ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recexprlemopl Unicode version

Theorem recexprlemopl 7546
Description: The lower cut of  B is open. Lemma for recexpr 7559. (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 7543 . . 3  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
3 ltbtwnnqq 7336 . . . . . 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 519 . . . . . . . 8  |-  ( ( ( q  <Q  r  /\  r  <Q  y )  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  q  <Q  r )
6 19.8a 1570 . . . . . . . . . 10  |-  ( ( r  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  E. y
( r  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) )
71recexprlemell 7543 . . . . . . . . . 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 468 . . . . . . . 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 2558 . . . . 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 1578 . . 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 1005 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 963    = wceq 1335   E.wex 1472    e. wcel 2128   {cab 2143   E.wrex 2436   <.cop 3563   class class class wbr 3966   ` cfv 5171   1stc1st 6087   2ndc2nd 6088   Q.cnq 7201   *Qcrq 7205    <Q cltq 7206   P.cnp 7212
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-eprel 4250  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-irdg 6318  df-1o 6364  df-oadd 6368  df-omul 6369  df-er 6481  df-ec 6483  df-qs 6487  df-ni 7225  df-pli 7226  df-mi 7227  df-lti 7228  df-plpq 7265  df-mpq 7266  df-enq 7268  df-nqqs 7269  df-plqqs 7270  df-mqqs 7271  df-1nqqs 7272  df-rq 7273  df-ltnqqs 7274
This theorem is referenced by:  recexprlemrnd  7550
  Copyright terms: Public domain W3C validator