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Theorem recexprlemopu 7435
Description: The upper cut of  B is open. Lemma for recexpr 7446. (Contributed by Jim Kingdon, 28-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
Assertion
Ref Expression
recexprlemopu  |-  ( ( A  e.  P.  /\  r  e.  Q.  /\  r  e.  ( 2nd `  B
) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
Distinct variable groups:    r, q, x, y, A    B, q,
r, x, y

Proof of Theorem recexprlemopu
StepHypRef Expression
1 recexpr.1 . . . 4  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
21recexprlemelu 7431 . . 3  |-  ( r  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
3 ltbtwnnqq 7223 . . . . . 6  |-  ( y 
<Q  r  <->  E. q  e.  Q.  ( y  <Q  q  /\  q  <Q  r ) )
43biimpi 119 . . . . 5  |-  ( y 
<Q  r  ->  E. q  e.  Q.  ( y  <Q 
q  /\  q  <Q  r ) )
5 simplr 519 . . . . . . . 8  |-  ( ( ( y  <Q  q  /\  q  <Q  r )  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  q  <Q  r )
6 19.8a 1569 . . . . . . . . . 10  |-  ( ( y  <Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  E. y
( y  <Q  q  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) )
71recexprlemelu 7431 . . . . . . . . . 10  |-  ( q  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
86, 7sylibr 133 . . . . . . . . 9  |-  ( ( y  <Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  q  e.  ( 2nd `  B
) )
98adantlr 468 . . . . . . . 8  |-  ( ( ( y  <Q  q  /\  q  <Q  r )  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  q  e.  ( 2nd `  B ) )
105, 9jca 304 . . . . . . 7  |-  ( ( ( y  <Q  q  /\  q  <Q  r )  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  ( q  <Q  r  /\  q  e.  ( 2nd `  B
) ) )
1110expcom 115 . . . . . 6  |-  ( ( *Q `  y )  e.  ( 1st `  A
)  ->  ( (
y  <Q  q  /\  q  <Q  r )  ->  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) ) ) )
1211reximdv 2533 . . . . 5  |-  ( ( *Q `  y )  e.  ( 1st `  A
)  ->  ( E. q  e.  Q.  (
y  <Q  q  /\  q  <Q  r )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) ) )
134, 12mpan9 279 . . . 4  |-  ( ( y  <Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
1413exlimiv 1577 . . 3  |-  ( E. y ( y  <Q 
r  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
152, 14sylbi 120 . 2  |-  ( r  e.  ( 2nd `  B
)  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
16153ad2ant3 1004 1  |-  ( ( A  e.  P.  /\  r  e.  Q.  /\  r  e.  ( 2nd `  B
) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   {cab 2125   E.wrex 2417   <.cop 3530   class class class wbr 3929   ` cfv 5123   1stc1st 6036   2ndc2nd 6037   Q.cnq 7088   *Qcrq 7092    <Q cltq 7093   P.cnp 7099
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161
This theorem is referenced by:  recexprlemrnd  7437
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