ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nqpru Unicode version

Theorem nqpru 7383
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 7306 . . . . . 6  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prnminu 7320 . . . . . 6  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  E. x  e.  ( 2nd `  B ) x 
<Q  A )
31, 2sylan 281 . . . . 5  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  E. x  e.  ( 2nd `  B ) x 
<Q  A )
4 elprnqu 7313 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
51, 4sylan 281 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  ->  x  e.  Q. )
65ad2ant2r 501 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  x  e.  Q. )
7 simprl 521 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  /\  ( x  e.  ( 2nd `  B )  /\  x  <Q  A ) )  ->  x  e.  ( 2nd `  B ) )
8 vex 2692 . . . . . . . . . . . 12  |-  x  e. 
_V
9 breq1 3939 . . . . . . . . . . . 12  |-  ( l  =  x  ->  (
l  <Q  A  <->  x  <Q  A ) )
108, 9elab 2831 . . . . . . . . . . 11  |-  ( x  e.  { l  |  l  <Q  A }  <->  x 
<Q  A )
1110biimpri 132 . . . . . . . . . 10  |-  ( x 
<Q  A  ->  x  e. 
{ l  |  l 
<Q  A } )
12 ltnqex 7380 . . . . . . . . . . . 12  |-  { l  |  l  <Q  A }  e.  _V
13 gtnqex 7381 . . . . . . . . . . . 12  |-  { u  |  A  <Q  u }  e.  _V
1412, 13op1st 6051 . . . . . . . . . . 11  |-  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  =  { l  |  l 
<Q  A }
1514eleq2i 2207 . . . . . . . . . 10  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  x  e.  { l  |  l  <Q  A }
)
1611, 15sylibr 133 . . . . . . . . 9  |-  ( x 
<Q  A  ->  x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) )
1716ad2antll 483 . . . . . . . 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 1570 . . . . . . . 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 1215 . . . . . . 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 2423 . . . . . . 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 133 . . . . . 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 7313 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  A  e.  Q. )
231, 22sylan 281 . . . . . . . 8  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  A  e.  Q. )
24 nqprlu 7378 . . . . . . . . 9  |-  ( A  e.  Q.  ->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
25 ltdfpr 7337 . . . . . . . . 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 284 . . . . . . . 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 280 . . . . . . 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 274 . . . . . 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 166 . . . . 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 2557 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  ( 2nd `  B ) )  ->  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
3130ex 114 . . 3  |-  ( B  e.  P.  ->  ( A  e.  ( 2nd `  B )  ->  B  <P 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) )
3231adantl 275 . 2  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( A  e.  ( 2nd `  B )  ->  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
3326ancoms 266 . . . . 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 294 . . . 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 183 . . . . . . . 8  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  <->  x  <Q  A )
3635biimpi 119 . . . . . . 7  |-  ( x  e.  ( 1st `  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  x  <Q  A )
3736ad2antll 483 . . . . . 6  |-  ( ( x  e.  Q.  /\  ( x  e.  ( 2nd `  B )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) ) )  ->  x  <Q  A )
3837adantl 275 . . . . 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 524 . . . . . 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 529 . . . . . 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 7316 . . . . . . 7  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  x  e.  ( 2nd `  B ) )  -> 
( x  <Q  A  ->  A  e.  ( 2nd `  B ) ) )
421, 41sylan 281 . . . . . 6  |-  ( ( B  e.  P.  /\  x  e.  ( 2nd `  B ) )  -> 
( x  <Q  A  ->  A  e.  ( 2nd `  B ) ) )
4339, 40, 42syl2anc 409 . . . . 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 2557 . . 3  |-  ( ( ( A  e.  Q.  /\  B  e.  P. )  /\  B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  ->  A  e.  ( 2nd `  B ) )
4645ex 114 . 2  |-  ( ( A  e.  Q.  /\  B  e.  P. )  ->  ( B  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  ->  A  e.  ( 2nd `  B
) ) )
4732, 46impbid 128 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 103    <-> wb 104   E.wex 1469    e. wcel 1481   {cab 2126   E.wrex 2418   <.cop 3534   class class class wbr 3936   ` cfv 5130   1stc1st 6043   2ndc2nd 6044   Q.cnq 7111    <Q cltq 7116   P.cnp 7122    <P cltp 7126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-inp 7297  df-iltp 7301
This theorem is referenced by:  prplnqu  7451  caucvgprprlemmu  7526  caucvgprprlemopu  7530  caucvgprprlemexbt  7537  caucvgprprlem2  7541  suplocexprlemloc  7552
  Copyright terms: Public domain W3C validator