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Theorem recexprlemlol 6724
Description: The lower cut of  B is lower. Lemma for recexpr 6736. (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
recexprlemlol  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  ( E. r  e. 
Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) )  ->  q  e.  ( 1st `  B ) ) )
Distinct variable groups:    r, q, x, y, A    B, q,
r, x, y

Proof of Theorem recexprlemlol
StepHypRef Expression
1 ltsonq 6496 . . . . . . . . 9  |-  <Q  Or  Q.
2 ltrelnq 6463 . . . . . . . . 9  |-  <Q  C_  ( Q.  X.  Q. )
31, 2sotri 4720 . . . . . . . 8  |-  ( ( q  <Q  r  /\  r  <Q  y )  -> 
q  <Q  y )
43ex 108 . . . . . . 7  |-  ( q 
<Q  r  ->  ( r 
<Q  y  ->  q  <Q 
y ) )
54anim1d 319 . . . . . 6  |-  ( q 
<Q  r  ->  ( ( r  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  (
q  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) ) )
65eximdv 1760 . . . . 5  |-  ( q 
<Q  r  ->  ( E. y ( r  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  E. y
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) ) )
7 recexpr.1 . . . . . 6  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
87recexprlemell 6720 . . . . 5  |-  ( r  e.  ( 1st `  B
)  <->  E. y ( r 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
97recexprlemell 6720 . . . . 5  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
106, 8, 93imtr4g 194 . . . 4  |-  ( q 
<Q  r  ->  ( r  e.  ( 1st `  B
)  ->  q  e.  ( 1st `  B ) ) )
1110imp 115 . . 3  |-  ( ( q  <Q  r  /\  r  e.  ( 1st `  B ) )  -> 
q  e.  ( 1st `  B ) )
1211rexlimivw 2429 . 2  |-  ( E. r  e.  Q.  (
q  <Q  r  /\  r  e.  ( 1st `  B
) )  ->  q  e.  ( 1st `  B
) )
1312a1i 9 1  |-  ( ( A  e.  P.  /\  q  e.  Q. )  ->  ( E. r  e. 
Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  B ) )  ->  q  e.  ( 1st `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   {cab 2026   E.wrex 2307   <.cop 3378   class class class wbr 3764   ` cfv 4902   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   *Qcrq 6382    <Q cltq 6383   P.cnp 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-lti 6405  df-enq 6445  df-nqqs 6446  df-ltnqqs 6451
This theorem is referenced by:  recexprlemrnd  6727
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