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Theorem ltexprlemlol 7417
Description: The lower cut of our constructed difference is lower. Lemma for ltexpri 7428. (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 519 . . . . . 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 529 . . . . . . 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 111 . . . . . 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 520 . . . . . . . 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 518 . . . . . . . . 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 7320 . . . . . . . . . . . 12  |-  <P  C_  ( P.  X.  P. )
76brel 4591 . . . . . . . . . . 11  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
87simpld 111 . . . . . . . . . 10  |-  ( A 
<P  B  ->  A  e. 
P. )
9 prop 7290 . . . . . . . . . . 11  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
10 elprnqu 7297 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
119, 10sylan 281 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
128, 11sylan 281 . . . . . . . . 9  |-  ( ( A  <P  B  /\  y  e.  ( 2nd `  A ) )  -> 
y  e.  Q. )
135, 3, 12syl2anc 408 . . . . . . . 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 7217 . . . . . . . 8  |-  ( ( q  <Q  r  /\  y  e.  Q. )  ->  ( y  +Q  q
)  <Q  ( y  +Q  r ) )
154, 13, 14syl2anc 408 . . . . . . 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 113 . . . . . . . . 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 113 . . . . . . . 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 7290 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
20 prcdnql 7299 . . . . . . . . 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 281 . . . . . . . 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 408 . . . . . . 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 308 . . . . 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 1579 . . . 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 7413 . . . . . . . . 9  |-  ( r  e.  ( 1st `  C
)  <->  ( r  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  r )  e.  ( 1st `  B ) ) ) )
28 19.42v 1878 . . . . . . . . 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 186 . . . . . . . 8  |-  ( r  e.  ( 1st `  C
)  <->  E. y ( r  e.  Q.  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  r )  e.  ( 1st `  B
) ) ) )
3029anbi2i 452 . . . . . . 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 1878 . . . . . . 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 186 . . . . . 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 452 . . . . 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 1878 . . . . 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 186 . . . 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 7413 . . . . 5  |-  ( q  e.  ( 1st `  C
)  <->  ( q  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  q )  e.  ( 1st `  B ) ) ) )
37 19.42v 1878 . . . . 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 186 . . . 4  |-  ( q  e.  ( 1st `  C
)  <->  E. y ( q  e.  Q.  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  q )  e.  ( 1st `  B
) ) ) )
3925, 35, 383imtr4i 200 . . 3  |-  ( ( ( A  <P  B  /\  q  e.  Q. )  /\  ( q  <Q  r  /\  r  e.  ( 1st `  C ) ) )  ->  q  e.  ( 1st `  C ) )
4039ex 114 . 2  |-  ( ( A  <P  B  /\  q  e.  Q. )  ->  ( ( q  <Q 
r  /\  r  e.  ( 1st `  C ) )  ->  q  e.  ( 1st `  C ) ) )
4140rexlimdvw 2553 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 103    = wceq 1331   E.wex 1468    e. wcel 1480   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7095    +Q cplq 7097    <Q cltq 7100   P.cnp 7106    <P cltp 7110
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-ltnqqs 7168  df-inp 7281  df-iltp 7285
This theorem is referenced by:  ltexprlemrnd  7420
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