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Theorem ltexprlempr 7384
Description: Our constructed difference is a positive real. Lemma for ltexpri 7389. (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 7376 . . 3  |-  ( A 
<P  B  ->  ( E. q  e.  Q.  q  e.  ( 1st `  C
)  /\  E. r  e.  Q.  r  e.  ( 2nd `  C ) ) )
3 ssrab2 3152 . . . . . 6  |-  { x  e.  Q.  |  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) }  C_  Q.
4 nqex 7139 . . . . . . 7  |-  Q.  e.  _V
54elpw2 4052 . . . . . 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 3152 . . . . . 6  |-  { x  e.  Q.  |  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) }  C_  Q.
84elpw2 4052 . . . . . 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 4541 . . . . 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 422 . . . 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 2190 . . 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 7381 . . 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 7382 . . 3  |-  ( A 
<P  B  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  C
)  /\  q  e.  ( 2nd `  C ) ) )
161ltexprlemloc 7383 . . 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 1146 . 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 7252 . 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 413 1  |-  ( A 
<P  B  ->  C  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   A.wral 2393   E.wrex 2394   {crab 2397    C_ wss 3041   ~Pcpw 3480   <.cop 3500   class class class wbr 3899    X. cxp 4507   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056    +Q cplq 7058    <Q cltq 7061   P.cnp 7067    <P cltp 7071
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-2o 6282  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-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iltp 7246
This theorem is referenced by:  ltexprlemfl  7385  ltexprlemrl  7386  ltexprlemfu  7387  ltexprlemru  7388  ltexpri  7389
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