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Theorem nqprloc 6797
Description: A cut produced from a rational is located. Lemma for nqprlu 6799. (Contributed by Jim Kingdon, 8-Dec-2019.)
Assertion
Ref Expression
nqprloc  |-  ( A  e.  Q.  ->  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  { x  |  x 
<Q  A }  \/  r  e.  { x  |  A  <Q  x } ) ) )
Distinct variable group:    x, A, r, q

Proof of Theorem nqprloc
StepHypRef Expression
1 nqtri3or 6648 . . . . . . 7  |-  ( ( q  e.  Q.  /\  A  e.  Q. )  ->  ( q  <Q  A  \/  q  =  A  \/  A  <Q  q ) )
21ancoms 264 . . . . . 6  |-  ( ( A  e.  Q.  /\  q  e.  Q. )  ->  ( q  <Q  A  \/  q  =  A  \/  A  <Q  q ) )
32ad2antrr 472 . . . . 5  |-  ( ( ( ( A  e. 
Q.  /\  q  e.  Q. )  /\  r  e.  Q. )  /\  q  <Q  r )  ->  (
q  <Q  A  \/  q  =  A  \/  A  <Q  q ) )
4 vex 2605 . . . . . . . . . 10  |-  q  e. 
_V
5 breq1 3796 . . . . . . . . . 10  |-  ( x  =  q  ->  (
x  <Q  A  <->  q  <Q  A ) )
64, 5elab 2739 . . . . . . . . 9  |-  ( q  e.  { x  |  x  <Q  A }  <->  q 
<Q  A )
76biimpri 131 . . . . . . . 8  |-  ( q 
<Q  A  ->  q  e. 
{ x  |  x 
<Q  A } )
87orcd 685 . . . . . . 7  |-  ( q 
<Q  A  ->  ( q  e.  { x  |  x  <Q  A }  \/  r  e.  { x  |  A  <Q  x }
) )
98a1i 9 . . . . . 6  |-  ( ( ( ( A  e. 
Q.  /\  q  e.  Q. )  /\  r  e.  Q. )  /\  q  <Q  r )  ->  (
q  <Q  A  ->  (
q  e.  { x  |  x  <Q  A }  \/  r  e.  { x  |  A  <Q  x }
) ) )
10 simpr 108 . . . . . . . 8  |-  ( ( ( ( A  e. 
Q.  /\  q  e.  Q. )  /\  r  e.  Q. )  /\  q  <Q  r )  ->  q  <Q  r )
11 breq1 3796 . . . . . . . 8  |-  ( q  =  A  ->  (
q  <Q  r  <->  A  <Q  r ) )
1210, 11syl5ibcom 153 . . . . . . 7  |-  ( ( ( ( A  e. 
Q.  /\  q  e.  Q. )  /\  r  e.  Q. )  /\  q  <Q  r )  ->  (
q  =  A  ->  A  <Q  r ) )
13 vex 2605 . . . . . . . . 9  |-  r  e. 
_V
14 breq2 3797 . . . . . . . . 9  |-  ( x  =  r  ->  ( A  <Q  x  <->  A  <Q  r ) )
1513, 14elab 2739 . . . . . . . 8  |-  ( r  e.  { x  |  A  <Q  x }  <->  A 
<Q  r )
16 olc 665 . . . . . . . 8  |-  ( r  e.  { x  |  A  <Q  x }  ->  ( q  e.  {
x  |  x  <Q  A }  \/  r  e. 
{ x  |  A  <Q  x } ) )
1715, 16sylbir 133 . . . . . . 7  |-  ( A 
<Q  r  ->  ( q  e.  { x  |  x  <Q  A }  \/  r  e.  { x  |  A  <Q  x }
) )
1812, 17syl6 33 . . . . . 6  |-  ( ( ( ( A  e. 
Q.  /\  q  e.  Q. )  /\  r  e.  Q. )  /\  q  <Q  r )  ->  (
q  =  A  -> 
( q  e.  {
x  |  x  <Q  A }  \/  r  e. 
{ x  |  A  <Q  x } ) ) )
19 ltsonq 6650 . . . . . . . . . 10  |-  <Q  Or  Q.
20 ltrelnq 6617 . . . . . . . . . 10  |-  <Q  C_  ( Q.  X.  Q. )
2119, 20sotri 4750 . . . . . . . . 9  |-  ( ( A  <Q  q  /\  q  <Q  r )  ->  A  <Q  r )
2221, 17syl 14 . . . . . . . 8  |-  ( ( A  <Q  q  /\  q  <Q  r )  -> 
( q  e.  {
x  |  x  <Q  A }  \/  r  e. 
{ x  |  A  <Q  x } ) )
2322expcom 114 . . . . . . 7  |-  ( q 
<Q  r  ->  ( A 
<Q  q  ->  ( q  e.  { x  |  x  <Q  A }  \/  r  e.  { x  |  A  <Q  x }
) ) )
2423adantl 271 . . . . . 6  |-  ( ( ( ( A  e. 
Q.  /\  q  e.  Q. )  /\  r  e.  Q. )  /\  q  <Q  r )  ->  ( A  <Q  q  ->  (
q  e.  { x  |  x  <Q  A }  \/  r  e.  { x  |  A  <Q  x }
) ) )
259, 18, 243jaod 1236 . . . . 5  |-  ( ( ( ( A  e. 
Q.  /\  q  e.  Q. )  /\  r  e.  Q. )  /\  q  <Q  r )  ->  (
( q  <Q  A  \/  q  =  A  \/  A  <Q  q )  -> 
( q  e.  {
x  |  x  <Q  A }  \/  r  e. 
{ x  |  A  <Q  x } ) ) )
263, 25mpd 13 . . . 4  |-  ( ( ( ( A  e. 
Q.  /\  q  e.  Q. )  /\  r  e.  Q. )  /\  q  <Q  r )  ->  (
q  e.  { x  |  x  <Q  A }  \/  r  e.  { x  |  A  <Q  x }
) )
2726ex 113 . . 3  |-  ( ( ( A  e.  Q.  /\  q  e.  Q. )  /\  r  e.  Q. )  ->  ( q  <Q 
r  ->  ( q  e.  { x  |  x 
<Q  A }  \/  r  e.  { x  |  A  <Q  x } ) ) )
2827ralrimiva 2435 . 2  |-  ( ( A  e.  Q.  /\  q  e.  Q. )  ->  A. r  e.  Q.  ( q  <Q  r  ->  ( q  e.  {
x  |  x  <Q  A }  \/  r  e. 
{ x  |  A  <Q  x } ) ) )
2928ralrimiva 2435 1  |-  ( A  e.  Q.  ->  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  { x  |  x 
<Q  A }  \/  r  e.  { x  |  A  <Q  x } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662    \/ w3o 919    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   class class class wbr 3793   Q.cnq 6532    <Q cltq 6537
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-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-mi 6558  df-lti 6559  df-enq 6599  df-nqqs 6600  df-ltnqqs 6605
This theorem is referenced by:  nqprxx  6798
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