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Theorem resqrexlemoverl 9619

Description: Lemma for resqrex 9624. Every term in the sequence is an overestimate compared with the limit

. Although this theorem is stated in terms of a particular sequence the proof could be adapted for any decreasing convergent sequence. (Contributed by Jim Kingdon, 9-Aug-2021.)

**Hypotheses**
Ref	Expression
resqrexlemex.seq
resqrexlemex.a
resqrexlemex.agt0
resqrexlemgt0.rr
resqrexlemgt0.lim	$/\$
resqrexlemoverl.k

**Assertion**
Ref	Expression
resqrexlemoverl

Distinct variable groups:

Allowed substitution hints:

(

)

(

)

Proof of Theorem resqrexlemoverl Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resqrexlemex.seq	. . . . . . . . . . 11
2		resqrexlemex.a	. . . . . . . . . . 11
3		resqrexlemex.agt0	. . . . . . . . . . 11
4	1, 2, 3	resqrexlemf 9605	. . . . . . . . . 10
5		resqrexlemoverl.k	. . . . . . . . . 10
6	4, 5	ffvelrnd 5303	. . . . . . . . 9
7	6	rpred 8622	. . . . . . . 8
8		resqrexlemgt0.rr	. . . . . . . 8
9		difrp 8619	. . . . . . . 8 $/\$
10	7, 8, 9	syl2anc 391	. . . . . . 7
11	10	biimpa 280	. . . . . 6 $/\$
12		resqrexlemgt0.lim	. . . . . . 7 $/\$
13	12	adantr 261	. . . . . 6 $/\$ $/\$
14		oveq2 5520	. . . . . . . . . 10
15	14	breq2d 3776	. . . . . . . . 9
16		oveq2 5520	. . . . . . . . . 10
17	16	breq2d 3776	. . . . . . . . 9
18	15, 17	anbi12d 442	. . . . . . . 8 $/\$ $/\$
19	18	rexralbidv 2350	. . . . . . 7 $/\$ $/\$
20	19	rspcv 2652	. . . . . 6 $/\$ $/\$
21	11, 13, 20	sylc 56	. . . . 5 $/\$ $/\$
22		fveq2 5178	. . . . . . 7
23	22	raleqdv 2511	. . . . . 6 $/\$ $/\$
24	23	cbvrexv 2534	. . . . 5 $/\$ $/\$
25	21, 24	sylib 127	. . . 4 $/\$ $/\$
26		simprl 483	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
27	26	nnzd 8359	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
28	27	adantr 261	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
29	5	ad2antrr 457	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
30	29	nnzd 8359	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
31	30	adantr 261	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
32		simpr 103	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
33		eluz2 8479	. . . . . . . . . 10 $/\$ $/\$
34	28, 31, 32, 33	syl3anbrc 1088	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
35		simprr 484	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
36	35	adantr 261	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		fveq2 5178	. . . . . . . . . . . 12
38	37	breq1d 3774	. . . . . . . . . . 11
39	37	oveq1d 5527	. . . . . . . . . . . 12
40	39	breq2d 3776	. . . . . . . . . . 11
41	38, 40	anbi12d 442	. . . . . . . . . 10 $/\$ $/\$
42	41	rspcv 2652	. . . . . . . . 9 $/\$ $/\$
43	34, 36, 42	sylc 56	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	43	simprd 107	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
45	6	ad2antrr 457	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
46	45	rpcnd 8624	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
47	46	adantr 261	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
48	8	ad2antrr 457	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
49	48	recnd 7054	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
50	49	adantr 261	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
51	47, 50	pncan3d 7325	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
52	44, 51	breqtrd 3788	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
53	8	ad3antrrr 461	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
54	53	ltnrd 7129	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
55	52, 54	pm2.21fal 1264	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
56	2	ad3antrrr 461	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
57	3	ad3antrrr 461	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
58	5	ad3antrrr 461	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
59	26	adantr 261	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
60		simpr 103	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
61	1, 56, 57, 58, 59, 60	resqrexlemdecn 9610	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
62	7	ad3antrrr 461	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
63	4	ad2antrr 457	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
64	63, 26	ffvelrnd 5303	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
65	64	rpred 8622	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
66	65	adantr 261	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
67		uzid 8487	. . . . . . . . . . . . . 14
68	27, 67	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
69		fveq2 5178	. . . . . . . . . . . . . . . 16
70	69	breq1d 3774	. . . . . . . . . . . . . . 15
71	69	oveq1d 5527	. . . . . . . . . . . . . . . 16
72	71	breq2d 3776	. . . . . . . . . . . . . . 15
73	70, 72	anbi12d 442	. . . . . . . . . . . . . 14 $/\$ $/\$
74	73	rspcv 2652	. . . . . . . . . . . . 13 $/\$ $/\$
75	68, 35, 74	sylc 56	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
76	75	simprd 107	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
77	64	rpcnd 8624	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
78	77, 49, 46	addsubassd 7342	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
79	76, 78	breqtrrd 3790	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
80	7	ad2antrr 457	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
81	65, 48	readdcld 7055	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
82	80, 48, 81	ltaddsub2d 7537	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
83	79, 82	mpbird 156	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
84	80, 65, 48	ltadd1d 7529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
85	83, 84	mpbird 156	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
86	85	adantr 261	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
87	62, 66, 86	ltnsymd 7136	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
88	61, 87	pm2.21fal 1264	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
89		zlelttric 8290	. . . . . 6 $/\$ $\/$
90	27, 30, 89	syl2anc 391	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\/$
91	55, 88, 90	mpjaodan 711	. . . 4 $/\$ $/\$ $/\$ $/\$
92	25, 91	rexlimddv 2437	. . 3 $/\$
93	92	inegd 1263	. 2
94	8, 7	lenltd 7134	. 2
95	93, 94	mpbird 156	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 97

wb 98 $\/$ wo 629

wceq 1243

wfal 1248

wcel 1393

wral 2306

wrex 2307

csn 3375 class class class wbr 3764

cxp 4343

wf 4898

cfv 4902 (class class class)co 5512

cmpt2 5514

cc 6887

cr 6888

cc0 6889

c1 6890

caddc 6892

clt 7060

cle 7061

cmin 7182

cdiv 7651

cn 7914

c2 7964

cz 8245

cuz 8473

crp 8583

cseq 9211

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-2 7973 df-3 7974 df-4 7975 df-n0 8182 df-z 8246 df-uz 8474 df-rp 8584 df-iseq 9212 df-iexp 9255

This theorem is referenced by: resqrexlemglsq 9620

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