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Theorem ltexprlemupu 7435
Description: The upper cut of our constructed difference is upper. Lemma for ltexpri 7444. (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 520 . . . . . 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 530 . . . . . . 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 111 . . . . . 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 521 . . . . . . . 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 519 . . . . . . . . 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 529 . . . . . . . . . 10  |-  ( ( q  <Q  r  /\  ( q  e.  Q.  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )  -> 
y  e.  ( 1st `  A ) )
76adantl 275 . . . . . . . . 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 7336 . . . . . . . . . . . . 13  |-  <P  C_  ( P.  X.  P. )
98brel 4598 . . . . . . . . . . . 12  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
109simpld 111 . . . . . . . . . . 11  |-  ( A 
<P  B  ->  A  e. 
P. )
11 prop 7306 . . . . . . . . . . 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 7312 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
1412, 13sylan 281 . . . . . . . . 9  |-  ( ( A  <P  B  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
155, 7, 14syl2anc 409 . . . . . . . 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 7233 . . . . . . . 8  |-  ( ( q  <Q  r  /\  y  e.  Q. )  ->  ( y  +Q  q
)  <Q  ( y  +Q  r ) )
174, 15, 16syl2anc 409 . . . . . . 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 113 . . . . . . . . 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 113 . . . . . . . 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 7306 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
22 prcunqu 7316 . . . . . . . . 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 281 . . . . . . . 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 409 . . . . . . 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 308 . . . . 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 1580 . . . 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 7430 . . . . . . . . 9  |-  ( q  e.  ( 2nd `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  q )  e.  ( 2nd `  B ) ) ) )
30 19.42v 1879 . . . . . . . . 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 186 . . . . . . . 8  |-  ( q  e.  ( 2nd `  C
)  <->  E. y ( q  e.  Q.  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  q )  e.  ( 2nd `  B
) ) ) )
3231anbi2i 453 . . . . . . 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 1879 . . . . . . 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 186 . . . . . 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 453 . . . . 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 1879 . . . . 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 186 . . . 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 7430 . . . . 5  |-  ( r  e.  ( 2nd `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  r )  e.  ( 2nd `  B ) ) ) )
39 19.42v 1879 . . . . 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 186 . . . 4  |-  ( r  e.  ( 2nd `  C
)  <->  E. y ( r  e.  Q.  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  r )  e.  ( 2nd `  B
) ) ) )
4127, 37, 403imtr4i 200 . . 3  |-  ( ( ( A  <P  B  /\  r  e.  Q. )  /\  ( q  <Q  r  /\  q  e.  ( 2nd `  C ) ) )  ->  r  e.  ( 2nd `  C ) )
4241ex 114 . 2  |-  ( ( A  <P  B  /\  r  e.  Q. )  ->  ( ( q  <Q 
r  /\  q  e.  ( 2nd `  C ) )  ->  r  e.  ( 2nd `  C ) ) )
4342rexlimdvw 2556 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 103    = wceq 1332   E.wex 1469    e. wcel 1481   E.wrex 2418   {crab 2421   <.cop 3534   class class class wbr 3936   ` cfv 5130  (class class class)co 5781   1stc1st 6043   2ndc2nd 6044   Q.cnq 7111    +Q cplq 7113    <Q cltq 7116   P.cnp 7122    <P cltp 7126
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-ltnqqs 7184  df-inp 7297  df-iltp 7301
This theorem is referenced by:  ltexprlemrnd  7436
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