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Theorem recexprlemopl 7587
Description: The lower cut of  B is open. Lemma for recexpr 7600. (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 7584 . . 3  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
3 ltbtwnnqq 7377 . . . . . 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 524 . . . . . . . 8  |-  ( ( ( q  <Q  r  /\  r  <Q  y )  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  q  <Q  r )
6 19.8a 1583 . . . . . . . . . 10  |-  ( ( r  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  E. y
( r  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) )
71recexprlemell 7584 . . . . . . . . . 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 473 . . . . . . . 8  |-  ( ( ( q  <Q  r  /\  r  <Q  y )  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  r  e.  ( 1st `  B ) )
105, 9jca 304 . . . . . . 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 2571 . . . . 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 279 . . . 4  |-  ( ( q  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) ) )
1413exlimiv 1591 . . 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 1015 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 973    = wceq 1348   E.wex 1485    e. wcel 2141   {cab 2156   E.wrex 2449   <.cop 3586   class class class wbr 3989   ` cfv 5198   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   *Qcrq 7246    <Q cltq 7247   P.cnp 7253
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315
This theorem is referenced by:  recexprlemrnd  7591
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