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Theorem resqrexlemnmsq 10104
Description: Lemma for resqrex 10113. The difference between the squares of two terms of the sequence. (Contributed by Mario Carneiro and Jim Kingdon, 30-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 )
resqrexlemnmsq.n  |-  ( ph  ->  N  e.  NN )
resqrexlemnmsq.m  |-  ( ph  ->  M  e.  NN )
resqrexlemnmsq.nm  |-  ( ph  ->  N  <_  M )
Assertion
Ref Expression
resqrexlemnmsq  |-  ( ph  ->  ( ( ( F `
 N ) ^
2 )  -  (
( F `  M
) ^ 2 ) )  <  ( ( ( F `  1
) ^ 2 )  /  ( 4 ^ ( N  -  1 ) ) ) )
Distinct variable groups:    y, A, z    ph, y, z    y, M, z    y, N, z
Allowed substitution hints:    F( y, z)

Proof of Theorem resqrexlemnmsq
StepHypRef Expression
1 resqrexlemex.seq . . . . . . . 8  |-  F  =  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ,  ( NN  X.  { ( 1  +  A ) } ) ,  RR+ )
2 resqrexlemex.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
3 resqrexlemex.agt0 . . . . . . . 8  |-  ( ph  ->  0  <_  A )
41, 2, 3resqrexlemf 10094 . . . . . . 7  |-  ( ph  ->  F : NN --> RR+ )
5 resqrexlemnmsq.n . . . . . . 7  |-  ( ph  ->  N  e.  NN )
64, 5ffvelrnd 5355 . . . . . 6  |-  ( ph  ->  ( F `  N
)  e.  RR+ )
76rpred 8906 . . . . 5  |-  ( ph  ->  ( F `  N
)  e.  RR )
87resqcld 9780 . . . 4  |-  ( ph  ->  ( ( F `  N ) ^ 2 )  e.  RR )
98recnd 7261 . . 3  |-  ( ph  ->  ( ( F `  N ) ^ 2 )  e.  CC )
10 resqrexlemnmsq.m . . . . . . 7  |-  ( ph  ->  M  e.  NN )
114, 10ffvelrnd 5355 . . . . . 6  |-  ( ph  ->  ( F `  M
)  e.  RR+ )
1211rpred 8906 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  RR )
1312resqcld 9780 . . . 4  |-  ( ph  ->  ( ( F `  M ) ^ 2 )  e.  RR )
1413recnd 7261 . . 3  |-  ( ph  ->  ( ( F `  M ) ^ 2 )  e.  CC )
152recnd 7261 . . 3  |-  ( ph  ->  A  e.  CC )
169, 14, 15nnncan2d 7573 . 2  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  =  ( ( ( F `  N
) ^ 2 )  -  ( ( F `
 M ) ^
2 ) ) )
178, 2resubcld 7604 . . . 4  |-  ( ph  ->  ( ( ( F `
 N ) ^
2 )  -  A
)  e.  RR )
1813, 2resubcld 7604 . . . 4  |-  ( ph  ->  ( ( ( F `
 M ) ^
2 )  -  A
)  e.  RR )
1917, 18resubcld 7604 . . 3  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  e.  RR )
20 1nn 8169 . . . . . . . 8  |-  1  e.  NN
2120a1i 9 . . . . . . 7  |-  ( ph  ->  1  e.  NN )
224, 21ffvelrnd 5355 . . . . . 6  |-  ( ph  ->  ( F `  1
)  e.  RR+ )
23 2z 8512 . . . . . . 7  |-  2  e.  ZZ
2423a1i 9 . . . . . 6  |-  ( ph  ->  2  e.  ZZ )
2522, 24rpexpcld 9778 . . . . 5  |-  ( ph  ->  ( ( F ` 
1 ) ^ 2 )  e.  RR+ )
26 4nn 8314 . . . . . . . 8  |-  4  e.  NN
2726a1i 9 . . . . . . 7  |-  ( ph  ->  4  e.  NN )
2827nnrpd 8905 . . . . . 6  |-  ( ph  ->  4  e.  RR+ )
295nnzd 8601 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
30 1zzd 8511 . . . . . . 7  |-  ( ph  ->  1  e.  ZZ )
3129, 30zsubcld 8607 . . . . . 6  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
3228, 31rpexpcld 9778 . . . . 5  |-  ( ph  ->  ( 4 ^ ( N  -  1 ) )  e.  RR+ )
3325, 32rpdivcld 8924 . . . 4  |-  ( ph  ->  ( ( ( F `
 1 ) ^
2 )  /  (
4 ^ ( N  -  1 ) ) )  e.  RR+ )
3433rpred 8906 . . 3  |-  ( ph  ->  ( ( ( F `
 1 ) ^
2 )  /  (
4 ^ ( N  -  1 ) ) )  e.  RR )
351, 2, 3resqrexlemover 10097 . . . . . 6  |-  ( (
ph  /\  M  e.  NN )  ->  A  < 
( ( F `  M ) ^ 2 ) )
3610, 35mpdan 412 . . . . 5  |-  ( ph  ->  A  <  ( ( F `  M ) ^ 2 ) )
37 difrp 8903 . . . . . 6  |-  ( ( A  e.  RR  /\  ( ( F `  M ) ^ 2 )  e.  RR )  ->  ( A  < 
( ( F `  M ) ^ 2 )  <->  ( ( ( F `  M ) ^ 2 )  -  A )  e.  RR+ ) )
382, 13, 37syl2anc 403 . . . . 5  |-  ( ph  ->  ( A  <  (
( F `  M
) ^ 2 )  <-> 
( ( ( F `
 M ) ^
2 )  -  A
)  e.  RR+ )
)
3936, 38mpbid 145 . . . 4  |-  ( ph  ->  ( ( ( F `
 M ) ^
2 )  -  A
)  e.  RR+ )
4017, 39ltsubrpd 8939 . . 3  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  <  ( ( ( F `  N
) ^ 2 )  -  A ) )
411, 2, 3resqrexlemcalc3 10103 . . . 4  |-  ( (
ph  /\  N  e.  NN )  ->  ( ( ( F `  N
) ^ 2 )  -  A )  <_ 
( ( ( F `
 1 ) ^
2 )  /  (
4 ^ ( N  -  1 ) ) ) )
425, 41mpdan 412 . . 3  |-  ( ph  ->  ( ( ( F `
 N ) ^
2 )  -  A
)  <_  ( (
( F `  1
) ^ 2 )  /  ( 4 ^ ( N  -  1 ) ) ) )
4319, 17, 34, 40, 42ltletrd 7646 . 2  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  <  ( ( ( F `  1
) ^ 2 )  /  ( 4 ^ ( N  -  1 ) ) ) )
4416, 43eqbrtrrd 3827 1  |-  ( ph  ->  ( ( ( F `
 N ) ^
2 )  -  (
( F `  M
) ^ 2 ) )  <  ( ( ( F `  1
) ^ 2 )  /  ( 4 ^ ( N  -  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434   {csn 3416   class class class wbr 3805    X. cxp 4389   ` cfv 4952  (class class class)co 5563    |-> cmpt2 5565   RRcr 7094   0cc0 7095   1c1 7096    + caddc 7098    < clt 7267    <_ cle 7268    - cmin 7398    / cdiv 7879   NNcn 8158   2c2 8208   4c4 8210   ZZcz 8484   RR+crp 8867    seqcseq 9573   ^cexp 9624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-rp 8868  df-iseq 9574  df-iexp 9625
This theorem is referenced by:  resqrexlemnm  10105
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