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Theorem resqrexlemlo 9611
Description: Lemma for resqrex 9624. A (variable) lower bound for each term of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2021.)
Hypotheses
Ref Expression
resqrexlemex.seq  |-  F  =  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ,  ( NN  X.  { ( 1  +  A ) } ) ,  RR+ )
resqrexlemex.a  |-  ( ph  ->  A  e.  RR )
resqrexlemex.agt0  |-  ( ph  ->  0  <_  A )
Assertion
Ref Expression
resqrexlemlo  |-  ( (
ph  /\  N  e.  NN )  ->  ( 1  /  ( 2 ^ N ) )  < 
( F `  N
) )
Distinct variable groups:    y, A, z    ph, y, z
Allowed substitution hints:    F( y, z)    N( y, z)

Proof of Theorem resqrexlemlo
Dummy variables  k  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5520 . . . . . 6  |-  ( w  =  1  ->  (
2 ^ w )  =  ( 2 ^ 1 ) )
21oveq2d 5528 . . . . 5  |-  ( w  =  1  ->  (
1  /  ( 2 ^ w ) )  =  ( 1  / 
( 2 ^ 1 ) ) )
3 fveq2 5178 . . . . 5  |-  ( w  =  1  ->  ( F `  w )  =  ( F ` 
1 ) )
42, 3breq12d 3777 . . . 4  |-  ( w  =  1  ->  (
( 1  /  (
2 ^ w ) )  <  ( F `
 w )  <->  ( 1  /  ( 2 ^ 1 ) )  < 
( F `  1
) ) )
54imbi2d 219 . . 3  |-  ( w  =  1  ->  (
( ph  ->  ( 1  /  ( 2 ^ w ) )  < 
( F `  w
) )  <->  ( ph  ->  ( 1  /  (
2 ^ 1 ) )  <  ( F `
 1 ) ) ) )
6 oveq2 5520 . . . . . 6  |-  ( w  =  k  ->  (
2 ^ w )  =  ( 2 ^ k ) )
76oveq2d 5528 . . . . 5  |-  ( w  =  k  ->  (
1  /  ( 2 ^ w ) )  =  ( 1  / 
( 2 ^ k
) ) )
8 fveq2 5178 . . . . 5  |-  ( w  =  k  ->  ( F `  w )  =  ( F `  k ) )
97, 8breq12d 3777 . . . 4  |-  ( w  =  k  ->  (
( 1  /  (
2 ^ w ) )  <  ( F `
 w )  <->  ( 1  /  ( 2 ^ k ) )  < 
( F `  k
) ) )
109imbi2d 219 . . 3  |-  ( w  =  k  ->  (
( ph  ->  ( 1  /  ( 2 ^ w ) )  < 
( F `  w
) )  <->  ( ph  ->  ( 1  /  (
2 ^ k ) )  <  ( F `
 k ) ) ) )
11 oveq2 5520 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  (
2 ^ w )  =  ( 2 ^ ( k  +  1 ) ) )
1211oveq2d 5528 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (
1  /  ( 2 ^ w ) )  =  ( 1  / 
( 2 ^ (
k  +  1 ) ) ) )
13 fveq2 5178 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  ( F `  w )  =  ( F `  ( k  +  1 ) ) )
1412, 13breq12d 3777 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
( 1  /  (
2 ^ w ) )  <  ( F `
 w )  <->  ( 1  /  ( 2 ^ ( k  +  1 ) ) )  < 
( F `  (
k  +  1 ) ) ) )
1514imbi2d 219 . . 3  |-  ( w  =  ( k  +  1 )  ->  (
( ph  ->  ( 1  /  ( 2 ^ w ) )  < 
( F `  w
) )  <->  ( ph  ->  ( 1  /  (
2 ^ ( k  +  1 ) ) )  <  ( F `
 ( k  +  1 ) ) ) ) )
16 oveq2 5520 . . . . . 6  |-  ( w  =  N  ->  (
2 ^ w )  =  ( 2 ^ N ) )
1716oveq2d 5528 . . . . 5  |-  ( w  =  N  ->  (
1  /  ( 2 ^ w ) )  =  ( 1  / 
( 2 ^ N
) ) )
18 fveq2 5178 . . . . 5  |-  ( w  =  N  ->  ( F `  w )  =  ( F `  N ) )
1917, 18breq12d 3777 . . . 4  |-  ( w  =  N  ->  (
( 1  /  (
2 ^ w ) )  <  ( F `
 w )  <->  ( 1  /  ( 2 ^ N ) )  < 
( F `  N
) ) )
2019imbi2d 219 . . 3  |-  ( w  =  N  ->  (
( ph  ->  ( 1  /  ( 2 ^ w ) )  < 
( F `  w
) )  <->  ( ph  ->  ( 1  /  (
2 ^ N ) )  <  ( F `
 N ) ) ) )
21 2cnd 7988 . . . . . . . 8  |-  ( ph  ->  2  e.  CC )
2221exp1d 9376 . . . . . . 7  |-  ( ph  ->  ( 2 ^ 1 )  =  2 )
23 2rp 8588 . . . . . . 7  |-  2  e.  RR+
