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Theorem ltexprlemrnd 7582
Description: Our constructed difference is rounded. Lemma for ltexpri 7590. (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
ltexprlemrnd  |-  ( 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 ) ) ) ) )
Distinct variable groups:    x, y, q, r, A    x, B, y, q, r    x, C, y, q, r

Proof of Theorem ltexprlemrnd
StepHypRef Expression
1 ltexprlem.1 . . . . . 6  |-  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 ) ) } >.
21ltexprlemopl 7578 . . . . 5  |-  ( ( A  <P  B  /\  q  e.  Q.  /\  q  e.  ( 1st `  C
) )  ->  E. r  e.  Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  C ) ) )
323expia 1205 . . . 4  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( q  e.  ( 1st `  C )  ->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) ) )
41ltexprlemlol 7579 . . . 4  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( E. r  e. 
Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  C ) )  ->  q  e.  ( 1st `  C ) ) )
53, 4impbid 129 . . 3  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( q  e.  ( 1st `  C )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) ) )
65ralrimiva 2550 . 2  |-  ( A 
<P  B  ->  A. q  e.  Q.  ( q  e.  ( 1st `  C
)  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) ) )
71ltexprlemopu 7580 . . . . 5  |-  ( ( A  <P  B  /\  r  e.  Q.  /\  r  e.  ( 2nd `  C
) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  C ) ) )
873expia 1205 . . . 4  |-  ( ( A  <P  B  /\  r  e.  Q. )  ->  ( r  e.  ( 2nd `  C )  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) ) )
91ltexprlemupu 7581 . . . 4  |-  ( ( A  <P  B  /\  r  e.  Q. )  ->  ( E. q  e. 
Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  C ) )  ->  r  e.  ( 2nd `  C ) ) )
108, 9impbid 129 . . 3  |-  ( ( A  <P  B  /\  r  e.  Q. )  ->  ( r  e.  ( 2nd `  C )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) ) )
1110ralrimiva 2550 . 2  |-  ( A 
<P  B  ->  A. r  e.  Q.  ( r  e.  ( 2nd `  C
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) ) )
126, 11jca 306 1  |-  ( 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 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3594   class class class wbr 4000   ` cfv 5211  (class class class)co 5868   1stc1st 6132   2ndc2nd 6133   Q.cnq 7257    +Q cplq 7259    <Q cltq 7262    <P cltp 7272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4285  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-1o 6410  df-oadd 6414  df-omul 6415  df-er 6528  df-ec 6530  df-qs 6534  df-ni 7281  df-pli 7282  df-mi 7283  df-lti 7284  df-plpq 7321  df-mpq 7322  df-enq 7324  df-nqqs 7325  df-plqqs 7326  df-mqqs 7327  df-1nqqs 7328  df-ltnqqs 7330  df-inp 7443  df-iltp 7447
This theorem is referenced by:  ltexprlempr  7585
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