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Theorem nqprl 6803
Description: Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by 
<P. (Contributed by Jim Kingdon, 8-Jul-2020.)
Assertion
Ref Expression
nqprl  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( A  e.  ( 1st `  B )  <->  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  <P  B ) )
Distinct variable group:    A, l, u
Allowed substitution hints:    B( u, l)

Proof of Theorem nqprl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 prop 6727 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnmaxl 6740 . . . . . 6  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  A  e.  ( 1st `  B ) )  ->  E. x  e.  ( 1st `  B ) A 
<Q  x )
31, 2sylan 277 . . . . 5  |-  ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  ->  E. x  e.  ( 1st `  B ) A 
<Q  x )
4 elprnql 6733 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 1st `  B ) )  ->  x  e.  Q. )
51, 4sylan 277 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  x  e.  ( 1st `  B ) )  ->  x  e.  Q. )
65ad2ant2r 493 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  /\  ( x  e.  ( 1st `  B )  /\  A  <Q  x ) )  ->  x  e.  Q. )
7 vex 2605 . . . . . . . . . . . 12  |-  x  e. 
_V
8 breq2 3797 . . . . . . . . . . . 12  |-  ( u  =  x  ->  ( A  <Q  u  <->  A  <Q  x ) )
97, 8elab 2739 . . . . . . . . . . 11  |-  ( x  e.  { u  |  A  <Q  u }  <->  A 
<Q  x )
109biimpri 131 . . . . . . . . . 10  |-  ( A 
<Q  x  ->  x  e. 
{ u  |  A  <Q  u } )
11 ltnqex 6801 . . . . . . . . . . . 12  |-  { l  |  l  <Q  A }  e.  _V
12 gtnqex 6802 . . . . . . . . . . . 12  |-  { u  |  A  <Q  u }  e.  _V
1311, 12op2nd 5805 . . . . . . . . . . 11  |-  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  =  { u  |  A  <Q  u }
1413eleq2i 2146 . . . . . . . . . 10  |-  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  x  e.  { u  |  A  <Q  u }
)
1510, 14sylibr 132 . . . . . . . . 9  |-  ( A 
<Q  x  ->  x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) )
1615ad2antll 475 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  /\  ( x  e.  ( 1st `  B )  /\  A  <Q  x ) )  ->  x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
17 simprl 498 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  /\  ( x  e.  ( 1st `  B )  /\  A  <Q  x ) )  ->  x  e.  ( 1st `  B ) )
18 19.8a 1523 . . . . . . . 8  |-  ( ( x  e.  Q.  /\  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B ) ) )  ->  E. x
( x  e.  Q.  /\  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B ) ) ) )
196, 16, 17, 18syl12anc 1168 . . . . . . 7  |-  ( ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  /\  ( x  e.  ( 1st `  B )  /\  A  <Q  x ) )  ->  E. x ( x  e.  Q.  /\  (
x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B ) ) ) )
20 df-rex 2355 . . . . . . 7  |-  ( E. x  e.  Q.  (
x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B ) )  <->  E. x ( x  e. 
Q.  /\  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B
) ) ) )
2119, 20sylibr 132 . . . . . 6  |-  ( ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  /\  ( x  e.  ( 1st `  B )  /\  A  <Q  x ) )  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B ) ) )
22 elprnql 6733 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  A  e.  ( 1st `  B ) )  ->  A  e.  Q. )
231, 22sylan 277 . . . . . . . 8  |-  ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  ->  A  e.  Q. )
24 simpl 107 . . . . . . . 8  |-  ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  ->  B  e.  P. )
25 nqprlu 6799 . . . . . . . . 9  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
26 ltdfpr 6758 . . . . . . . . 9  |-  ( (
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  e.  P.  /\  B  e.  P. )  ->  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B
) ) ) )
2725, 26sylan 277 . . . . . . . 8  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B
) ) ) )
2823, 24, 27syl2anc 403 . . . . . . 7  |-  ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  -> 
( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B
) ) ) )
2928adantr 270 . . . . . 6  |-  ( ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  /\  ( x  e.  ( 1st `  B )  /\  A  <Q  x ) )  ->  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B
) ) ) )
3021, 29mpbird 165 . . . . 5  |-  ( ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  /\  ( x  e.  ( 1st `  B )  /\  A  <Q  x ) )  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B )
313, 30rexlimddv 2482 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  ( 1st `  B ) )  ->  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  <P  B )
3231ex 113 . . 3  |-  ( B  e.  P.  ->  ( A  e.  ( 1st `  B )  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B ) )
3332adantl 271 . 2  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( A  e.  ( 1st `  B )  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B ) )
3427biimpa 290 . . . 4  |-  ( ( ( A  e.  Q.  /\  B  e.  P. )  /\  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B )  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B ) ) )
3514, 9bitri 182 . . . . . . . 8  |-  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  A  <Q  x )
3635biimpi 118 . . . . . . 7  |-  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  A  <Q  x )
3736ad2antrl 474 . . . . . 6  |-  ( ( x  e.  Q.  /\  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B ) ) )  ->  A  <Q  x )
3837adantl 271 . . . . 5  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  P. )  /\  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B )  /\  ( x  e. 
Q.  /\  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B
) ) ) )  ->  A  <Q  x
)
39 simpllr 501 . . . . . 6  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  P. )  /\  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B )  /\  ( x  e. 
Q.  /\  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B
) ) ) )  ->  B  e.  P. )
40 simprrr 507 . . . . . 6  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  P. )  /\  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B )  /\  ( x  e. 
Q.  /\  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B
) ) ) )  ->  x  e.  ( 1st `  B ) )
41 prcdnql 6736 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 1st `  B ) )  -> 
( A  <Q  x  ->  A  e.  ( 1st `  B ) ) )
421, 41sylan 277 . . . . . 6  |-  ( ( B  e.  P.  /\  x  e.  ( 1st `  B ) )  -> 
( A  <Q  x  ->  A  e.  ( 1st `  B ) ) )
4339, 40, 42syl2anc 403 . . . . 5  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  P. )  /\  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B )  /\  ( x  e. 
Q.  /\  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B
) ) ) )  ->  ( A  <Q  x  ->  A  e.  ( 1st `  B ) ) )
4438, 43mpd 13 . . . 4  |-  ( ( ( ( A  e. 
Q.  /\  B  e.  P. )  /\  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B )  /\  ( x  e. 
Q.  /\  ( x  e.  ( 2nd `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  /\  x  e.  ( 1st `  B
) ) ) )  ->  A  e.  ( 1st `  B ) )
4534, 44rexlimddv 2482 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  P. )  /\  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B )  ->  A  e.  ( 1st `  B ) )
4645ex 113 . 2  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B  ->  A  e.  ( 1st `  B ) ) )
4733, 46impbid 127 1  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( A  e.  ( 1st `  B )  <->  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  <P  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   E.wex 1422    e. wcel 1434   {cab 2068   E.wrex 2350   <.cop 3409   class class class wbr 3793   ` cfv 4932   1stc1st 5796   2ndc2nd 5797   Q.cnq 6532    <Q cltq 6537   P.cnp 6543    <P cltp 6547
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-inp 6718  df-iltp 6722
This theorem is referenced by:  caucvgprlemcanl  6896  cauappcvgprlem1  6911  archrecpr  6916  caucvgprlem1  6931  caucvgprprlemml  6946  caucvgprprlemopl  6949
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