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Theorem nqpru 6708
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 6631 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 6645 . . . . . 6  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  E. x  e.  ( 2nd `  B ) x 
<Q  A )
31, 2sylan 271 . . . . 5  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  E. x  e.  ( 2nd `  B ) x 
<Q  A )
4 elprnqu 6638 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
51, 4sylan 271 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
65ad2ant2r 486 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  x  e.  Q. )
7 simprl 491 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  x  e.  ( 2nd `  B ) )
8 vex 2577 . . . . . . . . . . . 12  |-  x  e. 
_V
9 breq1 3795 . . . . . . . . . . . 12  |-  ( l  =  x  ->  (
l  <Q  A  <->  x  <Q  A ) )
108, 9elab 2710 . . . . . . . . . . 11  |-  ( x  e.  { l  |  l  <Q  A }  <->  x 
<Q  A )
1110biimpri 128 . . . . . . . . . 10  |-  ( x 
<Q  A  ->  x  e. 
{ l  |  l 
<Q  A } )
12 ltnqex 6705 . . . . . . . . . . . 12  |-  { l  |  l  <Q  A }  e.  _V
13 gtnqex 6706 . . . . . . . . . . . 12  |-  { u  |  A  <Q  u }  e.  _V
1412, 13op1st 5801 . . . . . . . . . . 11  |-  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  =  { l  |  l 
<Q  A }
1514eleq2i 2120 . . . . . . . . . 10  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  x  e.  { l  |  l  <Q  A }
)
1611, 15sylibr 141 . . . . . . . . 9  |-  ( x 
<Q  A  ->  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) )
1716ad2antll 468 . . . . . . . 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 1498 . . . . . . . 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 1144 . . . . . . 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 2329 . . . . . . 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 141 . . . . . 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 6638 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  A  e.  Q. )
231, 22sylan 271 . . . . . . . 8  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  A  e.  Q. )
24 nqprlu 6703 . . . . . . . . 9  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
25 ltdfpr 6662 . . . . . . . . 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 274 . . . . . . . 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 270 . . . . . . 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 265 . . . . . 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 160 . . . . 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 2454 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
3130ex 112 . . 3  |-  ( B  e.  P.  ->  ( A  e.  ( 2nd `  B )  ->  B  <P 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) )
3231adantl 266 . 2  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( A  e.  ( 2nd `  B )  ->  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
3326ancoms 259 . . . . 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 284 . . . 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 177 . . . . . . . 8  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  x  <Q  A )
3635biimpi 117 . . . . . . 7  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  x  <Q  A )
3736ad2antll 468 . . . . . 6  |-  ( ( x  e.  Q.  /\  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) )  ->  x  <Q  A )
3837adantl 266 . . . . 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 494 . . . . . 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 499 . . . . . 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 6641 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  -> 
( x  <Q  A  ->  A  e.  ( 2nd `  B ) ) )
421, 41sylan 271 . . . . . 6  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  -> 
( x  <Q  A  ->  A  e.  ( 2nd `  B ) ) )
4339, 40, 42syl2anc 397 . . . . 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 2454 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  P. )  /\  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  A  e.  ( 2nd `  B ) )
4645ex 112 . 2  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  ->  A  e.  ( 2nd `  B
) ) )
4732, 46impbid 124 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 101    <-> wb 102   E.wex 1397    e. wcel 1409   {cab 2042   E.wrex 2324   <.cop 3406   class class class wbr 3792   ` cfv 4930   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    <Q cltq 6441   P.cnp 6447    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622  df-iltp 6626
This theorem is referenced by:  prplnqu  6776  caucvgprprlemmu  6851  caucvgprprlemopu  6855  caucvgprprlemexbt  6862  caucvgprprlem2  6866
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