2422, 23syl6eqel 2128 . . . . . 6  |-  ( ph  ->  ( 2 ^ 1 )  e.  RR+ )
2524rprecred 8634 . . . . 5  |-  ( ph  ->  ( 1  /  (
2 ^ 1 ) )  e.  RR )
26 1red 7042 . . . . 5  |-  ( ph  ->  1  e.  RR )
27 resqrexlemex.a . . . . . 6  |-  ( ph  ->  A  e.  RR )
2826, 27readdcld 7055 . . . . 5  |-  ( ph  ->  ( 1  +  A
)  e.  RR )
2922oveq2d 5528 . . . . . 6  |-  ( ph  ->  ( 1  /  (
2 ^ 1 ) )  =  ( 1  /  2 ) )
30 halflt1 8142 . . . . . 6  |-  ( 1  /  2 )  <  1
3129, 30syl6eqbr 3801 . . . . 5  |-  ( ph  ->  ( 1  /  (
2 ^ 1 ) )  <  1 )
32 resqrexlemex.agt0 . . . . . 6  |-  ( ph  ->  0  <_  A )
3326, 27addge01d 7524 . . . . . 6  |-  ( ph  ->  ( 0  <_  A  <->  1  <_  ( 1  +  A ) ) )
3432, 33mpbid 135 . . . . 5  |-  ( ph  ->  1  <_  ( 1  +  A ) )
3525, 26, 28, 31, 34ltletrd 7420 . . . 4  |-  ( ph  ->  ( 1  /  (
2 ^ 1 ) )  <  ( 1  +  A ) )
36 resqrexlemex.seq . . . . 5  |-  F  =  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ,  ( NN  X.  { ( 1  +  A ) } ) ,  RR+ )
3736, 27, 32resqrexlemf1 9606 . . . 4  |-  ( ph  ->  ( F `  1
)  =  ( 1  +  A ) )
3835, 37breqtrrd 3790 . . 3  |-  ( ph  ->  ( 1  /  (
2 ^ 1 ) )  <  ( F `
 1 ) )
3923a1i 9 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
2  e.  RR+ )
40 nnz 8264 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  ZZ )
4140ad2antlr 458 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
k  e.  ZZ )
4239, 41rpexpcld 9404 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( 2 ^ k
)  e.  RR+ )
4342rpcnd 8624 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( 2 ^ k
)  e.  CC )
44 2cnd 7988 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
2  e.  CC )
4542rpap0d 8628 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( 2 ^ k
) #  0 )
4639rpap0d 8628 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
2 #  0 )
4743, 44, 45, 46recdivap2d 7783 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( ( 1  / 
( 2 ^ k
) )  /  2
)  =  ( 1  /  ( ( 2 ^ k )  x.  2 ) ) )
48 nnnn0 8188 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  NN0 )
4948ad2antlr 458 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
k  e.  NN0 )
5044, 49expp1d 9382 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( 2 ^ (
k  +  1 ) )  =  ( ( 2 ^ k )  x.  2 ) )
5150oveq2d 5528 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( 1  /  (
2 ^ ( k  +  1 ) ) )  =  ( 1  /  ( ( 2 ^ k )  x.  2 ) ) )
5247, 51eqtr4d 2075 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( ( 1  / 
( 2 ^ k
) )  /  2
)  =  ( 1  /  ( 2 ^ ( k  +  1 ) ) ) )
5342rprecred 8634 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( 1  /  (
2 ^ k ) )  e.  RR )
5436, 27, 32resqrexlemf 9605 . . . . . . . . . . . . 13  |-  ( ph  ->  F : NN --> RR+ )
5554ffvelrnda 5302 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  RR+ )
5655rpred 8622 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  RR )
5756adantr 261 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( F `  k
)  e.  RR )
5827adantr 261 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  A  e.  RR )
5958, 55rerpdivcld 8654 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  ( A  /  ( F `  k ) )  e.  RR )
