Theorem resqrexlemnmsq 10945

Description: Lemma for resqrex 10954. 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

Step	Hyp	Ref	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 )
4	1, 2, 3	resqrexlemf 10935	. . . . . . 7 |- ( ph -> F : NN --> RR+ )
5		resqrexlemnmsq.n	. . . . . . 7 |- ( ph -> N e. NN )
6	4, 5	ffvelrnd 5615	. . . . . 6 |- ( ph -> ( F ` N ) e. RR+ )
7	6	rpred 9623	. . . . 5 |- ( ph -> ( F ` N ) e. RR )
8	7	resqcld 10603	. . . 4 |- ( ph -> ( ( F ` N ) ^ 2 ) e. RR )
9	8	recnd 7918	. . 3 |- ( ph -> ( ( F ` N ) ^ 2 ) e. CC )
10		resqrexlemnmsq.m	. . . . . . 7 |- ( ph -> M e. NN )
11	4, 10	ffvelrnd 5615	. . . . . 6 |- ( ph -> ( F ` M ) e. RR+ )
12	11	rpred 9623	. . . . 5 |- ( ph -> ( F ` M ) e. RR )
13	12	resqcld 10603	. . . 4 |- ( ph -> ( ( F ` M ) ^ 2 ) e. RR )
14	13	recnd 7918	. . 3 |- ( ph -> ( ( F ` M ) ^ 2 ) e. CC )
15	2	recnd 7918	. . 3 |- ( ph -> A e. CC )
16	9, 14, 15	nnncan2d 8235	. 2 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) = ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) )
17	8, 2	resubcld 8270	. . . 4 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - A ) e. RR )
18	13, 2	resubcld 8270	. . . 4 |- ( ph -> ( ( ( F ` M ) ^ 2 ) - A ) e. RR )
19	17, 18	resubcld 8270	. . 3 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) e. RR )
20		1nn 8859	. . . . . . . 8 |- 1 e. NN
21	20	a1i 9	. . . . . . 7 |- ( ph -> 1 e. NN )
22	4, 21	ffvelrnd 5615	. . . . . 6 |- ( ph -> ( F ` 1 ) e. RR+ )
23		2z 9210	. . . . . . 7 |- 2 e. ZZ
24	23	a1i 9	. . . . . 6 |- ( ph -> 2 e. ZZ )
25	22, 24	rpexpcld 10601	. . . . 5 |- ( ph -> ( ( F ` 1 ) ^ 2 ) e. RR+ )
26		4nn 9011	. . . . . . . 8 |- 4 e. NN
27	26	a1i 9	. . . . . . 7 |- ( ph -> 4 e. NN )
28	27	nnrpd 9621	. . . . . 6 |- ( ph -> 4 e. RR+ )
29	5	nnzd 9303	. . . . . . 7 |- ( ph -> N e. ZZ )
30		1zzd 9209	. . . . . . 7 |- ( ph -> 1 e. ZZ )
31	29, 30	zsubcld 9309	. . . . . 6 |- ( ph -> ( N - 1 ) e. ZZ )
32	28, 31	rpexpcld 10601	. . . . 5 |- ( ph -> ( 4 ^ ( N - 1 ) ) e. RR+ )
33	25, 32	rpdivcld 9641	. . . 4 |- ( ph -> ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) e. RR+ )
34	33	rpred 9623	. . 3 |- ( ph -> ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) e. RR )
35	1, 2, 3	resqrexlemover 10938	. . . . . 6 |- ( ( ph /\ M e. NN ) -> A < ( ( F ` M ) ^ 2 ) )
36	10, 35	mpdan 418	. . . . 5 |- ( ph -> A < ( ( F ` M ) ^ 2 ) )
37		difrp 9619	. . . . . 6 |- ( ( A e. RR /\ ( ( F ` M ) ^ 2 ) e. RR ) -> ( A < ( ( F ` M ) ^ 2 ) <-> ( ( ( F ` M ) ^ 2 ) - A ) e. RR+ ) )
38	2, 13, 37	syl2anc 409	. . . . 5 |- ( ph -> ( A < ( ( F ` M ) ^ 2 ) <-> ( ( ( F ` M ) ^ 2 ) - A ) e. RR+ ) )
39	36, 38	mpbid 146	. . . 4 |- ( ph -> ( ( ( F ` M ) ^ 2 ) - A ) e. RR+ )
40	17, 39	ltsubrpd 9656	. . 3 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) < ( ( ( F ` N ) ^ 2 ) - A ) )
41	1, 2, 3	resqrexlemcalc3 10944	. . . 4 |- ( ( ph /\ N e. NN ) -> ( ( ( F ` N ) ^ 2 ) - A ) <_ ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
42	5, 41	mpdan 418	. . 3 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - A ) <_ ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
43	19, 17, 34, 40, 42	ltletrd 8312	. 2 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
44	16, 43	eqbrtrrd 4000	1 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1342   e. wcel 2135   {csn 3570   class class class wbr 3976   X. cxp 4596   ` cfv 5182  (class class class)co 5836   e. cmpo 5838   RRcr 7743   0cc0 7744   1c1 7745   + caddc 7747   < clt 7924   <_ cle 7925   - cmin 8060   / cdiv 8559   NNcn 8848   2c2 8899   4c4 8901   ZZcz 9182   RR+crp 9580   seqcseq 10370   ^cexp 10444

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-frec 6350  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560  df-inn 8849  df-2 8907  df-3 8908  df-4 8909  df-n0 9106  df-z 9183  df-uz 9458  df-rp 9581  df-seqfrec 10371  df-exp 10445

This theorem is referenced by:  resqrexlemnm  10946