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Theorem ltexprlemlol 7159
Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 7170. (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
ltexprlemlol  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( E. r  e. 
Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  C ) )  ->  q  e.  ( 1st `  C ) ) )
Distinct variable groups:    x, y, q, r, A    x, B, y, q, r    x, C, y, q, r

Proof of Theorem ltexprlemlol
StepHypRef Expression
1 simplr 497 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  q  e.  Q. )
2 simprrr 507 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) )
32simpld 110 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  y  e.  ( 2nd `  A ) )
4 simprl 498 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  q  <Q  r
)
5 simpll 496 . . . . . . . . 9  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  A  <P  B )
6 ltrelpr 7062 . . . . . . . . . . . 12  |-  <P  C_  ( P.  X.  P. )
76brel 4490 . . . . . . . . . . 11  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
87simpld 110 . . . . . . . . . 10  |-  ( A 
<P  B  ->  A  e. 
P. )
9 prop 7032 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
10 elprnqu 7039 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
119, 10sylan 277 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
128, 11sylan 277 . . . . . . . . 9  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
135, 3, 12syl2anc 403 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  y  e.  Q. )
14 ltanqi 6959 . . . . . . . 8  |-  ( ( q  <Q  r  /\  y  e.  Q. )  ->  ( y  +Q  q
)  <Q  ( y  +Q  r ) )
154, 13, 14syl2anc 403 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( y  +Q  q )  <Q  (
y  +Q  r ) )
167simprd 112 . . . . . . . . 9  |-  ( A 
<P  B  ->  B  e. 
P. )
175, 16syl 14 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  B  e.  P. )
182simprd 112 . . . . . . . 8  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( y  +Q  r )  e.  ( 1st `  B ) )
19 prop 7032 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
20 prcdnql 7041 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  ( y  +Q  r
)  e.  ( 1st `  B ) )  -> 
( ( y  +Q  q )  <Q  (
y  +Q  r )  ->  ( y  +Q  q )  e.  ( 1st `  B ) ) )
2119, 20sylan 277 . . . . . . . 8  |-  ( ( B  e.  P.  /\  ( y  +Q  r
)  e.  ( 1st `  B ) )  -> 
( ( y  +Q  q )  <Q  (
y  +Q  r )  ->  ( y  +Q  q )  e.  ( 1st `  B ) ) )
2217, 18, 21syl2anc 403 . . . . . . 7  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( ( y  +Q  q )  <Q 
( y  +Q  r
)  ->  ( y  +Q  q )  e.  ( 1st `  B ) ) )
2315, 22mpd 13 . . . . . 6  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( y  +Q  q )  e.  ( 1st `  B ) )
241, 3, 23jca32 303 . . . . 5  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
2524eximi 1536 . . . 4  |-  ( E. y ( ( A 
<P  B  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  (
r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  ->  E. y ( q  e.  Q.  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  q )  e.  ( 1st `  B
) ) ) )
26 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 ) ) } >.
2726ltexprlemell 7155 . . . . . . . . 9  |-  ( r  e.  ( 1st `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )
28 19.42v 1834 . . . . . . . . 9  |-  ( E. y ( r  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) )  <->  ( r  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )
2927, 28bitr4i 185 . . . . . . . 8  |-  ( r  e.  ( 1st `  C
)  <->  E. y ( r  e.  Q.  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  r )  e.  ( 1st `  B
) ) ) )
3029anbi2i 445 . . . . . . 7  |-  ( ( q  <Q  r  /\  r  e.  ( 1st `  C ) )  <->  ( q  <Q  r  /\  E. y
( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
31 19.42v 1834 . . . . . . 7  |-  ( E. y ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )  <->  ( q  <Q  r  /\  E. y
( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
3230, 31bitr4i 185 . . . . . 6  |-  ( ( q  <Q  r  /\  r  e.  ( 1st `  C ) )  <->  E. y
( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )
3332anbi2i 445 . . . . 5  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )  <->  ( ( A 
<P  B  /\  q  e.  Q. )  /\  E. y ( q  <Q 
r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) ) )
34 19.42v 1834 . . . . 5  |-  ( E. y ( ( A 
<P  B  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  (
r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) )  <-> 
( ( A  <P  B  /\  q  e.  Q. )  /\  E. y ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) ) )
3533, 34bitr4i 185 . . . 4  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )  <->  E. y ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  ( r  e.  Q.  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) ) ) )
3626ltexprlemell 7155 . . . . 5  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
37 19.42v 1834 . . . . 5  |-  ( E. y ( q  e. 
Q.  /\  ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) )  <->  ( q  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
3836, 37bitr4i 185 . . . 4  |-  ( q  e.  ( 1st `  C
)  <->  E. y ( q  e.  Q.  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  q )  e.  ( 1st `  B
) ) ) )
3925, 35, 383imtr4i 199 . . 3  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )  ->  q  e.  ( 1st `  C ) )
4039ex 113 . 2  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( ( q  <Q 
r  /\  r  e.  ( 1st `  C ) )  ->  q  e.  ( 1st `  C ) ) )
4140rexlimdvw 2492 1  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( E. r  e. 
Q.  ( q  <Q 
r  /\  r  e.  ( 1st `  C ) )  ->  q  e.  ( 1st `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   E.wrex 2360   {crab 2363   <.cop 3449   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   1stc1st 5909   2ndc2nd 5910   Q.cnq 6837    +Q cplq 6839    <Q cltq 6842   P.cnp 6848    <P cltp 6852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-ltnqqs 6910  df-inp 7023  df-iltp 7027
This theorem is referenced by:  ltexprlemrnd  7162
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