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Theorem ltexprlempr 6706
Description: Our constructed difference is a positive real. Lemma for ltexpri 6711. (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 6698 . . 3  |-  ( A 
<P  B  ->  ( E. q  e.  Q.  q  e.  ( 1st `  C
)  /\  E. r  e.  Q.  r  e.  ( 2nd `  C ) ) )
3 ssrab2 3025 . . . . . 6  |-  { x  e.  Q.  |  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) }  C_  Q.
4 nqex 6461 . . . . . . 7  |-  Q.  e.  _V
54elpw2 3911 . . . . . 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 134 . . . . 5  |-  { x  e.  Q.  |  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) }  e.  ~P Q.
7 ssrab2 3025 . . . . . 6  |-  { x  e.  Q.  |  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }  C_  Q.
84elpw2 3911 . . . . . 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 134 . . . . 5  |-  { x  e.  Q.  |  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }  e.  ~P Q.
10 opelxpi 4376 . . . . 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 402 . . . 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 2110 . . 3  |-  C  e.  ( ~P Q.  X.  ~P Q. )
132, 12jctil 295 . 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 6703 . . 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 6704 . . 3  |-  ( A 
<P  B  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) ) )
161ltexprlemloc 6705 . . 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 1084 . 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 6574 . 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 394 1  |-  ( A 
<P  B  ->  C  e. 
P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    <-> wb 98    \/ wo 629    /\ w3a 885    = wceq 1243   E.wex 1381    e. wcel 1393   A.wral 2306   E.wrex 2307   {crab 2310    C_ wss 2917   ~Pcpw 3359   <.cop 3378   class class class wbr 3764    X. cxp 4343   ` cfv 4902  (class class class)co 5512   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378    +Q cplq 6380    <Q cltq 6383   P.cnp 6389    <P cltp 6393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iltp 6568
This theorem is referenced by:  ltexprlemfl  6707  ltexprlemrl  6708  ltexprlemfu  6709  ltexprlemru  6710  ltexpri  6711
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