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Theorem ltexprlempr 7543
Description: Our constructed difference is a positive real. Lemma for ltexpri 7548. (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 7535 . . 3  |-  ( A 
<P  B  ->  ( E. q  e.  Q.  q  e.  ( 1st `  C
)  /\  E. r  e.  Q.  r  e.  ( 2nd `  C ) ) )
3 ssrab2 3225 . . . . . 6  |-  { x  e.  Q.  |  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) }  C_  Q.
4 nqex 7298 . . . . . . 7  |-  Q.  e.  _V
54elpw2 4133 . . . . . 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 3225 . . . . . 6  |-  { x  e.  Q.  |  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }  C_  Q.
84elpw2 4133 . . . . . 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 4633 . . . . 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 423 . . . 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 2237 . . 3  |-  C  e.  ( ~P Q.  X.  ~P Q. )
132, 12jctil 310 . 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 7540 . . 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 7541 . . 3  |-  ( A 
<P  B  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) ) )
161ltexprlemloc 7542 . . 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 1166 . 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 7411 . 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 414 1  |-  ( A 
<P  B  ->  C  e. 
P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    /\ w3a 967    = wceq 1342   E.wex 1479    e. wcel 2135   A.wral 2442   E.wrex 2443   {crab 2446    C_ wss 3114   ~Pcpw 3556   <.cop 3576   class class class wbr 3979    X. cxp 4599   ` cfv 5185  (class class class)co 5839   1stc1st 6101   2ndc2nd 6102   Q.cnq 7215    +Q cplq 7217    <Q cltq 7220   P.cnp 7226    <P cltp 7230
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4094  ax-sep 4097  ax-nul 4105  ax-pow 4150  ax-pr 4184  ax-un 4408  ax-setind 4511  ax-iinf 4562
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2726  df-sbc 2950  df-csb 3044  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3408  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-int 3822  df-iun 3865  df-br 3980  df-opab 4041  df-mpt 4042  df-tr 4078  df-eprel 4264  df-id 4268  df-po 4271  df-iso 4272  df-iord 4341  df-on 4343  df-suc 4346  df-iom 4565  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-res 4613  df-ima 4614  df-iota 5150  df-fun 5187  df-fn 5188  df-f 5189  df-f1 5190  df-fo 5191  df-f1o 5192  df-fv 5193  df-ov 5842  df-oprab 5843  df-mpo 5844  df-1st 6103  df-2nd 6104  df-recs 6267  df-irdg 6332  df-1o 6378  df-2o 6379  df-oadd 6382  df-omul 6383  df-er 6495  df-ec 6497  df-qs 6501  df-ni 7239  df-pli 7240  df-mi 7241  df-lti 7242  df-plpq 7279  df-mpq 7280  df-enq 7282  df-nqqs 7283  df-plqqs 7284  df-mqqs 7285  df-1nqqs 7286  df-rq 7287  df-ltnqqs 7288  df-enq0 7359  df-nq0 7360  df-0nq0 7361  df-plq0 7362  df-mq0 7363  df-inp 7401  df-iltp 7405
This theorem is referenced by:  ltexprlemfl  7544  ltexprlemrl  7545  ltexprlemfu  7546  ltexprlemru  7547  ltexpri  7548
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