 2540 . . . 4  |-  F/_ e { <. q ,  s
>.  |  ( q  e.  QQ  /\  s  =  ( ( L `  (numer `  q ) ) 
./  ( L `  (denom `  q ) ) ) ) }
18 simpr 448 . . . . . . . . . 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 10537 . . . . . . . . . . . 12  |-  ZZ  C_  QQ
20 simplrl 737 . . . . . . . . . . . 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 3306 . . . . . . . . . . 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 738 . . . . . . . . . . . . 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 3292 . . . . . . . . . . . 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 3306 . . . . . . . . . . 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 3293 . . . . . . . . . . . 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 3789 . . . . . . . . . . . . 13  |-  ( y  e.  { 0 }  <-> 
y  =  0 )
2726necon3bbii 2598 . . . . . . . . . . . 12  |-  ( -.  y  e.  { 0 }  <->  y  =/=  0
)
2825, 27sylib 189 . . . . . . . . . . 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 10551 . . . . . . . . . . 11  |-  ( ( x  e.  QQ  /\  y  e.  QQ  /\  y  =/=  0 )  ->  (
x  /  y )  e.  QQ )
3021, 24, 28, 29syl3anc 1184 . . . . . . . . . 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 735 . . . . . . . . . . 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 736 . . . . . . . . . . 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 24318 . . . . . . . . . . . 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 2409 . . . . . . . . . . 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 1191 . . . . . . . . . 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 6065 . . . . . . . . . . 11  |-  ( x  /  y )  e. 
_V
37 ovex 6065 . . . . . . . . . . 11  |-  ( ( L `  x ) 
./  ( L `  y ) )  e. 
_V
38 opeq12 3946 . . . . . . . . . . . . 13  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  ->  <. q ,  s >.  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.
)
3938eqeq2d 2415 . . . . . . . . . . . 12  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( e  =  <. q ,  s >.  <->  e  =  <. ( x  /  y
) ,  ( ( L `  x ) 
./  ( L `  y ) ) >.
) )
40 simpl 444 . . . . . . . . . . . . . 14  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
q  =  ( x  /  y ) )
4140eleq1d 2470 . . . . . . . . . . . . 13  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( q  e.  QQ  <->  ( x  /  y )  e.  QQ ) )
42 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
s  =  ( ( L `  x ) 
./  ( L `  y ) ) )
4340fveq2d 5691 . . . . . . . . . . . . . . . 16  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
(numer `  q )  =  (numer `  ( x  /  y ) ) )
4443fveq2d 5691 . . . . . . . . . . . . . . 15  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( L `  (numer `  q ) )  =  ( L `  (numer `  ( x  /  y
) ) ) )
4540fveq2d 5691 . . . . . . . . . . . . . . . 16  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
(denom `  q )  =  (denom `  ( x  /  y ) ) )
4645fveq2d 5691 . . . . . . . . . . . . . . 15  |-  ( ( q  =  ( x  /  y )  /\  s  =  ( ( L `  x )  ./  ( L `  y
) ) )  -> 
( L `  (denom `  q ) )  =  ( L `  (denom `  ( x  /  y
) ) ) )
4744, 46oveq12d 6058 . . . . . . . . . . . . . 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 2418 . . . . . . . . . . . . 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 692 . . . . . . . . . . . 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 692 . . . . . . . . . . 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 3004 . . . . . . . . . 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 1182 . . . . . . . . 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 424 . . . . . . . 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 2797 . . . . . . 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 419 . . . . . 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 741 . . . . . . . . . 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 13087 . . . . . . . . . 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 13088 . . . . . . . . . . . 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 10330 . . . . . . . . . 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 9988 . . . . . . . . . . 11  |-  ( (denom `  q )  e.  NN  ->  (denom `  q )  =/=  0 )
64 elsni 3798 . . . . . . . . . . . 12  |-  ( (denom `  q )  e.  {
0 }  ->  (denom `  q )  =  0 )