6059adantr 261 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( A  /  ( F `  k )
)  e.  RR )
6157, 60readdcld 7055 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( ( F `  k )  +  ( A  /  ( F `
 k ) ) )  e.  RR )
62 simpr 103 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( 1  /  (
2 ^ k ) )  <  ( F `
 k ) )
6332adantr 261 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN )  ->  0  <_  A )
6458, 55, 63divge0d 8663 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  0  <_ 
( A  /  ( F `  k )
) )
6556, 59addge01d 7524 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  ( 0  <_  ( A  / 
( F `  k
) )  <->  ( F `  k )  <_  (
( F `  k
)  +  ( A  /  ( F `  k ) ) ) ) )
6664, 65mpbid 135 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  <_ 
( ( F `  k )  +  ( A  /  ( F `
 k ) ) ) )
6766adantr 261 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( F `  k
)  <_  ( ( F `  k )  +  ( A  / 
( F `  k
) ) ) )
6853, 57, 61, 62, 67ltletrd 7420 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( 1  /  (
2 ^ k ) )  <  ( ( F `  k )  +  ( A  / 
( F `  k
) ) ) )
6953, 61, 39, 68ltdiv1dd 8680 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( ( 1  / 
( 2 ^ k
) )  /  2
)  <  ( (
( F `  k
)  +  ( A  /  ( F `  k ) ) )  /  2 ) )
7036, 27, 32resqrexlemfp1 9607 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 ( k  +  1 ) )  =  ( ( ( F `
 k )  +  ( A  /  ( F `  k )
) )  /  2
) )
7170adantr 261 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( F `  (
k  +  1 ) )  =  ( ( ( F `  k
)  +  ( A  /  ( F `  k ) ) )  /  2 ) )
7269, 71breqtrrd 3790 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( ( 1  / 
( 2 ^ k
) )  /  2
)  <  ( F `  ( k  +  1 ) ) )
7352, 72eqbrtrrd 3786 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
1  /  ( 2 ^ k ) )  <  ( F `  k ) )  -> 
( 1  /  (
2 ^ ( k  +  1 ) ) )  <  ( F `
 ( k  +  1 ) ) )
7473ex 108 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( 1  /  ( 2 ^ k ) )  <  ( F `  k )  ->  (
1  /  ( 2 ^ ( k  +  1 ) ) )  <  ( F `  ( k  +  1 ) ) ) )
7574expcom 109 . . . 4  |-  ( k  e.  NN  ->  ( ph  ->  ( ( 1  /  ( 2 ^ k ) )  < 
( F `  k
)  ->  ( 1  /  ( 2 ^ ( k  +  1 ) ) )  < 
( F `  (
k  +  1 ) ) ) ) )
7675a2d 23 . . 3  |-  ( k  e.  NN  ->  (
( ph  ->  ( 1  /  ( 2 ^ k ) )  < 
( F `  k
) )  ->  ( ph  ->  ( 1  / 
( 2 ^ (
k  +  1 ) ) )  <  ( F `  ( k  +  1 ) ) ) ) )
775, 10, 15, 20, 38, 76nnind 7930 . 2  |-  ( N  e.  NN  ->  ( ph  ->  ( 1  / 
( 2 ^ N
) )  <  ( F `  N )
) )
7877impcom 116 1  |-  ( (
ph  /\  N  e.  NN )  ->  ( 1  /  ( 2 ^ N ) )  < 
( F `  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   {csn 3375   class class class wbr 3764    X. cxp 4343   ` cfv 4902  (class class class)co 5512    |-> cmpt2 5514   RRcr 6888   0cc0 6889   1c1 6890    + caddc 6892    x. cmul 6894    < clt 7060    <_ cle 7061    / cdiv 7651   NNcn 7914   2c2 7964   NN0cn0 8181   ZZcz 8245   RR+crp 8583    seqcseq 9211   ^cexp 9254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-inn 7915  df-2 7973  df-n0 8182  df-z 8246  df-uz 8474  df-rp 8584  df-iseq 9212  df-iexp 9255
This theorem is referenced by:  resqrexlemnm  9616
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