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Theorem ltexprlemupu 7578
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 7587. (Contributed by Jim Kingdon, 21-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
ltexprlemupu  |-  ( ( A  <P  B  /\  r  e.  Q. )  ->  ( E. q  e. 
Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  C ) )  ->  r  e.  ( 2nd `  C ) ) )
Distinct variable groups:    x, y, q, r, A    x, B, y, q, r    x, C, y, q, r

Proof of Theorem ltexprlemupu
StepHypRef Expression
1 simplr 528 . . . . . 6  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  r  e.  Q. )
2 simprrr 540 . . . . . . 7  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) )
32simpld 112 . . . . . 6  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  y  e.  ( 1st `  A ) )
4 simprl 529 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  q  <Q  r
)
5 simpll 527 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  A  <P  B )
6 simprrl 539 . . . . . . . . . 10  |-  ( ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
y  e.  ( 1st `  A ) )
76adantl 277 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  y  e.  ( 1st `  A ) )
8 ltrelpr 7479 . . . . . . . . . . . . 13  |-  <P  C_  ( P.  X.  P. )
98brel 4672 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
109simpld 112 . . . . . . . . . . 11  |-  ( A 
<P  B  ->  A  e. 
P. )
11 prop 7449 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
1210, 11syl 14 . . . . . . . . . 10  |-  ( A 
<P  B  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
13 elprnql 7455 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
1412, 13sylan 283 . . . . . . . . 9  |-  ( ( A  <P  B  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
155, 7, 14syl2anc 411 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  y  e.  Q. )
16 ltanqi 7376 . . . . . . . 8  |-  ( ( q  <Q  r  /\  y  e.  Q. )  ->  ( y  +Q  q
)  <Q  ( y  +Q  r ) )
174, 15, 16syl2anc 411 . . . . . . 7  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  ( y  +Q  q )  <Q  (
y  +Q  r ) )
189simprd 114 . . . . . . . . 9  |-  ( A 
<P  B  ->  B  e. 
P. )
195, 18syl 14 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  B  e.  P. )
202simprd 114 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  ( y  +Q  q )  e.  ( 2nd `  B ) )
21 prop 7449 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
22 prcunqu 7459 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  ( y  +Q  q
)  e.  ( 2nd `  B ) )  -> 
( ( y  +Q  q )  <Q  (
y  +Q  r )  ->  ( y  +Q  r )  e.  ( 2nd `  B ) ) )
2321, 22sylan 283 . . . . . . . 8  |-  ( ( B  e.  P.  /\  ( y  +Q  q
)  e.  ( 2nd `  B ) )  -> 
( ( y  +Q  q )  <Q  (
y  +Q  r )  ->  ( y  +Q  r )  e.  ( 2nd `  B ) ) )
2419, 20, 23syl2anc 411 . . . . . . 7  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  ( ( y  +Q  q )  <Q 
( y  +Q  r
)  ->  ( y  +Q  r )  e.  ( 2nd `  B ) ) )
2517, 24mpd 13 . . . . . 6  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  ( y  +Q  r )  e.  ( 2nd `  B ) )
261, 3, 25jca32 310 . . . . 5  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
2726eximi 1598 . . . 4  |-  ( E. y ( ( A 
<P  B  /\  r  e.  Q. )  /\  (
q  <Q  r  /\  (
q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  ->  E. y ( r  e.  Q.  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  r )  e.  ( 2nd `  B
) ) ) )
28 ltexprlem.1 . . . . . . . . . 10  |-  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 ) ) } >.
2928ltexprlemelu 7573 . . . . . . . . 9  |-  ( q  e.  ( 2nd `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )
30 19.42v 1904 . . . . . . . . 9  |-  ( E. y ( q  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) )  <->  ( q  e.  Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )
3129, 30bitr4i 187 . . . . . . . 8  |-  ( q  e.  ( 2nd `  C
)  <->  E. y ( q  e.  Q.  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  q )  e.  ( 2nd `  B
) ) ) )
3231anbi2i 457 . . . . . . 7  |-  ( ( q  <Q  r  /\  q  e.  ( 2nd `  C ) )  <->  ( q  <Q  r  /\  E. y
( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
33 19.42v 1904 . . . . . . 7  |-  ( E. y ( q  <Q 
r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )  <->  ( q  <Q  r  /\  E. y
( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
3432, 33bitr4i 187 . . . . . 6  |-  ( ( q  <Q  r  /\  q  e.  ( 2nd `  C ) )  <->  E. y
( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )
3534anbi2i 457 . . . . 5  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) )  <->  ( ( A 
<P  B  /\  r  e.  Q. )  /\  E. y ( q  <Q 
r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) ) )
36 19.42v 1904 . . . . 5  |-  ( E. y ( ( A 
<P  B  /\  r  e.  Q. )  /\  (
q  <Q  r  /\  (
q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) )  <-> 
( ( A  <P  B  /\  r  e.  Q. )  /\  E. y ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) ) )
3735, 36bitr4i 187 . . . 4  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) )  <->  E. y ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) ) ) )
3828ltexprlemelu 7573 . . . . 5  |-  ( r  e.  ( 2nd `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
39 19.42v 1904 . . . . 5  |-  ( E. y ( r  e. 
Q.  /\  ( y  e.  ( 1st `  A
)  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) )  <->  ( r  e.  Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
4038, 39bitr4i 187 . . . 4  |-  ( r  e.  ( 2nd `  C
)  <->  E. y ( r  e.  Q.  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  r )  e.  ( 2nd `  B
) ) ) )
4127, 37, 403imtr4i 201 . . 3  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) )  ->  r  e.  ( 2nd `  C ) )
4241ex 115 . 2  |-  ( ( A  <P  B  /\  r  e.  Q. )  ->  ( ( q  <Q 
r  /\  q  e.  ( 2nd `  C ) )  ->  r  e.  ( 2nd `  C ) ) )
4342rexlimdvw 2596 1  |-  ( ( A  <P  B  /\  r  e.  Q. )  ->  ( E. q  e. 
Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  C ) )  ->  r  e.  ( 2nd `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353   E.wex 1490    e. wcel 2146   E.wrex 2454   {crab 2457   <.cop 3592   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   1stc1st 6129   2ndc2nd 6130   Q.cnq 7254    +Q cplq 7256    <Q cltq 7259   P.cnp 7265    <P cltp 7269
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-ltnqqs 7327  df-inp 7440  df-iltp 7444
This theorem is referenced by:  ltexprlemrnd  7579
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