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Theorem resqrexlemgt0 11024
Description: Lemma for resqrex 11030. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.)
Hypotheses
Ref Expression
resqrexlemex.seq  |-  F  =  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ,  ( NN  X.  { ( 1  +  A ) } ) )
resqrexlemex.a  |-  ( ph  ->  A  e.  RR )
resqrexlemex.agt0  |-  ( ph  ->  0  <_  A )
resqrexlemgt0.rr  |-  ( ph  ->  L  e.  RR )
resqrexlemgt0.lim  |-  ( ph  ->  A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) ) )
Assertion
Ref Expression
resqrexlemgt0  |-  ( ph  ->  0  <_  L )
Distinct variable groups:    y, A, z   
e, F    e, L, i, j    ph, i, j   
z, j, ph    ph, y
Allowed substitution hints:    ph( e)    A( e,
i, j)    F( y,
z, i, j)    L( y, z)

Proof of Theorem resqrexlemgt0
StepHypRef Expression
1 oveq2 5882 . . . . . . . . 9  |-  ( e  =  -u L  ->  ( L  +  e )  =  ( L  +  -u L ) )
21breq2d 4015 . . . . . . . 8  |-  ( e  =  -u L  ->  (
( F `  i
)  <  ( L  +  e )  <->  ( F `  i )  <  ( L  +  -u L ) ) )
3 oveq2 5882 . . . . . . . . 9  |-  ( e  =  -u L  ->  (
( F `  i
)  +  e )  =  ( ( F `
 i )  + 
-u L ) )
43breq2d 4015 . . . . . . . 8  |-  ( e  =  -u L  ->  ( L  <  ( ( F `
 i )  +  e )  <->  L  <  ( ( F `  i
)  +  -u L
) ) )
52, 4anbi12d 473 . . . . . . 7  |-  ( e  =  -u L  ->  (
( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) )  <-> 
( ( F `  i )  <  ( L  +  -u L )  /\  L  <  (
( F `  i
)  +  -u L
) ) ) )
65rexralbidv 2503 . . . . . 6  |-  ( e  =  -u L  ->  ( E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) )  <->  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
( ( F `  i )  <  ( L  +  -u L )  /\  L  <  (
( F `  i
)  +  -u L
) ) ) )
7 resqrexlemgt0.lim . . . . . . 7  |-  ( ph  ->  A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) ) )
87adantr 276 . . . . . 6  |-  ( (
ph  /\  L  <  0 )  ->  A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) ) )
9 resqrexlemgt0.rr . . . . . . . . 9  |-  ( ph  ->  L  e.  RR )
109adantr 276 . . . . . . . 8  |-  ( (
ph  /\  L  <  0 )  ->  L  e.  RR )
1110renegcld 8335 . . . . . . 7  |-  ( (
ph  /\  L  <  0 )  ->  -u L  e.  RR )
129lt0neg1d 8470 . . . . . . . 8  |-  ( ph  ->  ( L  <  0  <->  0  <  -u L ) )
1312biimpa 296 . . . . . . 7  |-  ( (
ph  /\  L  <  0 )  ->  0  <  -u L )
1411, 13elrpd 9691 . . . . . 6  |-  ( (
ph  /\  L  <  0 )  ->  -u L  e.  RR+ )
156, 8, 14rspcdva 2846 . . . . 5  |-  ( (
ph  /\  L  <  0 )  ->  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
( ( F `  i )  <  ( L  +  -u L )  /\  L  <  (
( F `  i
)  +  -u L
) ) )
16 simpl 109 . . . . . . . 8  |-  ( ( ( F `  i
)  <  ( L  +  -u L )  /\  L  <  ( ( F `
 i )  + 
-u L ) )  ->  ( F `  i )  <  ( L  +  -u L ) )
1710recnd 7984 . . . . . . . . . 10  |-  ( (
ph  /\  L  <  0 )  ->  L  e.  CC )
1817negidd 8256 . . . . . . . . 9  |-  ( (
ph  /\  L  <  0 )  ->  ( L  +  -u L )  =  0 )
1918breq2d 4015 . . . . . . . 8  |-  ( (
ph  /\  L  <  0 )  ->  (
( F `  i
)  <  ( L  +  -u L )  <->  ( F `  i )  <  0
) )
2016, 19imbitrid 154 . . . . . . 7  |-  ( (
ph  /\  L  <  0 )  ->  (
( ( F `  i )  <  ( L  +  -u L )  /\  L  <  (
( F `  i
)  +  -u L
) )  ->  ( F `  i )  <  0 ) )
2120ralimdv 2545 . . . . . 6  |-  ( (
ph  /\  L  <  0 )  ->  ( A. i  e.  ( ZZ>=
`  j ) ( ( F `  i
)  <  ( L  +  -u L )  /\  L  <  ( ( F `
 i )  + 
-u L ) )  ->  A. i  e.  (
ZZ>= `  j ) ( F `  i )  <  0 ) )
2221reximdv 2578 . . . . 5  |-  ( (
ph  /\  L  <  0 )  ->  ( E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( L  +  -u L )  /\  L  <  ( ( F `  i )  +  -u L ) )  ->  E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( F `
 i )  <  0 ) )
2315, 22mpd 13 . . . 4  |-  ( (
ph  /\  L  <  0 )  ->  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
( F `  i
)  <  0 )
24 0red 7957 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  NN  /\  i  e.  ( ZZ>= `  j )
