Theorem abspef01tlubi 8660

Description: An upper bound on the absolute value of the infinite tail of the series expansion of the exponential function on the punctured closed unit disc projected onto the real or imaginary axis. (Contributed by Paul Chapman, 19-Jan-2008.)

Hypotheses

Ref	Expression
abspef01tlub.1	|- F = {<.j, y>. | (j e. NN0 /\ y = (((_i x. A)^j) / (!` j)))}
abspef01tlub.2	|- (P = Re \/ P = Im)

Assertion

Ref	Expression
abspef01tlubi	|- ((A e. (0(,]1) /\ M e. NN) -> (abs` (P` sum_k e. (ZZ>=` M)(F` k))) <_ ((A^M) x. ((M + 1) / ((!` M) x. M))))

Distinct variable groups:   A,j,k,y   k,F   j,M,k,y

Proof of Theorem abspef01tlubi

Step	Hyp	Ref	Expression
1		abspef01tlub.2	. . . . 5 |- (P = Re \/ P = Im)
2		fveq1 4680	. . . . . . . . 9 |- (P = Re -> (P` sum_k e. (ZZ>=` M)(F` k)) = (Re` sum_k e. (ZZ>=` M)(F` k)))
3	2	adantr 425	. . . . . . . 8 |- ((P = Re /\ (A e. (0(,]1) /\ M e. NN)) -> (P` sum_k e. (ZZ>=` M)(F` k)) = (Re` sum_k e. (ZZ>=` M)(F` k)))
4		abspef01tlub.1	. . . . . . . . . . . 12 |- F = {<.j, y>. | (j e. NN0 /\ y = (((_i x. A)^j) / (!` j)))}
5	4	eftlcl 8641	. . . . . . . . . . 11 |- (((_i x. A) e. CC /\ M e. NN) -> sum_k e. (ZZ>=` M)(F` k) e. CC)
6		mulcl 6456	. . . . . . . . . . . 12 |- ((_i e. CC /\ A e. CC) -> (_i x. A) e. CC)
7		axicn 6423	. . . . . . . . . . . 12 |- _i e. CC
8		0re 6603	. . . . . . . . . . . . . . 15 |- 0 e. RR
9		1re 6598	. . . . . . . . . . . . . . 15 |- 1 e. RR
10		elioc2 7558	. . . . . . . . . . . . . . 15 |- ((0 e. RR /\ 1 e. RR) -> (A e. (0(,]1) <-> (A e. RR /\ 0 < A /\ A <_ 1)))
11	8, 9, 10	mp2an 761	. . . . . . . . . . . . . 14 |- (A e. (0(,]1) <-> (A e. RR /\ 0 < A /\ A <_ 1))
12	11	simp1bi 891	. . . . . . . . . . . . 13 |- (A e. (0(,]1) -> A e. RR)
13	12	recnd 6468	. . . . . . . . . . . 12 |- (A e. (0(,]1) -> A e. CC)
14	6, 7, 13	sylancr 526	. . . . . . . . . . 11 |- (A e. (0(,]1) -> (_i x. A) e. CC)
15	5, 14	sylan 497	. . . . . . . . . 10 |- ((A e. (0(,]1) /\ M e. NN) -> sum_k e. (ZZ>=` M)(F` k) e. CC)
16		recl 8007	. . . . . . . . . 10 |- (sum_k e. (ZZ>=` M)(F` k) e. CC -> (Re` sum_k e. (ZZ>=` M)(F` k)) e. RR)
17	15, 16	syl 12	. . . . . . . . 9 |- ((A e. (0(,]1) /\ M e. NN) -> (Re` sum_k e. (ZZ>=` M)(F` k)) e. RR)
18	17	adantl 424	. . . . . . . 8 |- ((P = Re /\ (A e. (0(,]1) /\ M e. NN)) -> (Re` sum_k e. (ZZ>=` M)(F` k)) e. RR)
19	3, 18	eqeltrd 1971	. . . . . . 7 |- ((P = Re /\ (A e. (0(,]1) /\ M e. NN)) -> (P` sum_k e. (ZZ>=` M)(F` k)) e. RR)
