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Theorem cauappcvgprlemlol 7646
Description: Lemma for cauappcvgpr 7661. 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 7364 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4679 . . . 4  |-  ( s 
<Q  r  ->  ( s  e.  Q.  /\  r  e.  Q. ) )
32simpld 112 . . 3  |-  ( s 
<Q  r  ->  s  e. 
Q. )
433ad2ant2 1019 . 2  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  s  e.  Q. )
5 oveq1 5882 . . . . . . . 8  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
65breq1d 4014 . . . . . . 7  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( r  +Q  q )  <Q  ( F `  q )
) )
76rexbidv 2478 . . . . . 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 5519 . . . . . . 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 7362 . . . . . . . . 9  |-  Q.  e.  _V
1110rabex 4148 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
1210rabex 4148 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
1311, 12op1st 6147 . . . . . . 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 2198 . . . . . 6  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
157, 14elrab2 2897 . . . . 5  |-  ( r  e.  ( 1st `  L
)  <->  ( r  e. 
Q.  /\  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
) )
1615simprbi 275 . . . 4  |-  ( r  e.  ( 1st `  L
)  ->  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
)
17163ad2ant3 1020 . . 3  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
)
18 simpll2 1037 . . . . . . 7  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  s  <Q  r )
19 ltanqg 7399 . . . . . . . . 9  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
2019adantl 277 . . . . . . . 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 488 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  s  e.  Q. )
222simprd 114 . . . . . . . . . 10  |-  ( s 
<Q  r  ->  r  e. 
Q. )
23223ad2ant2 1019 . . . . . . . . 9  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  r  e.  Q. )
2423ad2antrr 488 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  r  e.  Q. )
25 simplr 528 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  q  e.  Q. )
26 addcomnqg 7380 . . . . . . . . 9  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
2726adantl 277 . . . . . . . 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 6044 . . . . . . 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 147 . . . . . 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 7397 . . . . . . 7  |-  <Q  Or  Q.
3130, 1sotri 5025 . . . . . 6  |-  ( ( ( s  +Q  q
)  <Q  ( r  +Q  q )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  (
s  +Q  q ) 
<Q  ( F `  q
) )
3229, 31sylancom 420 . . . . 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 115 . . . 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 2579 . . 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 5882 . . . . 5  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
3736breq1d 4014 . . . 4  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
3837rexbidv 2478 . . 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 2897 . 2  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
404, 35, 39sylanbrc 417 1  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3596   class class class wbr 4004   -->wf 5213   ` cfv 5217  (class class class)co 5875   1stc1st 6139   Q.cnq 7279    +Q cplq 7281    <Q cltq 7284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-ltnqqs 7352
This theorem is referenced by:  cauappcvgprlemrnd  7649
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