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Theorem recexprlempr 7408
Description:  B is a positive real. Lemma for recexpr 7414. (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 7400 . . 3  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  B )  /\  E. r  e.  Q.  r  e.  ( 2nd `  B
) ) )
3 ltrelnq 7141 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
43brel 4561 . . . . . . . . . 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 1562 . . . . . . 7  |-  ( E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) )  ->  x  e.  Q. )
87abssi 3142 . . . . . 6  |-  { x  |  E. y ( x 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) }  C_  Q.
9 nqex 7139 . . . . . . 7  |-  Q.  e.  _V
109elpw2 4052 . . . . . 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 4561 . . . . . . . . . 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 1562 . . . . . . 7  |-  ( E. y ( y  <Q  x  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  x  e.  Q. )
1615abssi 3142 . . . . . 6  |-  { x  |  E. y ( y 
<Q  x  /\  ( *Q `  y )  e.  ( 1st `  A
) ) }  C_  Q.
179elpw2 4052 . . . . . 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 4541 . . . . 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 2190 . . 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 7405 . . 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 7406 . . 3  |-  ( A  e.  P.  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  B
)  /\  q  e.  ( 2nd `  B ) ) )
251recexprlemloc 7407 . . 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 1146 . 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 7252 . 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 682    /\ w3a 947    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394    C_ wss 3041   ~Pcpw 3480   <.cop 3500   class class class wbr 3899    X. cxp 4507   ` cfv 5093   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056   *Qcrq 7060    <Q cltq 7061   P.cnp 7067
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-inp 7242
This theorem is referenced by:  recexprlem1ssl  7409  recexprlem1ssu  7410  recexprlemss1l  7411  recexprlemss1u  7412  recexprlemex  7413  recexpr  7414
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