20	19	ex 402	. . . . . 6 |- (P = Re -> ((A e. (0(,]1) /\ M e. NN) -> (P` sum_k e. (ZZ>=` M)(F` k)) e. RR))
21		fveq1 4680	. . . . . . . . 9 |- (P = Im -> (P` sum_k e. (ZZ>=` M)(F` k)) = (Im` sum_k e. (ZZ>=` M)(F` k)))
22	21	adantr 425	. . . . . . . 8 |- ((P = Im /\ (A e. (0(,]1) /\ M e. NN)) -> (P` sum_k e. (ZZ>=` M)(F` k)) = (Im` sum_k e. (ZZ>=` M)(F` k)))
23		imcl 8008	. . . . . . . . . 10 |- (sum_k e. (ZZ>=` M)(F` k) e. CC -> (Im` sum_k e. (ZZ>=` M)(F` k)) e. RR)
24	15, 23	syl 12	. . . . . . . . 9 |- ((A e. (0(,]1) /\ M e. NN) -> (Im` sum_k e. (ZZ>=` M)(F` k)) e. RR)
25	24	adantl 424	. . . . . . . 8 |- ((P = Im /\ (A e. (0(,]1) /\ M e. NN)) -> (Im` sum_k e. (ZZ>=` M)(F` k)) e. RR)
26	22, 25	eqeltrd 1971	. . . . . . 7 |- ((P = Im /\ (A e. (0(,]1) /\ M e. NN)) -> (P` sum_k e. (ZZ>=` M)(F` k)) e. RR)
27	26	ex 402	. . . . . 6 |- (P = Im -> ((A e. (0(,]1) /\ M e. NN) -> (P` sum_k e. (ZZ>=` M)(F` k)) e. RR))
28	20, 27	jaoi 368	. . . . 5 |- ((P = Re \/ P = Im) -> ((A e. (0(,]1) /\ M e. NN) -> (P` sum_k e. (ZZ>=` M)(F` k)) e. RR))
29	1, 28	ax-mp 7	. . . 4 |- ((A e. (0(,]1) /\ M e. NN) -> (P` sum_k e. (ZZ>=` M)(F` k)) e. RR)
30	29	recnd 6468	. . 3 |- ((A e. (0(,]1) /\ M e. NN) -> (P` sum_k e. (ZZ>=` M)(F` k)) e. CC)
31		abscl 8084	. . 3 |- ((P` sum_k e. (ZZ>=` M)(F` k)) e. CC -> (abs` (P` sum_k e. (ZZ>=` M)(F` k))) e. RR)
32	30, 31	syl 12	. 2 |- ((A e. (0(,]1) /\ M e. NN) -> (abs` (P` sum_k e. (ZZ>=` M)(F` k))) e. RR)
33		abscl 8084	. . 3 |- (sum_k e. (ZZ>=` M)(F` k) e. CC -> (abs` sum_k e. (ZZ>=` M)(F` k)) e. RR)
34	15, 33	syl 12	. 2 |- ((A e. (0(,]1) /\ M e. NN) -> (abs` sum_k e. (ZZ>=` M)(F` k)) e. RR)
35		reexpcl 7823	. . . 4 |- ((A e. RR /\ M e. NN0) -> (A^M) e. RR)
36		nnnn0 7315	. . . 4 |- (M e. NN -> M e. NN0)
37	35, 12, 36	syl2an 503	. . 3 |- ((A e. (0(,]1) /\ M e. NN) -> (A^M) e. RR)
38		eftlubcl 8638	. . . 4 |- (M e. NN -> ((M + 1) / ((!` M) x. M)) e. RR)
39	38	adantl 424	. . 3 |- ((A e. (0(,]1) /\ M e. NN) -> ((M + 1) / ((!` M) x. M)) e. RR)
40		remulcl 6457	. . 3 |- (((A^M) e. RR /\ ((M + 1) / ((!` M) x. M)) e. RR) -> ((A^M) x. ((M + 1) / ((!` M) x. M))) e. RR)
41	37, 39, 40	syl11anc 524	. 2 |- ((A e. (0(,]1) /\ M e. NN) -> ((A^M) x. ((M + 1) / ((!` M) x. M))) e. RR)
42	2	fveq2d 4685	. . . . . 6 |- (P = Re -> (abs` (P` sum_k e. (ZZ>=` M)(F` k))) = (abs` (Re` sum_k e. (ZZ>=` M)(F` k))))
43	42	breq1d 3348	. . . . 5 |- (P = Re -> ((abs` (P` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k)) <-> (abs` (Re` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k))))