) )  ->  0  e.  RR )
25 eluznn 9598 . . . . . . . . . . . . 13  |-  ( ( j  e.  NN  /\  i  e.  ( ZZ>= `  j ) )  -> 
i  e.  NN )
26 resqrexlemex.seq . . . . . . . . . . . . . . 15  |-  F  =  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ,  ( NN  X.  { ( 1  +  A ) } ) )
27 resqrexlemex.a . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  RR )
28 resqrexlemex.agt0 . . . . . . . . . . . . . . 15  |-  ( ph  ->  0  <_  A )
2926, 27, 28resqrexlemf 11011 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : NN --> RR+ )
3029ffvelcdmda 5651 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  NN )  ->  ( F `
 i )  e.  RR+ )
3125, 30sylan2 286 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  NN  /\  i  e.  ( ZZ>= `  j )
) )  ->  ( F `  i )  e.  RR+ )
3231rpred 9694 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  NN  /\  i  e.  ( ZZ>= `  j )
) )  ->  ( F `  i )  e.  RR )
3331rpgt0d 9697 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  NN  /\  i  e.  ( ZZ>= `  j )
) )  ->  0  <  ( F `  i
) )
3424, 32, 33ltnsymd 8075 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  NN  /\  i  e.  ( ZZ>= `  j )
) )  ->  -.  ( F `  i )  <  0 )
3534pm2.21d 619 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  NN  /\  i  e.  ( ZZ>= `  j )
) )  ->  (
( F `  i
)  <  0  -> F.  ) )
3635anassrs 400 . . . . . . . 8  |-  ( ( ( ph  /\  j  e.  NN )  /\  i  e.  ( ZZ>= `  j )
)  ->  ( ( F `  i )  <  0  -> F.  )
)
3736ralimdva 2544 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN )  ->  ( A. i  e.  ( ZZ>= `  j ) ( F `
 i )  <  0  ->  A. i  e.  ( ZZ>= `  j ) F.  ) )
38 nnz 9270 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  ZZ )
39 uzid 9540 . . . . . . . . . 10  |-  ( j  e.  ZZ  ->  j  e.  ( ZZ>= `  j )
)
40 elex2 2753 . . . . . . . . . 10  |-  ( j  e.  ( ZZ>= `  j
)  ->  E. z 
z  e.  ( ZZ>= `  j ) )
41 r19.3rmv 3513 . . . . . . . . . 10  |-  ( E. z  z  e.  (
ZZ>= `  j )  -> 
( F.  <->  A. i  e.  ( ZZ>= `  j ) F.  ) )
4239, 40, 413syl 17 . . . . . . . . 9  |-  ( j  e.  ZZ  ->  ( F. 
<-> 
A. i  e.  (
ZZ>= `  j ) F.  ) )
4338, 42syl 14 . . . . . . . 8  |-  ( j  e.  NN  ->  ( F. 
<-> 
A. i  e.  (
ZZ>= `  j ) F.  ) )
4443adantl 277 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN )  ->  ( F.  <->  A. i  e.  ( ZZ>=
`  j ) F.  ) )
4537, 44sylibrd 169 . . . . . 6  |-  ( (
ph  /\  j  e.  NN )  ->  ( A. i  e.  ( ZZ>= `  j ) ( F `
 i )  <  0  -> F.  )
)
4645rexlimdva 2594 . . . . 5  |-  ( ph  ->  ( E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
( F `  i
)  <  0  -> F.  ) )
4746adantr 276 . . . 4  |-  ( (
ph  /\  L  <  0 )  ->  ( E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( F `
 i )  <  0  -> F.  )
)
4823, 47mpd 13 . . 3  |-  ( (
ph  /\  L  <  0 )  -> F.  )
4948inegd 1372 . 2  |-  ( ph  ->  -.  L  <  0
)
50 0re 7956 . . 3  |-  0  e.  RR
51 lenlt 8031 . . 3  |-  ( ( 0  e.  RR  /\  L  e.  RR )  ->  ( 0  <_  L  <->  -.  L  <  0 ) )
5250, 9, 51sylancr 414 . 2  |-  ( ph  ->  ( 0  <_  L  <->  -.  L  <  0 ) )
5349, 52mpbird 167 1  |-  ( ph  ->  0  <_  L )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   F. wfal 1358   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456   {csn 3592   class class class wbr 4003    X. cxp 4624   ` cfv 5216  (class class class)co 5874    e. cmpo 5876   RRcr 7809   0cc0 7810   1c1 7811    + caddc 7813    < clt 7990    <_ cle 7991   -ucneg 8127    / cdiv 8627   NNcn 8917   2c2 8968   ZZcz 9251   ZZ>=cuz 9526   RR+crp 9651    seqcseq 10442
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928
This theorem depends on definitions:  df-bi 117  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-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-n0 9175  df-z 9252  df-uz 9527  df-rp 9652  df-seqfrec 10443
This theorem is referenced by:  resqrexlemglsq  11026  resqrexlemex  11029
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