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Theorem cauappcvgprlemlol 7462
Description: Lemma for cauappcvgpr 7477. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f  |-  ( ph  ->  F : Q. --> Q. )
cauappcvgpr.app  |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
( F `  p
)  <Q  ( ( F `
 q )  +Q  ( p  +Q  q
) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  (
p  +Q  q ) ) ) )
cauappcvgpr.bnd  |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )
cauappcvgpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u } >.
Assertion
Ref Expression
cauappcvgprlemlol  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) )
Distinct variable groups:    A, p    L, p, q    ph, p, q    L, r, s    A, s, p    F, l, u, p, q, r, s    ph, r,
s
Allowed substitution hints:    ph( u, l)    A( u, r, q, l)    L( u, l)

Proof of Theorem cauappcvgprlemlol
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 7180 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4591 . . . 4  |-  ( s 
<Q  r  ->  ( s  e.  Q.  /\  r  e.  Q. ) )
32simpld 111 . . 3  |-  ( s 
<Q  r  ->  s  e. 
Q. )
433ad2ant2 1003 . 2  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  s  e.  Q. )
5 oveq1 5781 . . . . . . . 8  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
65breq1d 3939 . . . . . . 7  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( r  +Q  q )  <Q  ( F `  q )
) )
76rexbidv 2438 . . . . . 6  |-  ( l  =  r  ->  ( E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q )  <->  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
) )
8 cauappcvgpr.lim . . . . . . . 8  |-  L  = 
<. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u } >.
98fveq2i 5424 . . . . . . 7  |-  ( 1st `  L )  =  ( 1st `  <. { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) } ,  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u } >. )
10 nqex 7178 . . . . . . . . 9  |-  Q.  e.  _V
1110rabex 4072 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
1210rabex 4072 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
1311, 12op1st 6044 . . . . . . 7  |-  ( 1st `  <. { l  e. 
Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u } >. )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
149, 13eqtri 2160 . . . . . 6  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
157, 14elrab2 2843 . . . . 5  |-  ( r  e.  ( 1st `  L
)  <->  ( r  e. 
Q.  /\  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
) )
1615simprbi 273 . . . 4  |-  ( r  e.  ( 1st `  L
)  ->  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
)
17163ad2ant3 1004 . . 3  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
)
18 simpll2 1021 . . . . . . 7  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  s  <Q  r )
19 ltanqg 7215 . . . . . . . . 9  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
2019adantl 275 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  /\  q  e.  Q. )  /\  ( r  +Q  q
)  <Q  ( F `  q ) )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
214ad2antrr 479 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  s  e.  Q. )
222simprd 113 . . . . . . . . . 10  |-  ( s 
<Q  r  ->  r  e. 
Q. )
23223ad2ant2 1003 . . . . . . . . 9  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  r  e.  Q. )
2423ad2antrr 479 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  r  e.  Q. )
25 simplr 519 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  q  e.  Q. )
26 addcomnqg 7196 . . . . . . . . 9  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
2726adantl 275 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  /\  q  e.  Q. )  /\  ( r  +Q  q
)  <Q  ( F `  q ) )  /\  ( f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
2820, 21, 24, 25, 27caovord2d 5940 . . . . . . 7  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  (
s  <Q  r  <->  ( s  +Q  q )  <Q  (
r  +Q  q ) ) )
2918, 28mpbid 146 . . . . . 6  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  (
s  +Q  q ) 
<Q  ( r  +Q  q
) )
30 ltsonq 7213 . . . . . . 7  |-  <Q  Or  Q.
3130, 1sotri 4934 . . . . . 6  |-  ( ( ( s  +Q  q
)  <Q  ( r  +Q  q )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  (
s  +Q  q ) 
<Q  ( F `  q
) )
3229, 31sylancom 416 . . . . 5  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  (
s  +Q  q ) 
<Q  ( F `  q
) )
3332ex 114 . . . 4  |-  ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  ->  (
( r  +Q  q
)  <Q  ( F `  q )  ->  (
s  +Q  q ) 
<Q  ( F `  q
) ) )
3433reximdva 2534 . . 3  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  ( E. q  e.  Q.  (
r  +Q  q ) 
<Q  ( F `  q
)  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
3517, 34mpd 13 . 2  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
)
36 oveq1 5781 . . . . 5  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
3736breq1d 3939 . . . 4  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
3837rexbidv 2438 . . 3  |-  ( l  =  s  ->  ( E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q )  <->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
3938, 14elrab2 2843 . 2  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
404, 35, 39sylanbrc 413 1  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036   Q.cnq 7095    +Q cplq 7097    <Q cltq 7100
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-po 4218  df-iso 4219  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
This theorem is referenced by:  cauappcvgprlemrnd  7465
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