44		absrele 8121	. . . . . 6 |- (sum_k e. (ZZ>=` M)(F` k) e. CC -> (abs` (Re` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k)))
45	15, 44	syl 12	. . . . 5 |- ((A e. (0(,]1) /\ M e. NN) -> (abs` (Re` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k)))
46	43, 45	syl5bir 227	. . . 4 |- (P = Re -> ((A e. (0(,]1) /\ M e. NN) -> (abs` (P` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k))))
47	21	fveq2d 4685	. . . . . 6 |- (P = Im -> (abs` (P` sum_k e. (ZZ>=` M)(F` k))) = (abs` (Im` sum_k e. (ZZ>=` M)(F` k))))
48	47	breq1d 3348	. . . . 5 |- (P = Im -> ((abs` (P` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k)) <-> (abs` (Im` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k))))
49		absimle 8122	. . . . . 6 |- (sum_k e. (ZZ>=` M)(F` k) e. CC -> (abs` (Im` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k)))
50	15, 49	syl 12	. . . . 5 |- ((A e. (0(,]1) /\ M e. NN) -> (abs` (Im` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k)))
51	48, 50	syl5bir 227	. . . 4 |- (P = Im -> ((A e. (0(,]1) /\ M e. NN) -> (abs` (P` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k))))
52	46, 51	jaoi 368	. . 3 |- ((P = Re \/ P = Im) -> ((A e. (0(,]1) /\ M e. NN) -> (abs` (P` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k))))
53	1, 52	ax-mp 7	. 2 |- ((A e. (0(,]1) /\ M e. NN) -> (abs` (P` sum_k e. (ZZ>=` M)(F` k))) <_ (abs` sum_k e. (ZZ>=` M)(F` k)))
54	4	absef01tlubi 8650	. . . . 5 |- (((_i x. A) e. CC /\ (abs` (_i x. A)) e. (0(,]1) /\ M e. NN) -> (abs` sum_k e. (ZZ>=` M)(F` k)) <_ (((abs` (_i x. A))^M) x. ((M + 1) / ((!` M) x. M))))
55	54	3expa 1067	. . . 4 |- ((((_i x. A) e. CC /\ (abs` (_i x. A)) e. (0(,]1)) /\ M e. NN) -> (abs` sum_k e. (ZZ>=` M)(F` k)) <_ (((abs` (_i x. A))^M) x. ((M + 1) / ((!` M) x. M))))
56		absmul 8109	. . . . . . . . 9 |- ((_i e. CC /\ A e. CC) -> (abs` (_i x. A)) = ((abs` _i) x. (abs` A)))
57	56, 7, 13	sylancr 526	. . . . . . . 8 |- (A e. (0(,]1) -> (abs` (_i x. A)) = ((abs` _i) x. (abs` A)))
58		absi 8130	. . . . . . . . . 10 |- (abs` _i) = 1
59	58	opreq1i 4892	. . . . . . . . 9 |- ((abs` _i) x. (abs` A)) = (1 x. (abs` A))
60	59	a1i 8	. . . . . . . 8 |- (A e. (0(,]1) -> ((abs` _i) x. (abs` A)) = (1 x. (abs` A)))
61	11	simp2bi 892	. . . . . . . . . . . 12 |- (A e. (0(,]1) -> 0 < A)
62		ltle 6690	. . . . . . . . . . . . 13 |- ((0 e. RR /\ A e. RR) -> (0 < A -> 0 <_ A))
63	8, 62	mpan 759	. . . . . . . . . . . 12 |- (A e. RR -> (0 < A -> 0 <_ A))
