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Theorem recexprlempr 7189
Description:  B is a positive real. Lemma for recexpr 7195. (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 7181 . . 3  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  B )  /\  E. r  e.  Q.  r  e.  ( 2nd `  B
) ) )
3 ltrelnq 6922 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
43brel 4490 . . . . . . . . . 10  |-  ( x 
<Q  y  ->  ( x  e.  Q.  /\  y  e.  Q. ) )
54simpld 110 . . . . . . . . 9  |-  ( x 
<Q  y  ->  x  e. 
Q. )
65adantr 270 . . . . . . . 8  |-  ( ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) )  ->  x  e.  Q. )
76exlimiv 1534 . . . . . . 7  |-  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
87abssi 3096 . . . . . 6  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q.
9 nqex 6920 . . . . . . 7  |-  Q.  e.  _V
109elpw2 3993 . . . . . 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 144 . . . . 5  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  e.  ~P Q.
123brel 4490 . . . . . . . . . 10  |-  ( y 
<Q  x  ->  ( y  e.  Q.  /\  x  e.  Q. ) )
1312simprd 112 . . . . . . . . 9  |-  ( y 
<Q  x  ->  x  e. 
Q. )
1413adantr 270 . . . . . . . 8  |-  ( ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  x  e.  Q. )
1514exlimiv 1534 . . . . . . 7  |-  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  x  e.  Q. )
1615abssi 3096 . . . . . 6  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q.
179elpw2 3993 . . . . . 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 144 . . . . 5  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  e.  ~P Q.
19 opelxpi 4469 . . . . 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 417 . . . 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 2160 . . 3  |-  B  e.  ( ~P Q.  X.  ~P Q. )
222, 21jctil 305 . 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 7186 . . 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 7187 . . 3  |-  ( A  e.  P.  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  B
)  /\  q  e.  ( 2nd `  B ) ) )
251recexprlemloc 7188 . . 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 1123 . 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 7033 . 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 408 1  |-  ( A  e.  P.  ->  B  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360    C_ wss 2999   ~Pcpw 3429   <.cop 3449   class class class wbr 3845    X. cxp 4436   ` cfv 5015   1stc1st 5909   2ndc2nd 5910   Q.cnq 6837   *Qcrq 6841    <Q cltq 6842   P.cnp 6848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-inp 7023
This theorem is referenced by:  recexprlem1ssl  7190  recexprlem1ssu  7191  recexprlemss1l  7192  recexprlemss1u  7193  recexprlemex  7194  recexpr  7195
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