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Theorem recexprlempr 7583
Description:  B is a positive real. Lemma for recexpr 7589. (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 7575 . . 3  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  B )  /\  E. r  e.  Q.  r  e.  ( 2nd `  B
) ) )
3 ltrelnq 7316 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
43brel 4661 . . . . . . . . . 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 1591 . . . . . . 7  |-  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
87abssi 3222 . . . . . 6  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q.
9 nqex 7314 . . . . . . 7  |-  Q.  e.  _V
109elpw2 4141 . . . . . 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 4661 . . . . . . . . . 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 1591 . . . . . . 7  |-  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  x  e.  Q. )
1615abssi 3222 . . . . . 6  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q.
179elpw2 4141 . . . . . 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 4641 . . . . 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 424 . . . 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 2243 . . 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 7580 . . 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 7581 . . 3  |-  ( A  e.  P.  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  B
)  /\  q  e.  ( 2nd `  B ) ) )
251recexprlemloc 7582 . . 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 1172 . 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 7427 . 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 415 1  |-  ( A  e.  P.  ->  B  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 703    /\ w3a 973    = wceq 1348   E.wex 1485    e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449    C_ wss 3121   ~Pcpw 3564   <.cop 3584   class class class wbr 3987    X. cxp 4607   ` cfv 5196   1stc1st 6115   2ndc2nd 6116   Q.cnq 7231   *Qcrq 7235    <Q cltq 7236   P.cnp 7242
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-1o 6393  df-oadd 6397  df-omul 6398  df-er 6510  df-ec 6512  df-qs 6516  df-ni 7255  df-pli 7256  df-mi 7257  df-lti 7258  df-plpq 7295  df-mpq 7296  df-enq 7298  df-nqqs 7299  df-plqqs 7300  df-mqqs 7301  df-1nqqs 7302  df-rq 7303  df-ltnqqs 7304  df-inp 7417
This theorem is referenced by:  recexprlem1ssl  7584  recexprlem1ssu  7585  recexprlemss1l  7586  recexprlemss1u  7587  recexprlemex  7588  recexpr  7589
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