64	12, 61, 63	sylc 83	. . . . . . . . . . 11 |- (A e. (0(,]1) -> 0 <_ A)
65		absid 8113	. . . . . . . . . . 11 |- ((A e. RR /\ 0 <_ A) -> (abs` A) = A)
66	12, 64, 65	syl11anc 524	. . . . . . . . . 10 |- (A e. (0(,]1) -> (abs` A) = A)
67	66, 13	eqeltrd 1971	. . . . . . . . 9 |- (A e. (0(,]1) -> (abs` A) e. CC)
68		mulid2 6578	. . . . . . . . 9 |- ((abs` A) e. CC -> (1 x. (abs` A)) = (abs` A))
69	67, 68	syl 12	. . . . . . . 8 |- (A e. (0(,]1) -> (1 x. (abs` A)) = (abs` A))
70	57, 60, 69	3eqtrd 1929	. . . . . . 7 |- (A e. (0(,]1) -> (abs` (_i x. A)) = (abs` A))
71	70, 66	eqtrd 1925	. . . . . 6 |- (A e. (0(,]1) -> (abs` (_i x. A)) = A)
72		id 73	. . . . . 6 |- (A e. (0(,]1) -> A e. (0(,]1))
73	71, 72	eqeltrd 1971	. . . . 5 |- (A e. (0(,]1) -> (abs` (_i x. A)) e. (0(,]1))
74	14, 73	jca 310	. . . 4 |- (A e. (0(,]1) -> ((_i x. A) e. CC /\ (abs` (_i x. A)) e. (0(,]1)))
75	55, 74	sylan 497	. . 3 |- ((A e. (0(,]1) /\ M e. NN) -> (abs` sum_k e. (ZZ>=` M)(F` k)) <_ (((abs` (_i x. A))^M) x. ((M + 1) / ((!` M) x. M))))
76	71	opreq1d 4897	. . . . . 6 |- (A e. (0(,]1) -> ((abs` (_i x. A))^M) = (A^M))
77	76	opreq1d 4897	. . . . 5 |- (A e. (0(,]1) -> (((abs` (_i x. A))^M) x. ((M + 1) / ((!` M) x. M))) = ((A^M) x. ((M + 1) / ((!` M) x. M))))
78	77	breq2d 3350	. . . 4 |- (A e. (0(,]1) -> ((abs` sum_k e. (ZZ>=` M)(F` k)) <_ (((abs` (_i x. A))^M) x. ((M + 1) / ((!` M) x. M))) <-> (abs` sum_k e. (ZZ>=` M)(F` k)) <_ ((A^M) x. ((M + 1) / ((!` M) x. M)))))
79	78	adantr 425	. . 3 |- ((A e. (0(,]1) /\ M e. NN) -> ((abs` sum_k e. (ZZ>=` M)(F` k)) <_ (((abs` (_i x. A))^M) x. ((M + 1) / ((!` M) x. M))) <-> (abs` sum_k e. (ZZ>=` M)(F` k)) <_ ((A^M) x. ((M + 1) / ((!` M) x. M)))))
80	75, 79	mpbid 212	. 2 |- ((A e. (0(,]1) /\ M e. NN) -> (abs` sum_k e. (ZZ>=` M)(F` k)) <_ ((A^M) x. ((M + 1) / ((!` M) x. M))))
81	32, 34, 41, 53, 80	letrd 6696	1 |- ((A e. (0(,]1) /\ M e. NN) -> (abs` (P` sum_k e. (ZZ>=` M)(F` k))) <_ ((A^M) x. ((M + 1) / ((!` M) x. M))))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   / cdiv 6447   <_ cle 6448  NNcn 6449  NN0cn0 6450   < clt 6653  (,]cioc 7525  ZZ>=cuz 7586  ^cexp 7811  Recre 7997  Imcim 7998  abscabs 8000  !cfa 8183  sum_csu 8239

This theorem is referenced by:  sin01bndlem2 8734  cos01bndlem2 8736

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-fl 7463  df-ioc 7529  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-fac 8184  df-clim 8235  df-sum 8240  df-ef 8560

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