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Theorem recexprlempr 7447
Description:  B is a positive real. Lemma for recexpr 7453. (Contributed by Jim Kingdon, 27-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
recexprlempr  |-  ( A  e.  P.  ->  B  e.  P. )
Distinct variable groups:    x, y, A   
x, B, y

Proof of Theorem recexprlempr
Dummy variables  r  q are mutually distinct and distinct from all other variables.
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 ) ) }
>.
21recexprlemm 7439 . . 3  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  B )  /\  E. r  e.  Q.  r  e.  ( 2nd `  B
) ) )
3 ltrelnq 7180 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
43brel 4591 . . . . . . . . . 10  |-  ( x 
<Q  y  ->  ( x  e.  Q.  /\  y  e.  Q. ) )
54simpld 111 . . . . . . . . 9  |-  ( x 
<Q  y  ->  x  e. 
Q. )
65adantr 274 . . . . . . . 8  |-  ( ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  x  e.  Q. )
76exlimiv 1577 . . . . . . 7  |-  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
87abssi 3172 . . . . . 6  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q.
9 nqex 7178 . . . . . . 7  |-  Q.  e.  _V
109elpw2 4082 . . . . . 6  |-  ( { x  |  E. y
( x  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) }  e.  ~P Q.  <->  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q. )
118, 10mpbir 145 . . . . 5  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  e.  ~P Q.
123brel 4591 . . . . . . . . . 10  |-  ( y 
<Q  x  ->  ( y  e.  Q.  /\  x  e.  Q. ) )
1312simprd 113 . . . . . . . . 9  |-  ( y 
<Q  x  ->  x  e. 
Q. )
1413adantr 274 . . . . . . . 8  |-  ( ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  x  e.  Q. )
1514exlimiv 1577 . . . . . . 7  |-  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  x  e.  Q. )
1615abssi 3172 . . . . . 6  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q.
179elpw2 4082 . . . . . 6  |-  ( { x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }  e.  ~P Q.  <->  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q. )
1816, 17mpbir 145 . . . . 5  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  e.  ~P Q.
19 opelxpi 4571 . . . . 5  |-  ( ( { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) }  e.  ~P Q.  /\  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  e.  ~P Q. )  ->  <. { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) } ,  { x  |  E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.  e.  ( ~P Q.  X.  ~P Q. ) )
2011, 18, 19mp2an 422 . . . 4  |-  <. { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) } ,  { x  |  E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) ) } >.  e.  ( ~P Q.  X.  ~P Q. )
211, 20eqeltri 2212 . . 3  |-  B  e.  ( ~P Q.  X.  ~P Q. )
222, 21jctil 310 . 2  |-  ( A  e.  P.  ->  ( B  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e. 
Q.  q  e.  ( 1st `  B )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  B ) ) ) )
231recexprlemrnd 7444 . . 3  |-  ( A  e.  P.  ->  ( A. q  e.  Q.  ( q  e.  ( 1st `  B )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  B ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  B
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) ) )
241recexprlemdisj 7445 . . 3  |-  ( A  e.  P.  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  B
)  /\  q  e.  ( 2nd `  B ) ) )
251recexprlemloc 7446 . . 3  |-  ( A  e.  P.  ->  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  B
)  \/  r  e.  ( 2nd `  B
) ) ) )
2623, 24, 253jca 1161 . 2  |-  ( A  e.  P.  ->  (
( A. q  e. 
Q.  ( q  e.  ( 1st `  B
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  B ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  B
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  B )  /\  q  e.  ( 2nd `  B
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  B
)  \/  r  e.  ( 2nd `  B
) ) ) ) )
27 elnp1st2nd 7291 . 2  |-  ( B  e.  P.  <->  ( ( B  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e. 
Q.  q  e.  ( 1st `  B )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  B ) ) )  /\  (
( A. q  e. 
Q.  ( q  e.  ( 1st `  B
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  B ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  B
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  B )  /\  q  e.  ( 2nd `  B
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  B
)  \/  r  e.  ( 2nd `  B
) ) ) ) ) )
2822, 26, 27sylanbrc 413 1  |-  ( A  e.  P.  ->  B  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417    C_ wss 3071   ~Pcpw 3510   <.cop 3530   class class class wbr 3929    X. cxp 4537   ` cfv 5123   1stc1st 6036   2ndc2nd 6037   Q.cnq 7095   *Qcrq 7099    <Q cltq 7100   P.cnp 7106
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-inp 7281
This theorem is referenced by:  recexprlem1ssl  7448  recexprlem1ssu  7449  recexprlemss1l  7450  recexprlemss1u  7451  recexprlemex  7452  recexpr  7453
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