Theorem resqrexlemnmsq 11443

Description: Lemma for resqrex 11452. 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 11433	. . . . . . 7 |- ( ph -> F : NN --> RR+ )
5		resqrexlemnmsq.n	. . . . . . 7 |- ( ph -> N e. NN )
6	4, 5	ffvelcdmd 5739	. . . . . 6 |- ( ph -> ( F ` N ) e. RR+ )
7	6	rpred 9853	. . . . 5 |- ( ph -> ( F ` N ) e. RR )
8	7	resqcld 10881	. . . 4 |- ( ph -> ( ( F ` N ) ^ 2 ) e. RR )
9	8	recnd 8136	. . 3 |- ( ph -> ( ( F ` N ) ^ 2 ) e. CC )
10		resqrexlemnmsq.m	. . . . . . 7 |- ( ph -> M e. NN )
11	4, 10	ffvelcdmd 5739	. . . . . 6 |- ( ph -> ( F ` M ) e. RR+ )
12	11	rpred 9853	. . . . 5 |- ( ph -> ( F ` M ) e. RR )
13	12	resqcld 10881	. . . 4 |- ( ph -> ( ( F ` M ) ^ 2 ) e. RR )
14	13	recnd 8136	. . 3 |- ( ph -> ( ( F ` M ) ^ 2 ) e. CC )
15	2	recnd 8136	. . 3 |- ( ph -> A e. CC )
16	9, 14, 15	nnncan2d 8453	. 2 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) = ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) )
17	8, 2	resubcld 8488	. . . 4 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - A ) e. RR )
18	13, 2	resubcld 8488	. . . 4 |- ( ph -> ( ( ( F ` M ) ^ 2 ) - A ) e. RR )
19	17, 18	resubcld 8488	. . 3 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) e. RR )
20		1nn 9082	. . . . . . . 8 |- 1 e. NN
21	20	a1i 9	. . . . . . 7 |- ( ph -> 1 e. NN )
22	4, 21	ffvelcdmd 5739	. . . . . 6 |- ( ph -> ( F ` 1 ) e. RR+ )
23		2z 9435	. . . . . . 7 |- 2 e. ZZ
24	23	a1i 9	. . . . . 6 |- ( ph -> 2 e. ZZ )
25	22, 24	rpexpcld 10879	. . . . 5 |- ( ph -> ( ( F ` 1 ) ^ 2 ) e. RR+ )
26		4nn 9235	. . . . . . . 8 |- 4 e. NN
27	26	a1i 9	. . . . . . 7 |- ( ph -> 4 e. NN )
28	27	nnrpd 9851	. . . . . 6 |- ( ph -> 4 e. RR+ )
29	5	nnzd 9529	. . . . . . 7 |- ( ph -> N e. ZZ )
30		1zzd 9434	. . . . . . 7 |- ( ph -> 1 e. ZZ )
31	29, 30	zsubcld 9535	. . . . . 6 |- ( ph -> ( N - 1 ) e. ZZ )
32	28, 31	rpexpcld 10879	. . . . 5 |- ( ph -> ( 4 ^ ( N - 1 ) ) e. RR+ )
33	25, 32	rpdivcld 9871	. . . 4 |- ( ph -> ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) e. RR+ )
34	33	rpred 9853	. . 3 |- ( ph -> ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) e. RR )
35	1, 2, 3	resqrexlemover 11436	. . . . . 6 |- ( ( ph /\ M e. NN ) -> A < ( ( F ` M ) ^ 2 ) )
36	10, 35	mpdan 421	. . . . 5 |- ( ph -> A < ( ( F ` M ) ^ 2 ) )
37		difrp 9849	. . . . . 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 9886	. . 3 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) < ( ( ( F ` N ) ^ 2 ) - A ) )
41	1, 2, 3	resqrexlemcalc3 11442	. . . 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 8531	. 2 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
44	16, 43	eqbrtrrd 4083	1 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2178   {csn 3643   class class class wbr 4059   X. cxp 4691   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   < clt 8142   <_ cle 8143   - cmin 8278   / cdiv 8780   NNcn 9071   2c2 9122   4c4 9124   ZZcz 9407   RR+crp 9810   seqcseq 10629   ^cexp 10720

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-seqfrec 10630  df-exp 10721

This theorem is referenced by:  resqrexlemnm  11444