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Theorem recexprlemopl 7184
Description: The lower cut of  B is open. Lemma for recexpr 7197. (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 7181 . . 3  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
3 ltbtwnnqq 6974 . . . . . 6  |-  ( q 
<Q  y  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  <Q  y ) )
43biimpi 118 . . . . 5  |-  ( q 
<Q  y  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  <Q  y ) )
5 simpll 496 . . . . . . . 8  |-  ( ( ( q  <Q  r  /\  r  <Q  y )  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  q  <Q  r )
6 19.8a 1527 . . . . . . . . . 10  |-  ( ( r  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  E. y
( r  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) )
71recexprlemell 7181 . . . . . . . . . 10  |-  ( r  e.  ( 1st `  B
)  <->  E. y ( r 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
86, 7sylibr 132 . . . . . . . . 9  |-  ( ( r  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  r  e.  ( 1st `  B
) )
98adantll 460 . . . . . . . 8  |-  ( ( ( q  <Q  r  /\  r  <Q  y )  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  r  e.  ( 1st `  B ) )
105, 9jca 300 . . . . . . 7  |-  ( ( ( q  <Q  r  /\  r  <Q  y )  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  ( q  <Q  r  /\  r  e.  ( 1st `  B
) ) )
1110expcom 114 . . . . . 6  |-  ( ( *Q `  y )  e.  ( 2nd `  A
)  ->  ( (
q  <Q  r  /\  r  <Q  y )  ->  (
q  <Q  r  /\  r  e.  ( 1st `  B
) ) ) )
1211reximdv 2474 . . . . 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 275 . . . 4  |-  ( ( q  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) ) )
1413exlimiv 1534 . . 3  |-  ( E. y ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) ) )
152, 14sylbi 119 . 2  |-  ( q  e.  ( 1st `  B
)  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) ) )
16153ad2ant3 966 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 102    /\ w3a 924    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   E.wrex 2360   <.cop 3449   class class class wbr 3845   ` cfv 5015   1stc1st 5909   2ndc2nd 5910   Q.cnq 6839   *Qcrq 6843    <Q cltq 6844   P.cnp 6850
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912
This theorem is referenced by:  recexprlemrnd  7188
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