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Theorem nqpru 7582
Description: Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by 
<P. (Contributed by Jim Kingdon, 29-Nov-2020.)
Assertion
Ref Expression
nqpru  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( A  e.  ( 2nd `  B )  <-> 
B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
Distinct variable group:    A, l, u
Allowed substitution hints:    B( u, l)

Proof of Theorem nqpru
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 prop 7505 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 7519 . . . . . 6  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  E. x  e.  ( 2nd `  B ) x 
<Q  A )
31, 2sylan 283 . . . . 5  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  E. x  e.  ( 2nd `  B ) x 
<Q  A )
4 elprnqu 7512 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
51, 4sylan 283 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
65ad2ant2r 509 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  x  e.  Q. )
7 simprl 529 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  x  e.  ( 2nd `  B ) )
8 vex 2755 . . . . . . . . . . . 12  |-  x  e. 
_V
9 breq1 4021 . . . . . . . . . . . 12  |-  ( l  =  x  ->  (
l  <Q  A  <->  x  <Q  A ) )
108, 9elab 2896 . . . . . . . . . . 11  |-  ( x  e.  { l  |  l  <Q  A }  <->  x 
<Q  A )
1110biimpri 133 . . . . . . . . . 10  |-  ( x 
<Q  A  ->  x  e. 
{ l  |  l 
<Q  A } )
12 ltnqex 7579 . . . . . . . . . . . 12  |-  { l  |  l  <Q  A }  e.  _V
13 gtnqex 7580 . . . . . . . . . . . 12  |-  { u  |  A  <Q  u }  e.  _V
1412, 13op1st 6172 . . . . . . . . . . 11  |-  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  =  { l  |  l 
<Q  A }
1514eleq2i 2256 . . . . . . . . . 10  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  x  e.  { l  |  l  <Q  A }
)
1611, 15sylibr 134 . . . . . . . . 9  |-  ( x 
<Q  A  ->  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) )
1716ad2antll 491 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
18 19.8a 1601 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) )  ->  E. x ( x  e.  Q.  /\  (
x  e.  ( 2nd `  B )  /\  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) ) ) )
196, 7, 17, 18syl12anc 1247 . . . . . . 7  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  E. x ( x  e.  Q.  /\  (
x  e.  ( 2nd `  B )  /\  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) ) ) )
20 df-rex 2474 . . . . . . 7  |-  ( E. x  e.  Q.  (
x  e.  ( 2nd `  B )  /\  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) )  <->  E. x
( x  e.  Q.  /\  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
) ) )
2119, 20sylibr 134 . . . . . 6  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) )
22 elprnqu 7512 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  A  e.  Q. )
231, 22sylan 283 . . . . . . . 8  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  A  e.  Q. )
24 nqprlu 7577 . . . . . . . . 9  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
25 ltdfpr 7536 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  e.  P. )  ->  ( B  <P  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) ) )
2624, 25sylan2 286 . . . . . . . 8  |-  ( ( B  e.  P.  /\  A  e.  Q. )  ->  ( B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  B
)  /\  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
) ) )
2723, 26syldan 282 . . . . . . 7  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  -> 
( B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  B
)  /\  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
) ) )
2827adantr 276 . . . . . 6  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  ( B  <P  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) ) )
2921, 28mpbird 167 . . . . 5  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
303, 29rexlimddv 2612 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
3130ex 115 . . 3  |-  ( B  e.  P.  ->  ( A  e.  ( 2nd `  B )  ->  B  <P 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) )
3231adantl 277 . 2  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( A  e.  ( 2nd `  B )  ->  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
3326ancoms 268 . . . . 5  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  B
)  /\  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
) ) )
3433biimpa 296 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  P. )  /\  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) )
3515, 10bitri 184 . . . . . . . 8  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  x  <Q  A )
3635biimpi 120 . . . . . . 7  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  x  <Q  A )
3736ad2antll 491 . . . . . 6  |-  ( ( x  e.  Q.  /\  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) )  ->  x  <Q  A )
3837adantl 277 . . . . 5  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  P. )  /\  B  <P  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. )  /\  (
x  e.  Q.  /\  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) ) )  ->  x  <Q  A )
39 simpllr 534 . . . . . 6  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  P. )  /\  B  <P  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. )  /\  (
x  e.  Q.  /\  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) ) )  ->  B  e.  P. )
40 simprrl 539 . . . . . 6  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  P. )  /\  B  <P  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. )  /\  (
x  e.  Q.  /\  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) ) )  ->  x  e.  ( 2nd `  B ) )
41 prcunqu 7515 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  -> 
( x  <Q  A  ->  A  e.  ( 2nd `  B ) ) )
421, 41sylan 283 . . . . . 6  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  -> 
( x  <Q  A  ->  A  e.  ( 2nd `  B ) ) )
4339, 40, 42syl2anc 411 . . . . 5  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  P. )  /\  B  <P  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. )  /\  (
x  e.  Q.  /\  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) ) )  ->  ( x  <Q  A  ->  A  e.  ( 2nd `  B ) ) )
4438, 43mpd 13 . . . 4  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  P. )  /\  B  <P  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. )  /\  (
x  e.  Q.  /\  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) ) )  ->  A  e.  ( 2nd `  B ) )
4534, 44rexlimddv 2612 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  P. )  /\  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  A  e.  ( 2nd `  B ) )
4645ex 115 . 2  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  ->  A  e.  ( 2nd `  B
) ) )
4732, 46impbid 129 1  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( A  e.  ( 2nd `  B )  <-> 
B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   E.wex 1503    e. wcel 2160   {cab 2175   E.wrex 2469   <.cop 3610   class class class wbr 4018   ` cfv 5235   1stc1st 6164   2ndc2nd 6165   Q.cnq 7310    <Q cltq 7315   P.cnp 7321    <P cltp 7325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-inp 7496  df-iltp 7500
This theorem is referenced by:  prplnqu  7650  caucvgprprlemmu  7725  caucvgprprlemopu  7729  caucvgprprlemexbt  7736  caucvgprprlem2  7740  suplocexprlemloc  7751
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