Theorem resqrexlemnmsq 11361

Description: Lemma for resqrex 11370. 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 11351	. . . . . . 7 |- ( ph -> F : NN --> RR+ )
5		resqrexlemnmsq.n	. . . . . . 7 |- ( ph -> N e. NN )
6	4, 5	ffvelcdmd 5718	. . . . . 6 |- ( ph -> ( F ` N ) e. RR+ )
7	6	rpred 9820	. . . . 5 |- ( ph -> ( F ` N ) e. RR )
8	7	resqcld 10846	. . . 4 |- ( ph -> ( ( F ` N ) ^ 2 ) e. RR )
9	8	recnd 8103	. . 3 |- ( ph -> ( ( F ` N ) ^ 2 ) e. CC )
10		resqrexlemnmsq.m	. . . . . . 7 |- ( ph -> M e. NN )
11	4, 10	ffvelcdmd 5718	. . . . . 6 |- ( ph -> ( F ` M ) e. RR+ )
12	11	rpred 9820	. . . . 5 |- ( ph -> ( F ` M ) e. RR )
13	12	resqcld 10846	. . . 4 |- ( ph -> ( ( F ` M ) ^ 2 ) e. RR )
14	13	recnd 8103	. . 3 |- ( ph -> ( ( F ` M ) ^ 2 ) e. CC )
15	2	recnd 8103	. . 3 |- ( ph -> A e. CC )
16	9, 14, 15	nnncan2d 8420	. 2 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) = ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) )
17	8, 2	resubcld 8455	. . . 4 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - A ) e. RR )
18	13, 2	resubcld 8455	. . . 4 |- ( ph -> ( ( ( F ` M ) ^ 2 ) - A ) e. RR )
19	17, 18	resubcld 8455	. . 3 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) e. RR )
20		1nn 9049	. . . . . . . 8 |- 1 e. NN
21	20	a1i 9	. . . . . . 7 |- ( ph -> 1 e. NN )
22	4, 21	ffvelcdmd 5718	. . . . . 6 |- ( ph -> ( F ` 1 ) e. RR+ )
23		2z 9402	. . . . . . 7 |- 2 e. ZZ
24	23	a1i 9	. . . . . 6 |- ( ph -> 2 e. ZZ )
25	22, 24	rpexpcld 10844	. . . . 5 |- ( ph -> ( ( F ` 1 ) ^ 2 ) e. RR+ )
26		4nn 9202	. . . . . . . 8 |- 4 e. NN
27	26	a1i 9	. . . . . . 7 |- ( ph -> 4 e. NN )
28	27	nnrpd 9818	. . . . . 6 |- ( ph -> 4 e. RR+ )
29	5	nnzd 9496	. . . . . . 7 |- ( ph -> N e. ZZ )
30		1zzd 9401	. . . . . . 7 |- ( ph -> 1 e. ZZ )
31	29, 30	zsubcld 9502	. . . . . 6 |- ( ph -> ( N - 1 ) e. ZZ )
32	28, 31	rpexpcld 10844	. . . . 5 |- ( ph -> ( 4 ^ ( N - 1 ) ) e. RR+ )
33	25, 32	rpdivcld 9838	. . . 4 |- ( ph -> ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) e. RR+ )
34	33	rpred 9820	. . 3 |- ( ph -> ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) e. RR )
35	1, 2, 3	resqrexlemover 11354	. . . . . 6 |- ( ( ph /\ M e. NN ) -> A < ( ( F ` M ) ^ 2 ) )
36	10, 35	mpdan 421	. . . . 5 |- ( ph -> A < ( ( F ` M ) ^ 2 ) )
37		difrp 9816	. . . . . 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 9853	. . 3 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) < ( ( ( F ` N ) ^ 2 ) - A ) )
41	1, 2, 3	resqrexlemcalc3 11360	. . . 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 8498	. 2 |- ( ph -> ( ( ( ( F ` N ) ^ 2 ) - A ) - ( ( ( F ` M ) ^ 2 ) - A ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )
44	16, 43	eqbrtrrd 4069	1 |- ( ph -> ( ( ( F ` N ) ^ 2 ) - ( ( F ` M ) ^ 2 ) ) < ( ( ( F ` 1 ) ^ 2 ) / ( 4 ^ ( N - 1 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2176   {csn 3633   class class class wbr 4045   X. cxp 4674   ` cfv 5272  (class class class)co 5946   e. cmpo 5948   RRcr 7926   0cc0 7927   1c1 7928   + caddc 7930   < clt 8109   <_ cle 8110   - cmin 8245   / cdiv 8747   NNcn 9038   2c2 9089   4c4 9091   ZZcz 9374   RR+crp 9777   seqcseq 10594   ^cexp 10685

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-rp 9778  df-seqfrec 10595  df-exp 10686

This theorem is referenced by:  resqrexlemnm  11362