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Theorem recexprlemlol 7541
Description: The lower cut of  B is lower. Lemma for recexpr 7553. (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 7313 . . . . . . . . 9  |-  <Q  Or  Q.
2 ltrelnq 7280 . . . . . . . . 9  |-  <Q  C_  ( Q.  X.  Q. )
31, 2sotri 4980 . . . . . . . 8  |-  ( ( q  <Q  r  /\  r  <Q  y )  -> 
q  <Q  y )
43ex 114 . . . . . . 7  |-  ( q 
<Q  r  ->  ( r 
<Q  y  ->  q  <Q 
y ) )
54anim1d 334 . . . . . 6  |-  ( q 
<Q  r  ->  ( ( r  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  (
q  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) ) )
65eximdv 1860 . . . . 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 7537 . . . . 5  |-  ( r  e.  ( 1st `  B
)  <->  E. y ( r 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
97recexprlemell 7537 . . . . 5  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
106, 8, 93imtr4g 204 . . . 4  |-  ( q 
<Q  r  ->  ( r  e.  ( 1st `  B
)  ->  q  e.  ( 1st `  B ) ) )
1110imp 123 . . 3  |-  ( ( q  <Q  r  /\  r  e.  ( 1st `  B ) )  -> 
q  e.  ( 1st `  B ) )
1211rexlimivw 2570 . 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 103    = wceq 1335   E.wex 1472    e. wcel 2128   {cab 2143   E.wrex 2436   <.cop 3563   class class class wbr 3965   ` cfv 5169   1stc1st 6083   2ndc2nd 6084   Q.cnq 7195   *Qcrq 7199    <Q cltq 7200   P.cnp 7206
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 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546
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 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-oadd 6364  df-omul 6365  df-er 6477  df-ec 6479  df-qs 6483  df-ni 7219  df-mi 7221  df-lti 7222  df-enq 7262  df-nqqs 7263  df-ltnqqs 7268
This theorem is referenced by:  recexprlemrnd  7544
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