Theorem absef01tlubi 8650

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 disk. (Contributed by Paul Chapman, 19-Jan-2008.)

Hypothesis

Ref	Expression
ef1tllem.1	|- F = {<.j, y>. | (j e. NN0 /\ y = ((A^j) / (!` j)))}

Assertion

Ref	Expression
absef01tlubi	|- ((A e. CC /\ (abs` A) e. (0(,]1) /\ M e. NN) -> (abs` sum_k e. (ZZ>=` M)(F` k)) <_ (((abs` A)^M) x. ((M + 1) / ((!` M) x. M))))

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

Proof of Theorem absef01tlubi

Step	Hyp	Ref	Expression
1		opreq1 4889	. . . . . . . . . . . 12 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> (A^j) = (if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j))
2	1	opreq1d 4897	. . . . . . . . . . 11 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> ((A^j) / (!` j)) = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))
3	2	eqeq2d 1895	. . . . . . . . . 10 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> (y = ((A^j) / (!` j)) <-> y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j))))
4	3	anbi2d 678	. . . . . . . . 9 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> ((j e. NN0 /\ y = ((A^j) / (!` j))) <-> (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))))
5	4	opabbidv 3401	. . . . . . . 8 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> {<.j, y>. | (j e. NN0 /\ y = ((A^j) / (!` j)))} = {<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))})
6		ef1tllem.1	. . . . . . . 8 |- F = {<.j, y>. | (j e. NN0 /\ y = ((A^j) / (!` j)))}
7	5, 6	syl5eq 1940	. . . . . . 7 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> F = {<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))})
8	7	fveq1d 4683	. . . . . 6 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> (F` k) = ({<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}` k))
9	8	sumeq2sdv 8253	. . . . 5 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> sum_k e. (ZZ>=` M)(F` k) = sum_k e. (ZZ>=` M)({<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}` k))
10	9	fveq2d 4685	. . . 4 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> (abs` sum_k e. (ZZ>=` M)(F` k)) = (abs` sum_k e. (ZZ>=` M)({<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}` k)))
11		fveq2 4681	. . . . . 6 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> (abs` A) = (abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)))
12	11	opreq1d 4897	. . . . 5 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> ((abs` A)^M) = ((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^M))
13	12	opreq1d 4897	. . . 4 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> (((abs` A)^M) x. ((M + 1) / ((!` M) x. M))) = (((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^M) x. ((M + 1) / ((!` M) x. M))))
14	10, 13	breq12d 3351	. . 3 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> ((abs` sum_k e. (ZZ>=` M)(F` k)) <_ (((abs` A)^M) x. ((M + 1) / ((!` M) x. M))) <-> (abs` sum_k e. (ZZ>=` M)({<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}` k)) <_ (((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^M) x. ((M + 1) / ((!` M) x. M)))))
15		fveq2 4681	. . . . . 6 |- (M = if(M e. NN, M, 1) -> (ZZ>=` M) = (ZZ>=` if(M e. NN, M, 1)))
16	15	sumeq1d 8250	. . . . 5 |- (M = if(M e. NN, M, 1) -> sum_k e. (ZZ>=` M)({<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}` k) = sum_k e. (ZZ>=` if(M e. NN, M, 1))({<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}` k))
17	16	fveq2d 4685	. . . 4 |- (M = if(M e. NN, M, 1) -> (abs` sum_k e. (ZZ>=` M)({<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}` k)) = (abs` sum_k e. (ZZ>=` if(M e. NN, M, 1))({<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}` k)))
18		opreq2 4890	. . . . 5 |- (M = if(M e. NN, M, 1) -> ((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^M) = ((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^if(M e. NN, M, 1)))
19		opreq1 4889	. . . . . 6 |- (M = if(M e. NN, M, 1) -> (M + 1) = (if(M e. NN, M, 1) + 1))
20		fveq2 4681	. . . . . . 7 |- (M = if(M e. NN, M, 1) -> (!` M) = (!` if(M e. NN, M, 1)))
21		id 73	. . . . . . 7 |- (M = if(M e. NN, M, 1) -> M = if(M e. NN, M, 1))
22	20, 21	opreq12d 4900	. . . . . 6 |- (M = if(M e. NN, M, 1) -> ((!` M) x. M) = ((!` if(M e. NN, M, 1)) x. if(M e. NN, M, 1)))
23	19, 22	opreq12d 4900	. . . . 5 |- (M = if(M e. NN, M, 1) -> ((M + 1) / ((!` M) x. M)) = ((if(M e. NN, M, 1) + 1) / ((!` if(M e. NN, M, 1)) x. if(M e. NN, M, 1))))
24	18, 23	opreq12d 4900	. . . 4 |- (M = if(M e. NN, M, 1) -> (((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^M) x. ((M + 1) / ((!` M) x. M))) = (((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^if(M e. NN, M, 1)) x. ((if(M e. NN, M, 1) + 1) / ((!` if(M e. NN, M, 1)) x. if(M e. NN, M, 1)))))
