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Theorem recexprlemlol 7603
Description: The lower cut of  B is lower. Lemma for recexpr 7615. (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 7375 . . . . . . . . 9  |-  <Q  Or  Q.
2 ltrelnq 7342 . . . . . . . . 9  |-  <Q  C_  ( Q.  X.  Q. )
31, 2sotri 5019 . . . . . . . 8  |-  ( ( q  <Q  r  /\  r  <Q  y )  -> 
q  <Q  y )
43ex 115 . . . . . . 7  |-  ( q 
<Q  r  ->  ( r 
<Q  y  ->  q  <Q 
y ) )
54anim1d 336 . . . . . 6  |-  ( q 
<Q  r  ->  ( ( r  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  (
q  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) ) )
65eximdv 1880 . . . . 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 7599 . . . . 5  |-  ( r  e.  ( 1st `  B
)  <->  E. y ( r 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
97recexprlemell 7599 . . . . 5  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
106, 8, 93imtr4g 205 . . . 4  |-  ( q 
<Q  r  ->  ( r  e.  ( 1st `  B
)  ->  q  e.  ( 1st `  B ) ) )
1110imp 124 . . 3  |-  ( ( q  <Q  r  /\  r  e.  ( 1st `  B ) )  -> 
q  e.  ( 1st `  B ) )
1211rexlimivw 2590 . 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 104    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   E.wrex 2456   <.cop 3594   class class class wbr 4000   ` cfv 5211   1stc1st 6132   2ndc2nd 6133   Q.cnq 7257   *Qcrq 7261    <Q cltq 7262   P.cnp 7268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4285  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-oadd 6414  df-omul 6415  df-er 6528  df-ec 6530  df-qs 6534  df-ni 7281  df-mi 7283  df-lti 7284  df-enq 7324  df-nqqs 7325  df-ltnqqs 7330
This theorem is referenced by:  recexprlemrnd  7606
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