Theorem resqrexlemnmsq 11199

Description: Lemma for resqrex 11208. 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 11189	. . . . . . 7 |- ( ph -> F : NN --> RR+ )
5		resqrexlemnmsq.n	. . . . . . 7 |- ( ph -> N e. NN )
6	4, 5	ffvelcdmd 5701	. . . . . 6 |- ( ph -> ( F ` N ) e. RR+ )
7	6	rpred 9788	. . . . 5 |- ( ph -> ( F ` N ) e. RR )
8	7	resqcld 10808	. . . 4 |- ( ph -> ( ( F ` N ) ^ 2 ) e. RR )
9	8	recnd 8072	. . 3 |- ( ph -> ( ( F ` N ) ^ 2 ) e. CC )
10		resqrexlemnmsq.m	. . . . . . 7 |- ( ph -> M e. NN )
11	4, 10	ffvelcdmd 5701	. . . . . 6 |- ( ph -> ( F ` M ) e. RR+ )
12	11	rpred 9788	. . . . 5 |- ( ph -> ( F ` M ) e. RR )
13	12	resqcld 10808	. . . 4 |- ( ph -> ( ( F ` M ) ^ 2 ) e. RR )
14	13	recnd 8072	. . 3 |- ( ph -> ( ( F ` M ) ^ 2 ) e. CC )
15	2	recnd 8072	. . 3 |- ( ph -> A e. CC )
16	9, 14, 15	nnncan2d 8389	. 2 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) = ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) )
17	8, 2	resubcld 8424	. . . 4 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - A ) e. RR )
18	13, 2	resubcld 8424	. . . 4 |- ( ph -> ( ( ( F ` M ) ^ 2 ) - A ) e. RR )
19	17, 18	resubcld 8424	. . 3 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) e. RR )
20		1nn 9018	. . . . . . . 8 |- 1 e. NN
21	20	a1i 9	. . . . . . 7 |- ( ph -> 1 e. NN )
22	4, 21	ffvelcdmd 5701	. . . . . 6 |- ( ph -> ( F ` 1 ) e. RR+ )
23		2z 9371	. . . . . . 7 |- 2 e. ZZ
24	23	a1i 9	. . . . . 6 |- ( ph -> 2 e. ZZ )
25	22, 24	rpexpcld 10806	. . . . 5 |- ( ph -> ( ( F ` 1 ) ^ 2 ) e. RR+ )
26		4nn 9171	. . . . . . . 8 |- 4 e. NN
27	26	a1i 9	. . . . . . 7 |- ( ph -> 4 e. NN )
28	27	nnrpd 9786	. . . . . 6 |- ( ph -> 4 e. RR+ )
29	5	nnzd 9464	. . . . . . 7 |- ( ph -> N e. ZZ )
30		1zzd 9370	. . . . . . 7 |- ( ph -> 1 e. ZZ )
31	29, 30	zsubcld 9470	. . . . . 6 |- ( ph -> ( N - 1 ) e. ZZ )
32	28, 31	rpexpcld 10806	. . . . 5 |- ( ph -> ( 4 ^ ( N - 1 ) ) e. RR+ )
33	25, 32	rpdivcld 9806	. . . 4 |- ( ph -> ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) e. RR+ )
34	33	rpred 9788	. . 3 |- ( ph -> ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) e. RR )
35	1, 2, 3	resqrexlemover 11192	. . . . . 6 |- ( ( ph /\ M e. NN ) -> A < ( ( F ` M ) ^ 2 ) )
36	10, 35	mpdan 421	. . . . 5 |- ( ph -> A < ( ( F ` M ) ^ 2 ) )
37		difrp 9784	. . . . . 6 |- ( ( A e. RR /\ ( ( F ` M ) ^ 2 ) e. RR ) -> ( A < ( ( F ` M ) ^ 2 ) <-> ( ( ( F ` M ) ^ 2 ) - A ) e. RR+ ) )
38	2, 13, 37	syl2anc 411	. . . . 5 |- ( ph -> ( A < ( ( F ` M ) ^ 2 ) <-> ( ( ( F ` M ) ^ 2 ) - A ) e. RR+ ) )
39	36, 38	mpbid 147	. . . 4 |- ( ph -> ( ( ( F ` M ) ^ 2 ) - A ) e. RR+ )
40	17, 39	ltsubrpd 9821	. . 3 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) < ( ( ( F ` N ) ^ 2 ) - A ) )
41	1, 2, 3	resqrexlemcalc3 11198	. . . 4 |- ( ( ph /\ N e. NN ) -> ( ( ( F ` N ) ^ 2 ) - A ) <_ ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
42	5, 41	mpdan 421	. . 3 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - A ) <_ ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
43	19, 17, 34, 40, 42	ltletrd 8467	. 2 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
44	16, 43	eqbrtrrd 4058	1 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   {csn 3623   class class class wbr 4034   X. cxp 4662   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   RRcr 7895   0cc0 7896   1c1 7897   + caddc 7899   < clt 8078   <_ cle 8079   - cmin 8214   / cdiv 8716   NNcn 9007   2c2 9058   4c4 9060   ZZcz 9343   RR+crp 9745   seqcseq 10556   ^cexp 10647

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-seqfrec 10557  df-exp 10648

This theorem is referenced by:  resqrexlemnm  11200