25	17, 24	breq12d 3351	. . 3 |- (M = if(M e. NN, M, 1) -> ((abs` sum_k e. (ZZ>=` M)({<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}` k)) <_ (((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^M) x. ((M + 1) / ((!` M) x. M))) <-> (abs` sum_k e. (ZZ>=` if(M e. NN, M, 1))({<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}` k)) <_ (((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^if(M e. NN, M, 1)) x. ((if(M e. NN, M, 1) + 1) / ((!` if(M e. NN, M, 1)) x. if(M e. NN, M, 1))))))
26		eqid 1884	. . . 4 |- {<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))} = {<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}
27		eqid 1884	. . . 4 |- {<.j, y>. | (j e. NN0 /\ y = (((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^j) / (!` j)))} = {<.j, y>. | (j e. NN0 /\ y = (((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^j) / (!` j)))}
28		eleq1 1957	. . . . . . 7 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> (A e. CC <-> if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) e. CC))
29	11	eleq1d 1963	. . . . . . 7 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> ((abs` A) e. (0(,]1) <-> (abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)) e. (0(,]1)))
30	28, 29	anbi12d 690	. . . . . 6 |- (A = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> ((A e. CC /\ (abs` A) e. (0(,]1)) <-> (if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) e. CC /\ (abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)) e. (0(,]1))))
31		eleq1 1957	. . . . . . 7 |- (1 = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> (1 e. CC <-> if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) e. CC))
32		fveq2 4681	. . . . . . . 8 |- (1 = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> (abs` 1) = (abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)))
33	32	eleq1d 1963	. . . . . . 7 |- (1 = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> ((abs` 1) e. (0(,]1) <-> (abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)) e. (0(,]1)))
34	31, 33	anbi12d 690	. . . . . 6 |- (1 = if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) -> ((1 e. CC /\ (abs` 1) e. (0(,]1)) <-> (if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) e. CC /\ (abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)) e. (0(,]1))))
35		ax1cn 6422	. . . . . . 7 |- 1 e. CC
36		0re 6603	. . . . . . . . . 10 |- 0 e. RR
37		1re 6598	. . . . . . . . . 10 |- 1 e. RR
38		lt01 6871	. . . . . . . . . 10 |- 0 < 1
39	36, 37, 38	ltleii 6756	. . . . . . . . 9 |- 0 <_ 1
40	37	absidi 8112	. . . . . . . . 9 |- (0 <_ 1 -> (abs` 1) = 1)
41	39, 40	ax-mp 7	. . . . . . . 8 |- (abs` 1) = 1
42	37	leidi 6790	. . . . . . . . 9 |- 1 <_ 1
43		elioc2 7558	. . . . . . . . . . 11 |- ((0 e. RR /\ 1 e. RR) -> (1 e. (0(,]1) <-> (1 e. RR /\ 0 < 1 /\ 1 <_ 1)))
44	36, 37, 43	mp2an 761	. . . . . . . . . 10 |- (1 e. (0(,]1) <-> (1 e. RR /\ 0 < 1 /\ 1 <_ 1))
45	44	biimpri 169	. . . . . . . . 9 |- ((1 e. RR /\ 0 < 1 /\ 1 <_ 1) -> 1 e. (0(,]1))
46	37, 38, 42, 45	mp3an 1191	. . . . . . . 8 |- 1 e. (0(,]1)
47	41, 46	eqeltri 1967	. . . . . . 7 |- (abs` 1) e. (0(,]1)
48	35, 47	pm3.2i 307	. . . . . 6 |- (1 e. CC /\ (abs` 1) e. (0(,]1))
49	30, 34, 48	elimhyp 3021	. . . . 5 |- (if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) e. CC /\ (abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)) e. (0(,]1))
50	49	simpli 347	. . . 4 |- if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1) e. CC
51	49	simpri 351	. . . 4 |- (abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)) e. (0(,]1)
52		1nn 7117	. . . . 5 |- 1 e. NN
53	52	elimel 3025	. . . 4 |- if(M e. NN, M, 1) e. NN
54	26, 27, 50, 51, 53	absef01tllem 8649	. . 3 |- (abs` sum_k e. (ZZ>=` if(M e. NN, M, 1))({<.j, y>. | (j e. NN0 /\ y = ((if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1)^j) / (!` j)))}` k)) <_ (((abs` if((A e. CC /\ (abs` A) e. (0(,]1)), A, 1))^if(M e. NN, M, 1)) x. ((if(M e. NN, M, 1) + 1) / ((!` if(M e. NN, M, 1)) x. if(M e. NN, M, 1))))
55	14, 25, 54	dedth2h 3015	. 2 |- (((A e. CC /\ (abs` A) e. (0(,]1)) /\ M e. NN) -> (abs` sum_k e. (ZZ>=` M)(F` k)) <_ (((abs` A)^M) x. ((M + 1) / ((!` M) x. M))))
56	55	3impa 1062	1 |- ((A e. CC /\ (abs` A) e. (0(,]1) /\ M e. NN) -> (abs` sum_k e. (ZZ>=` M)(F` k)) <_ (((abs` A)^M) x. ((M + 1) / ((!` M) x. M))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ifcif 2982   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   / cdiv 6447   <_ cle 6448  NNcn 6449  NN0cn0 6450   < clt 6653  (,]cioc 7525  ZZ>=cuz 7586  ^cexp 7811  abscabs 8000  !cfa 8183  sum_csu 8239

This theorem is referenced by:  abspef01tlubi 8660

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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