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Theorem nqpru 7526
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 7449 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 7463 . . . . . 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 7456 . . . . . . . . . 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 2738 . . . . . . . . . . . 12  |-  x  e. 
_V
9 breq1 4001 . . . . . . . . . . . 12  |-  ( l  =  x  ->  (
l  <Q  A  <->  x  <Q  A ) )
108, 9elab 2879 . . . . . . . . . . 11  |-  ( x  e.  { l  |  l  <Q  A }  <->  x 
<Q  A )
1110biimpri 133 . . . . . . . . . 10  |-  ( x 
<Q  A  ->  x  e. 
{ l  |  l 
<Q  A } )
12 ltnqex 7523 . . . . . . . . . . . 12  |-  { l  |  l  <Q  A }  e.  _V
13 gtnqex 7524 . . . . . . . . . . . 12  |-  { u  |  A  <Q  u }  e.  _V
1412, 13op1st 6137 . . . . . . . . . . 11  |-  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  =  { l  |  l 
<Q  A }
1514eleq2i 2242 . . . . . . . . . 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 1588 . . . . . . . 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 1236 . . . . . . 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 2459 . . . . . . 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 7456 . . . . . . . . 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 7521 . . . . . . . . 9  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
25 ltdfpr 7480 . . . . . . . . 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 2597 . . . 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 7459 . . . . . . 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 2597 . . 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 1490    e. wcel 2146   {cab 2161   E.wrex 2454   <.cop 3592   class class class wbr 3998   ` cfv 5208   1stc1st 6129   2ndc2nd 6130   Q.cnq 7254    <Q cltq 7259   P.cnp 7265    <P cltp 7269
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-inp 7440  df-iltp 7444
This theorem is referenced by:  prplnqu  7594  caucvgprprlemmu  7669  caucvgprprlemopu  7673  caucvgprprlemexbt  7680  caucvgprprlem2  7684  suplocexprlemloc  7695
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