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Theorem ltexprlempr 7380
Description: Our constructed difference is a positive real. Lemma for ltexpri 7385. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
Assertion
Ref Expression
ltexprlempr  |-  ( A 
<P  B  ->  C  e. 
P. )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y

Proof of Theorem ltexprlempr
Dummy variables  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexprlem.1 . . . 4  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
21ltexprlemm 7372 . . 3  |-  ( A 
<P  B  ->  ( E. q  e.  Q.  q  e.  ( 1st `  C
)  /\  E. r  e.  Q.  r  e.  ( 2nd `  C ) ) )
3 ssrab2 3150 . . . . . 6  |-  { x  e.  Q.  |  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) }  C_  Q.
4 nqex 7135 . . . . . . 7  |-  Q.  e.  _V
54elpw2 4050 . . . . . 6  |-  ( { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) }  e.  ~P Q. 
<->  { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) }  C_  Q. )
63, 5mpbir 145 . . . . 5  |-  { x  e.  Q.  |  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) }  e.  ~P Q.
7 ssrab2 3150 . . . . . 6  |-  { x  e.  Q.  |  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }  C_  Q.
84elpw2 4050 . . . . . 6  |-  ( { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }  e.  ~P Q. 
<->  { x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }  C_  Q. )
97, 8mpbir 145 . . . . 5  |-  { x  e.  Q.  |  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }  e.  ~P Q.
10 opelxpi 4539 . . . . 5  |-  ( ( { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) }  e.  ~P Q.  /\  { x  e. 
Q.  |  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }  e.  ~P Q. )  ->  <. { x  e.  Q.  |  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.  e.  ( ~P Q.  X.  ~P Q. ) )
116, 9, 10mp2an 420 . . . 4  |-  <. { x  e.  Q.  |  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.  e.  ( ~P Q.  X.  ~P Q. )
121, 11eqeltri 2188 . . 3  |-  C  e.  ( ~P Q.  X.  ~P Q. )
132, 12jctil 308 . 2  |-  ( A 
<P  B  ->  ( C  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  C )  /\  E. r  e.  Q.  r  e.  ( 2nd `  C
) ) ) )
141ltexprlemrnd 7377 . . 3  |-  ( A 
<P  B  ->  ( A. q  e.  Q.  (
q  e.  ( 1st `  C )  <->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  C ) ) )  /\  A. r  e.  Q.  (
r  e.  ( 2nd `  C )  <->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  C ) ) ) ) )
151ltexprlemdisj 7378 . . 3  |-  ( A 
<P  B  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) ) )
161ltexprlemloc 7379 . . 3  |-  ( A 
<P  B  ->  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) )
1714, 15, 163jca 1144 . 2  |-  ( A 
<P  B  ->  ( ( A. q  e.  Q.  ( q  e.  ( 1st `  C )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  C
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) ) )
18 elnp1st2nd 7248 . 2  |-  ( C  e.  P.  <->  ( ( C  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e. 
Q.  q  e.  ( 1st `  C )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  C ) ) )  /\  (
( A. q  e. 
Q.  ( q  e.  ( 1st `  C
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  C
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) ) )  /\  A. q  e.  Q.  -.  (
q  e.  ( 1st `  C )  /\  q  e.  ( 2nd `  C
) )  /\  A. q  e.  Q.  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  C
)  \/  r  e.  ( 2nd `  C
) ) ) ) ) )
1913, 17, 18sylanbrc 411 1  |-  ( A 
<P  B  ->  C  e. 
P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    /\ w3a 945    = wceq 1314   E.wex 1451    e. wcel 1463   A.wral 2391   E.wrex 2392   {crab 2395    C_ wss 3039   ~Pcpw 3478   <.cop 3498   class class class wbr 3897    X. cxp 4505   ` cfv 5091  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052    +Q cplq 7054    <Q cltq 7057   P.cnp 7063    <P cltp 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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iltp 7242
This theorem is referenced by:  ltexprlemfl  7381  ltexprlemrl  7382  ltexprlemfu  7383  ltexprlemru  7384  ltexpri  7385
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