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Theorem recexprlemopl 7397
Description: The lower cut of  B is open. Lemma for recexpr 7410. (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
recexprlemopl  |-  ( ( A  e.  P.  /\  q  e.  Q.  /\  q  e.  ( 1st `  B
) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) ) )
Distinct variable groups:    r, q, x, y, A    B, q,
r, x, y

Proof of Theorem recexprlemopl
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 ) ) }
>.
21recexprlemell 7394 . . 3  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
3 ltbtwnnqq 7187 . . . . . 6  |-  ( q 
<Q  y  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  <Q  y ) )
43biimpi 119 . . . . 5  |-  ( q 
<Q  y  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  <Q  y ) )
5 simpll 501 . . . . . . . 8  |-  ( ( ( q  <Q  r  /\  r  <Q  y )  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  q  <Q  r )
6 19.8a 1552 . . . . . . . . . 10  |-  ( ( r  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  E. y
( r  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) )
71recexprlemell 7394 . . . . . . . . . 10  |-  ( r  e.  ( 1st `  B
)  <->  E. y ( r 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
86, 7sylibr 133 . . . . . . . . 9  |-  ( ( r  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  r  e.  ( 1st `  B
) )
98adantll 465 . . . . . . . 8  |-  ( ( ( q  <Q  r  /\  r  <Q  y )  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  r  e.  ( 1st `  B ) )
105, 9jca 302 . . . . . . 7  |-  ( ( ( q  <Q  r  /\  r  <Q  y )  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( q  <Q  r  /\  r  e.  ( 1st `  B
) ) )
1110expcom 115 . . . . . 6  |-  ( ( *Q `  y )  e.  ( 2nd `  A
)  ->  ( (
q  <Q  r  /\  r  <Q  y )  ->  (
q  <Q  r  /\  r  e.  ( 1st `  B
) ) ) )
1211reximdv 2508 . . . . 5  |-  ( ( *Q `  y )  e.  ( 2nd `  A
)  ->  ( E. r  e.  Q.  (
q  <Q  r  /\  r  <Q  y )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) ) ) )
134, 12mpan9 277 . . . 4  |-  ( ( q  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) ) )
1413exlimiv 1560 . . 3  |-  ( E. y ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) ) )
152, 14sylbi 120 . 2  |-  ( q  e.  ( 1st `  B
)  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) ) )
16153ad2ant3 987 1  |-  ( ( A  e.  P.  /\  q  e.  Q.  /\  q  e.  ( 1st `  B
) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   E.wrex 2392   <.cop 3498   class class class wbr 3897   ` cfv 5091   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052   *Qcrq 7056    <Q cltq 7057   P.cnp 7063
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125
This theorem is referenced by:  recexprlemrnd  7401
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