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Theorem resqrexlemnmsq 11021
Description: Lemma for resqrex 11030. 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 ) } ) )
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 ) } ) )
2 resqrexlemex.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
3 resqrexlemex.agt0 . . . . . . . 8  |-  ( ph  ->  0  <_  A )
41, 2, 3resqrexlemf 11011 . . . . . . 7  |-  ( ph  ->  F : NN --> RR+ )
5 resqrexlemnmsq.n . . . . . . 7  |-  ( ph  ->  N  e.  NN )
64, 5ffvelcdmd 5652 . . . . . 6  |-  ( ph  ->  ( F `  N
)  e.  RR+ )
76rpred 9694 . . . . 5  |-  ( ph  ->  ( F `  N
)  e.  RR )
87resqcld 10676 . . . 4  |-  ( ph  ->  ( ( F `  N ) ^ 2 )  e.  RR )
98recnd 7984 . . 3  |-  ( ph  ->  ( ( F `  N ) ^ 2 )  e.  CC )
10 resqrexlemnmsq.m . . . . . . 7  |-  ( ph  ->  M  e.  NN )
114, 10ffvelcdmd 5652 . . . . . 6  |-  ( ph  ->  ( F `  M
)  e.  RR+ )
1211rpred 9694 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  RR )
1312resqcld 10676 . . . 4  |-  ( ph  ->  ( ( F `  M ) ^ 2 )  e.  RR )
1413recnd 7984 . . 3  |-  ( ph  ->  ( ( F `  M ) ^ 2 )  e.  CC )
152recnd 7984 . . 3  |-  ( ph  ->  A  e.  CC )
169, 14, 15nnncan2d 8301 . 2  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  =  ( ( ( F `  N
) ^ 2 )  -  ( ( F `
 M ) ^
2 ) ) )
178, 2resubcld 8336 . . . 4  |-  ( ph  ->  ( ( ( F `
 N ) ^
2 )  -  A
)  e.  RR )
1813, 2resubcld 8336 . . . 4  |-  ( ph  ->  ( ( ( F `
 M ) ^
2 )  -  A
)  e.  RR )
1917, 18resubcld 8336 . . 3  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  e.  RR )
20 1nn 8928 . . . . . . . 8  |-  1  e.  NN
2120a1i 9 . . . . . . 7  |-  ( ph  ->  1  e.  NN )
224, 21ffvelcdmd 5652 . . . . . 6  |-  ( ph  ->  ( F `  1
)  e.  RR+ )
23 2z 9279 . . . . . . 7  |-  2  e.  ZZ
2423a1i 9 . . . . . 6  |-  ( ph  ->  2  e.  ZZ )
2522, 24rpexpcld 10674 . . . . 5  |-  ( ph  ->  ( ( F ` 
1 ) ^ 2 )  e.  RR+ )
26 4nn 9080 . . . . . . . 8  |-  4  e.  NN
2726a1i 9 . . . . . . 7  |-  ( ph  ->  4  e.  NN )
2827nnrpd 9692 . . . . . 6  |-  ( ph  ->  4  e.  RR+ )
295nnzd 9372 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
30 1zzd 9278 . . . . . . 7  |-  ( ph  ->  1  e.  ZZ )
3129, 30zsubcld 9378 . . . . . 6  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
3228, 31rpexpcld 10674 . . . . 5  |-  ( ph  ->  ( 4 ^ ( N  -  1 ) )  e.  RR+ )
3325, 32rpdivcld 9712 . . . 4  |-  ( ph  ->  ( ( ( F `
 1 ) ^
2 )  /  (
4 ^ ( N  -  1 ) ) )  e.  RR+ )
3433rpred 9694 . . 3  |-  ( ph  ->  ( ( ( F `
 1 ) ^
2 )  /  (
4 ^ ( N  -  1 ) ) )  e.  RR )
351, 2, 3resqrexlemover 11014 . . . . . 6  |-  ( (
ph  /\  M  e.  NN )  ->  A  < 
( ( F `  M ) ^ 2 ) )
3610, 35mpdan 421 . . . . 5  |-  ( ph  ->  A  <  ( ( F `  M ) ^ 2 ) )
37 difrp 9690 . . . . . 6  |-  ( ( A  e.  RR  /\  ( ( F `  M ) ^ 2 )  e.  RR )  ->  ( A  < 
( ( F `  M ) ^ 2 )  <->  ( ( ( F `  M ) ^ 2 )  -  A )  e.  RR+ ) )
382, 13, 37syl2anc 411 . . . . 5  |-  ( ph  ->  ( A  <  (
( F `  M
) ^ 2 )  <-> 
( ( ( F `
 M ) ^
2 )  -  A
)  e.  RR+ )
)
3936, 38mpbid 147 . . . 4  |-  ( ph  ->  ( ( ( F `
 M ) ^
2 )  -  A
)  e.  RR+ )
4017, 39ltsubrpd 9727 . . 3  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  <  ( ( ( F `  N
) ^ 2 )  -  A ) )
411, 2, 3resqrexlemcalc3 11020 . . . 4  |-  ( (
ph  /\  N  e.  NN )  ->  ( ( ( F `  N
) ^ 2 )  -  A )  <_ 
( ( ( F `
 1 ) ^
2 )  /  (
4 ^ ( N  -  1 ) ) ) )
425, 41mpdan 421 . . 3  |-  ( ph  ->  ( ( ( F `
 N ) ^
2 )  -  A
)  <_  ( (
( F `  1
) ^ 2 )  /  ( 4 ^ ( N  -  1 ) ) ) )
4319, 17, 34, 40, 42ltletrd 8378 . 2  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  <  ( ( ( F `  1
) ^ 2 )  /  ( 4 ^ ( N  -  1 ) ) ) )
4416, 43eqbrtrrd 4027 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 105    = wceq 1353    e. wcel 2148   {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    - cmin 8126    / cdiv 8627   NNcn 8917   2c2 8968   4c4 8970   ZZcz 9251   RR+crp 9651    seqcseq 10442   ^cexp 10516
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-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-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-if 3535  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-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-rp 9652  df-seqfrec 10443  df-exp 10517
This theorem is referenced by:  resqrexlemnm  11022
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