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Theorem ltexprlemupu 6760
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 6769. (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 490 . . . . . 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 500 . . . . . . 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 109 . . . . . 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 491 . . . . . . . 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 489 . . . . . . . . 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 499 . . . . . . . . . 10  |-  ( ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
y  e.  ( 1st `  A ) )
76adantl 266 . . . . . . . . 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 6661 . . . . . . . . . . . . 13  |-  <P  C_  ( P.  X.  P. )
98brel 4420 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
109simpld 109 . . . . . . . . . . 11  |-  ( A 
<P  B  ->  A  e. 
P. )
11 prop 6631 . . . . . . . . . . 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 6637 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
1412, 13sylan 271 . . . . . . . . 9  |-  ( ( A  <P  B  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
155, 7, 14syl2anc 397 . . . . . . . 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 6558 . . . . . . . 8  |-  ( ( q  <Q  r  /\  y  e.  Q. )  ->  ( y  +Q  q
)  <Q  ( y  +Q  r ) )
174, 15, 16syl2anc 397 . . . . . . 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 111 . . . . . . . . 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 111 . . . . . . . 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 6631 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
22 prcunqu 6641 . . . . . . . . 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 271 . . . . . . . 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 397 . . . . . . 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 297 . . . . 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 1507 . . . 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 6755 . . . . . . . . 9  |-  ( q  e.  ( 2nd `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )
30 19.42v 1802 . . . . . . . . 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 180 . . . . . . . 8  |-  ( q  e.  ( 2nd `  C
)  <->  E. y ( q  e.  Q.  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  q )  e.  ( 2nd `  B
) ) ) )
3231anbi2i 438 . . . . . . 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 1802 . . . . . . 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 180 . . . . . 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 438 . . . . 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 1802 . . . . 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 180 . . . 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 6755 . . . . 5  |-  ( r  e.  ( 2nd `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
39 19.42v 1802 . . . . 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 180 . . . 4  |-  ( r  e.  ( 2nd `  C
)  <->  E. y ( r  e.  Q.  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  r )  e.  ( 2nd `  B
) ) ) )
4127, 37, 403imtr4i 194 . . 3  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) )  ->  r  e.  ( 2nd `  C ) )
4241ex 112 . 2  |-  ( ( A  <P  B  /\  r  e.  Q. )  ->  ( ( q  <Q 
r  /\  q  e.  ( 2nd `  C ) )  ->  r  e.  ( 2nd `  C ) ) )
4342rexlimdvw 2453 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 101    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    +Q cplq 6438    <Q cltq 6441   P.cnp 6447    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-ltnqqs 6509  df-inp 6622  df-iltp 6626
This theorem is referenced by:  ltexprlemrnd  6761
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