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Theorem nqpru 7109
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 7032 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 7046 . . . . . 6  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  E. x  e.  ( 2nd `  B ) x 
<Q  A )
31, 2sylan 277 . . . . 5  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  E. x  e.  ( 2nd `  B ) x 
<Q  A )
4 elprnqu 7039 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
51, 4sylan 277 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
65ad2ant2r 493 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  x  e.  Q. )
7 simprl 498 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  x  e.  ( 2nd `  B ) )
8 vex 2622 . . . . . . . . . . . 12  |-  x  e. 
_V
9 breq1 3848 . . . . . . . . . . . 12  |-  ( l  =  x  ->  (
l  <Q  A  <->  x  <Q  A ) )
108, 9elab 2760 . . . . . . . . . . 11  |-  ( x  e.  { l  |  l  <Q  A }  <->  x 
<Q  A )
1110biimpri 131 . . . . . . . . . 10  |-  ( x 
<Q  A  ->  x  e. 
{ l  |  l 
<Q  A } )
12 ltnqex 7106 . . . . . . . . . . . 12  |-  { l  |  l  <Q  A }  e.  _V
13 gtnqex 7107 . . . . . . . . . . . 12  |-  { u  |  A  <Q  u }  e.  _V
1412, 13op1st 5917 . . . . . . . . . . 11  |-  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  =  { l  |  l 
<Q  A }
1514eleq2i 2154 . . . . . . . . . 10  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  x  e.  { l  |  l  <Q  A }
)
1611, 15sylibr 132 . . . . . . . . 9  |-  ( x 
<Q  A  ->  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) )
1716ad2antll 475 . . . . . . . 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 1527 . . . . . . . 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 1172 . . . . . . 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 2365 . . . . . . 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 132 . . . . . 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 7039 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  A  e.  Q. )
231, 22sylan 277 . . . . . . . 8  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  A  e.  Q. )
24 nqprlu 7104 . . . . . . . . 9  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
25 ltdfpr 7063 . . . . . . . . 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 280 . . . . . . . 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 276 . . . . . . 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 270 . . . . . 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 165 . . . . 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 2493 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
3130ex 113 . . 3  |-  ( B  e.  P.  ->  ( A  e.  ( 2nd `  B )  ->  B  <P 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) )
3231adantl 271 . 2  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( A  e.  ( 2nd `  B )  ->  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
3326ancoms 264 . . . . 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 290 . . . 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 182 . . . . . . . 8  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  x  <Q  A )
3635biimpi 118 . . . . . . 7  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  x  <Q  A )
3736ad2antll 475 . . . . . 6  |-  ( ( x  e.  Q.  /\  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) )  ->  x  <Q  A )
3837adantl 271 . . . . 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 501 . . . . . 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 506 . . . . . 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 7042 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  -> 
( x  <Q  A  ->  A  e.  ( 2nd `  B ) ) )
421, 41sylan 277 . . . . . 6  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  -> 
( x  <Q  A  ->  A  e.  ( 2nd `  B ) ) )
4339, 40, 42syl2anc 403 . . . . 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 2493 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  P. )  /\  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  A  e.  ( 2nd `  B ) )
4645ex 113 . 2  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  ->  A  e.  ( 2nd `  B
) ) )
4732, 46impbid 127 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 102    <-> wb 103   E.wex 1426    e. wcel 1438   {cab 2074   E.wrex 2360   <.cop 3449   class class class wbr 3845   ` cfv 5015   1stc1st 5909   2ndc2nd 5910   Q.cnq 6837    <Q cltq 6842   P.cnp 6848    <P cltp 6852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-inp 7023  df-iltp 7027
This theorem is referenced by:  prplnqu  7177  caucvgprprlemmu  7252  caucvgprprlemopu  7256  caucvgprprlemexbt  7263  caucvgprprlem2  7267
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