Theorem resqrexlemnmsq 11702

Description: Lemma for resqrex 11711. 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 11692	. . . . . . 7 |- ( ph -> F : NN --> RR+ )
5		resqrexlemnmsq.n	. . . . . . 7 |- ( ph -> N e. NN )
6	4, 5	ffvelcdmd 5813	. . . . . 6 |- ( ph -> ( F ` N ) e. RR+ )
7	6	rpred 10029	. . . . 5 |- ( ph -> ( F ` N ) e. RR )
8	7	resqcld 11061	. . . 4 |- ( ph -> ( ( F ` N ) ^ 2 ) e. RR )
9	8	recnd 8302	. . 3 |- ( ph -> ( ( F ` N ) ^ 2 ) e. CC )
10		resqrexlemnmsq.m	. . . . . . 7 |- ( ph -> M e. NN )
11	4, 10	ffvelcdmd 5813	. . . . . 6 |- ( ph -> ( F ` M ) e. RR+ )
12	11	rpred 10029	. . . . 5 |- ( ph -> ( F ` M ) e. RR )
13	12	resqcld 11061	. . . 4 |- ( ph -> ( ( F ` M ) ^ 2 ) e. RR )
14	13	recnd 8302	. . 3 |- ( ph -> ( ( F ` M ) ^ 2 ) e. CC )
15	2	recnd 8302	. . 3 |- ( ph -> A e. CC )
16	9, 14, 15	nnncan2d 8619	. 2 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) = ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) )
17	8, 2	resubcld 8654	. . . 4 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - A ) e. RR )
18	13, 2	resubcld 8654	. . . 4 |- ( ph -> ( ( ( F ` M ) ^ 2 ) - A ) e. RR )
19	17, 18	resubcld 8654	. . 3 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) e. RR )
20		1nn 9248	. . . . . . . 8 |- 1 e. NN
21	20	a1i 9	. . . . . . 7 |- ( ph -> 1 e. NN )
22	4, 21	ffvelcdmd 5813	. . . . . 6 |- ( ph -> ( F ` 1 ) e. RR+ )
23		2z 9605	. . . . . . 7 |- 2 e. ZZ
24	23	a1i 9	. . . . . 6 |- ( ph -> 2 e. ZZ )
25	22, 24	rpexpcld 11059	. . . . 5 |- ( ph -> ( ( F ` 1 ) ^ 2 ) e. RR+ )
26		4nn 9401	. . . . . . . 8 |- 4 e. NN
27	26	a1i 9	. . . . . . 7 |- ( ph -> 4 e. NN )
28	27	nnrpd 10027	. . . . . 6 |- ( ph -> 4 e. RR+ )
29	5	nnzd 9699	. . . . . . 7 |- ( ph -> N e. ZZ )
30		1zzd 9604	. . . . . . 7 |- ( ph -> 1 e. ZZ )
31	29, 30	zsubcld 9705	. . . . . 6 |- ( ph -> ( N - 1 ) e. ZZ )
32	28, 31	rpexpcld 11059	. . . . 5 |- ( ph -> ( 4 ^ ( N - 1 ) ) e. RR+ )
33	25, 32	rpdivcld 10047	. . . 4 |- ( ph -> ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) e. RR+ )
34	33	rpred 10029	. . 3 |- ( ph -> ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) e. RR )
35	1, 2, 3	resqrexlemover 11695	. . . . . 6 |- ( ( ph /\ M e. NN ) -> A < ( ( F ` M ) ^ 2 ) )
36	10, 35	mpdan 421	. . . . 5 |- ( ph -> A < ( ( F ` M ) ^ 2 ) )
37		difrp 10025	. . . . . 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 10062	. . 3 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) < ( ( ( F ` N ) ^ 2 ) - A ) )
41	1, 2, 3	resqrexlemcalc3 11701	. . . 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 8697	. 2 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
44	16, 43	eqbrtrrd 4133	1 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203   {csn 3689   class class class wbr 4109   X. cxp 4747   ` cfv 5352  (class class class)co 6050   e. cmpo 6052   RRcr 8126   0cc0 8127   1c1 8128   + caddc 8130   < clt 8308   <_ cle 8309   - cmin 8444   / cdiv 8946   NNcn 9237   2c2 9288   4c4 9290   ZZcz 9577   RR+crp 9986   seqcseq 10809   ^cexp 10900

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-rp 9987  df-seqfrec 10810  df-exp 10901

This theorem is referenced by:  resqrexlemnm  11703