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Theorem resqrexlemnmsq 10820
Description: Lemma for resqrex 10829. 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 10810 . . . . . . 7  |-  ( ph  ->  F : NN --> RR+ )
5 resqrexlemnmsq.n . . . . . . 7  |-  ( ph  ->  N  e.  NN )
64, 5ffvelrnd 5563 . . . . . 6  |-  ( ph  ->  ( F `  N
)  e.  RR+ )
76rpred 9512 . . . . 5  |-  ( ph  ->  ( F `  N
)  e.  RR )
87resqcld 10480 . . . 4  |-  ( ph  ->  ( ( F `  N ) ^ 2 )  e.  RR )
98recnd 7817 . . 3  |-  ( ph  ->  ( ( F `  N ) ^ 2 )  e.  CC )
10 resqrexlemnmsq.m . . . . . . 7  |-  ( ph  ->  M  e.  NN )
114, 10ffvelrnd 5563 . . . . . 6  |-  ( ph  ->  ( F `  M
)  e.  RR+ )
1211rpred 9512 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  RR )
1312resqcld 10480 . . . 4  |-  ( ph  ->  ( ( F `  M ) ^ 2 )  e.  RR )
1413recnd 7817 . . 3  |-  ( ph  ->  ( ( F `  M ) ^ 2 )  e.  CC )
152recnd 7817 . . 3  |-  ( ph  ->  A  e.  CC )
169, 14, 15nnncan2d 8131 . 2  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  =  ( ( ( F `  N
) ^ 2 )  -  ( ( F `
 M ) ^
2 ) ) )
178, 2resubcld 8166 . . . 4  |-  ( ph  ->  ( ( ( F `
 N ) ^
2 )  -  A
)  e.  RR )
1813, 2resubcld 8166 . . . 4  |-  ( ph  ->  ( ( ( F `
 M ) ^
2 )  -  A
)  e.  RR )
1917, 18resubcld 8166 . . 3  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  e.  RR )
20 1nn 8754 . . . . . . . 8  |-  1  e.  NN
2120a1i 9 . . . . . . 7  |-  ( ph  ->  1  e.  NN )
224, 21ffvelrnd 5563 . . . . . 6  |-  ( ph  ->  ( F `  1
)  e.  RR+ )
23 2z 9105 . . . . . . 7  |-  2  e.  ZZ
2423a1i 9 . . . . . 6  |-  ( ph  ->  2  e.  ZZ )
2522, 24rpexpcld 10478 . . . . 5  |-  ( ph  ->  ( ( F ` 
1 ) ^ 2 )  e.  RR+ )
26 4nn 8906 . . . . . . . 8  |-  4  e.  NN
2726a1i 9 . . . . . . 7  |-  ( ph  ->  4  e.  NN )
2827nnrpd 9510 . . . . . 6  |-  ( ph  ->  4  e.  RR+ )
295nnzd 9195 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
30 1zzd 9104 . . . . . . 7  |-  ( ph  ->  1  e.  ZZ )
3129, 30zsubcld 9201 . . . . . 6  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
3228, 31rpexpcld 10478 . . . . 5  |-  ( ph  ->  ( 4 ^ ( N  -  1 ) )  e.  RR+ )
3325, 32rpdivcld 9530 . . . 4  |-  ( ph  ->  ( ( ( F `
 1 ) ^
2 )  /  (
4 ^ ( N  -  1 ) ) )  e.  RR+ )
3433rpred 9512 . . 3  |-  ( ph  ->  ( ( ( F `
 1 ) ^
2 )  /  (
4 ^ ( N  -  1 ) ) )  e.  RR )
351, 2, 3resqrexlemover 10813 . . . . . 6  |-  ( (
ph  /\  M  e.  NN )  ->  A  < 
( ( F `  M ) ^ 2 ) )
3610, 35mpdan 418 . . . . 5  |-  ( ph  ->  A  <  ( ( F `  M ) ^ 2 ) )
37 difrp 9508 . . . . . 6  |-  ( ( A  e.  RR  /\  ( ( F `  M ) ^ 2 )  e.  RR )  ->  ( A  < 
( ( F `  M ) ^ 2 )  <->  ( ( ( F `  M ) ^ 2 )  -  A )  e.  RR+ ) )
382, 13, 37syl2anc 409 . . . . 5  |-  ( ph  ->  ( A  <  (
( F `  M
) ^ 2 )  <-> 
( ( ( F `
 M ) ^
2 )  -  A
)  e.  RR+ )
)
3936, 38mpbid 146 . . . 4  |-  ( ph  ->  ( ( ( F `
 M ) ^
2 )  -  A
)  e.  RR+ )
4017, 39ltsubrpd 9545 . . 3  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  <  ( ( ( F `  N
) ^ 2 )  -  A ) )
411, 2, 3resqrexlemcalc3 10819 . . . 4  |-  ( (
ph  /\  N  e.  NN )  ->  ( ( ( F `  N
) ^ 2 )  -  A )  <_ 
( ( ( F `
 1 ) ^
2 )  /  (
4 ^ ( N  -  1 ) ) ) )
425, 41mpdan 418 . . 3  |-  ( ph  ->  ( ( ( F `
 N ) ^
2 )  -  A
)  <_  ( (
( F `  1
) ^ 2 )  /  ( 4 ^ ( N  -  1 ) ) ) )
4319, 17, 34, 40, 42ltletrd 8208 . 2  |-  ( ph  ->  ( ( ( ( F `  N ) ^ 2 )  -  A )  -  (
( ( F `  M ) ^ 2 )  -  A ) )  <  ( ( ( F `  1
) ^ 2 )  /  ( 4 ^ ( N  -  1 ) ) ) )
4416, 43eqbrtrrd 3959 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 104    = wceq 1332    e. wcel 1481   {csn 3531   class class class wbr 3936    X. cxp 4544   ` cfv 5130  (class class class)co 5781    e. cmpo 5783   RRcr 7642   0cc0 7643   1c1 7644    + caddc 7646    < clt 7823    <_ cle 7824    - cmin 7956    / cdiv 8455   NNcn 8743   2c2 8794   4c4 8796   ZZcz 9077   RR+crp 9469    seqcseq 10248   ^cexp 10322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-rp 9470  df-seqfrec 10249  df-exp 10323
This theorem is referenced by:  resqrexlemnm  10821
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