6564necon3ai 2607 . . . . . . . . . . 11  |-  ( (denom `  q )  =/=  0  ->  -.  (denom `  q
)  e.  { 0 } )
6661, 63, 653syl 19 . . . . . . . . . 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 3291 . . . . . . . . 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 733 . . . . . . . . . 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 13093 . . . . . . . . . . . 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 742 . . . . . . . . . . 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 3952 . . . . . . . . . 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 2436 . . . . . . . . 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 6047 . . . . . . . . . . . 12  |-  ( x  =  (numer `  q
)  ->  ( x  /  y )  =  ( (numer `  q
)  /  y ) )
75 fveq2 5687 . . . . . . . . . . . . 13  |-  ( x  =  (numer `  q
)  ->  ( L `  x )  =  ( L `  (numer `  q ) ) )
7675oveq1d 6055 . . . . . . . . . . . 12  |-  ( x  =  (numer `  q
)  ->  ( ( L `  x )  ./  ( L `  y
) )  =  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) )
7774, 76opeq12d 3952 . . . . . . . . . . 11  |-  ( x  =  (numer `  q
)  ->  <. ( x  /  y ) ,  ( ( L `  x )  ./  ( L `  y )
) >.  =  <. (
(numer `  q )  /  y ) ,  ( ( L `  (numer `  q ) ) 
./  ( L `  y ) ) >.
)
7877eqeq2d 2415 . . . . . . . . . 10  |-  ( x  =  (numer `  q
)  ->  ( e  =  <. ( x  / 
y ) ,  ( ( L `  x
)  ./  ( L `  y ) ) >.  <->  e  =  <. ( (numer `  q )  /  y
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) >. ) )
79 oveq2 6048 . . . . . . . . . . . 12  |-  ( y  =  (denom `  q
)  ->  ( (numer `  q )  /  y
)  =  ( (numer `  q )  /  (denom `  q ) ) )
80 fveq2 5687 . . . . . . . . . . . . 13  |-  ( y  =  (denom `  q
)  ->  ( L `  y )  =  ( L `  (denom `  q ) ) )
8180oveq2d 6056 . . . . . . . . . . . 12  |-  ( y  =  (denom `  q
)  ->  ( ( L `  (numer `  q
) )  ./  ( L `  y )
)  =  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) )
8279, 81opeq12d 3952 . . . . . . . . . . 11  |-  ( y  =  (denom `  q
)  ->  <. ( (numer `  q )  /  y
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  y ) ) >.  =  <. ( (numer `  q )  /  (denom `  q )
) ,  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) )
>. )
8382eqeq2d 2415 . . . . . . . . . 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 3020 . . . . . . . . 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 1184 . . . . . . . 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 1642 . . . . . . 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 205 . . . . . 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 806 . . . . 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 2392 . . . . 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 4422 . . . . 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 280 . . . 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 3326 . . 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 2404 . . . 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 6139 . . 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 4228 . . 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 2461 . 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 2440 1  |-  ( ( R  e.  DivRing  /\  (chr `  R )  =  0 )  ->  (QQHom `  R
)  =  ( q  e.  QQ  |->  ( ( L `  (numer `  q ) )  ./  ( L `  (denom `  q ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   E.wrex 2667   _Vcvv 2916    \ cdif 3277   {csn 3774   <.cop 3777   {copab 4225    e. cmpt 4226   `'ccnv 4836   ran crn 4838   "cima 4840   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   0cc0 8946    / cdiv 9633   NNcn 9956   ZZcz 10238   QQcq 10530  numercnumer 13080  denomcdenom 13081   Basecbs 13424   0gc0g 13678   1rcur 15617  Unitcui 15699  /rcdvr 15742   DivRingcdr 15790   ZRHomczrh 16733  chrcchr 16735  QQHomcqqh 24309
This theorem is referenced by:  qqhvval  24320  qqhf  24323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-numer 13082  df-denom 13083  df-gz 13253  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-mnd 14645  df-mhm 14693  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-od 15122  df-cmn 15369  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774  df-drng 15792  df-subrg 15821  df-cnfld 16659  df-zrh 16737  df-chr 16739  df-qqh 24310
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