Theorem efaddlem27 8626

Description: Lemma for efaddi 8628. Show that the convergence of the sequence of partial sum products H to exp` (A + B). The key theorem used is 2climnn 8362.

Hypotheses

Ref	Expression
efaddlem27.1	|- A e. CC
efaddlem27.2	|- B e. CC
efaddlem27.3	|- F = {<.n, y>. | (n e. NN0 /\ y = sum_j e. (0...n)(((A + B)^j) / (!` j)))}
efaddlem27.4	|- H = {<.n, y>. | (n e. NN0 /\ y = (sum_j e. (0...n)((A^j) / (!` j)) x. sum_k e. (0...n)((B^k) / (!` k))))}

Assertion

Ref	Expression
efaddlem27	|- H ~~> (exp` (A + B))

Distinct variable groups:   j,k,n,y,A   B,j,k,n,y

Proof of Theorem efaddlem27

Step	Hyp	Ref	Expression
1		efaddlem27.1	. . . . 5 |- A e. CC
2		efaddlem27.2	. . . . 5 |- B e. CC
3	1, 2	addcli 6473	. . . 4 |- (A + B) e. CC
4		efcl 8574	. . . 4 |- ((A + B) e. CC -> (exp` (A + B)) e. CC)
5	3, 4	ax-mp 7	. . 3 |- (exp` (A + B)) e. CC
6		nnnn0 7315	. . . . 5 |- (m e. NN -> m e. NN0)
7		opreq2 4890	. . . . . . . . 9 |- (n = m -> (0...n) = (0...m))
8	7	sumeq1d 8250	. . . . . . . 8 |- (n = m -> sum_j e. (0...n)(((A + B)^j) / (!` j)) = sum_j e. (0...m)(((A + B)^j) / (!` j)))
9		efaddlem27.3	. . . . . . . 8 |- F = {<.n, y>. | (n e. NN0 /\ y = sum_j e. (0...n)(((A + B)^j) / (!` j)))}
10		sumex 8241	. . . . . . . 8 |- sum_j e. (0...m)(((A + B)^j) / (!` j)) e. _V
11	8, 9, 10	fvopab4 4743	. . . . . . 7 |- (m e. NN0 -> (F` m) = sum_j e. (0...m)(((A + B)^j) / (!` j)))
12		eftcl 8565	. . . . . . . . . 10 |- (((A + B) e. CC /\ j e. NN0) -> (((A + B)^j) / (!` j)) e. CC)
13		elfznn0 7668	. . . . . . . . . 10 |- (j e. (0...m) -> j e. NN0)
14	12, 3, 13	sylancr 526	. . . . . . . . 9 |- (j e. (0...m) -> (((A + B)^j) / (!` j)) e. CC)
15	14	rgen 2159	. . . . . . . 8 |- A.j e. (0...m)(((A + B)^j) / (!` j)) e. CC
16		fsum0cl 8276	. . . . . . . 8 |- ((m e. NN0 /\ A.j e. (0...m)(((A + B)^j) / (!` j)) e. CC) -> sum_j e. (0...m)(((A + B)^j) / (!` j)) e. CC)
17	15, 16	mpan2 760	. . . . . . 7 |- (m e. NN0 -> sum_j e. (0...m)(((A + B)^j) / (!` j)) e. CC)
18	11, 17	eqeltrd 1971	. . . . . 6 |- (m e. NN0 -> (F` m) e. CC)
19	7	sumeq1d 8250	. . . . . . . . 9 |- (n = m -> sum_j e. (0...n)((A^j) / (!` j)) = sum_j e. (0...m)((A^j) / (!` j)))
20	7	sumeq1d 8250	. . . . . . . . 9 |- (n = m -> sum_k e. (0...n)((B^k) / (!` k)) = sum_k e. (0...m)((B^k) / (!` k)))
21	19, 20	opreq12d 4900	. . . . . . . 8 |- (n = m -> (sum_j e. (0...n)((A^j) / (!` j)) x. sum_k e. (0...n)((B^k) / (!` k))) = (sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))))
22		efaddlem27.4	. . . . . . . 8 |- H = {<.n, y>. | (n e. NN0 /\ y = (sum_j e. (0...n)((A^j) / (!` j)) x. sum_k e. (0...n)((B^k) / (!` k))))}
23		oprex 4907	. . . . . . . 8 |- (sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) e. _V
24	21, 22, 23	fvopab4 4743	. . . . . . 7 |- (m e. NN0 -> (H` m) = (sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))))
25		eftcl 8565	. . . . . . . . . . 11 |- ((A e. CC /\ j e. NN0) -> ((A^j) / (!` j)) e. CC)
26	25, 1, 13	sylancr 526	. . . . . . . . . 10 |- (j e. (0...m) -> ((A^j) / (!` j)) e. CC)
27	26	rgen 2159	. . . . . . . . 9 |- A.j e. (0...m)((A^j) / (!` j)) e. CC
28		fsum0cl 8276	. . . . . . . . 9 |- ((m e. NN0 /\ A.j e. (0...m)((A^j) / (!` j)) e. CC) -> sum_j e. (0...m)((A^j) / (!` j)) e. CC)
29	27, 28	mpan2 760	. . . . . . . 8 |- (m e. NN0 -> sum_j e. (0...m)((A^j) / (!` j)) e. CC)
30		eftcl 8565	. . . . . . . . . . 11 |- ((B e. CC /\ k e. NN0) -> ((B^k) / (!` k)) e. CC)
31		elfznn0 7668	. . . . . . . . . . 11 |- (k e. (0...m) -> k e. NN0)
32	30, 2, 31	sylancr 526	. . . . . . . . . 10 |- (k e. (0...m) -> ((B^k) / (!` k)) e. CC)
33	32	rgen 2159	. . . . . . . . 9 |- A.k e. (0...m)((B^k) / (!` k)) e. CC
34		fsum0cl 8276	. . . . . . . . 9 |- ((m e. NN0 /\ A.k e. (0...m)((B^k) / (!` k)) e. CC) -> sum_k e. (0...m)((B^k) / (!` k)) e. CC)
35	33, 34	mpan2 760	. . . . . . . 8 |- (m e. NN0 -> sum_k e. (0...m)((B^k) / (!` k)) e. CC)
36		mulcl 6456	. . . . . . . 8 |- ((sum_j e. (0...m)((A^j) / (!` j)) e. CC /\ sum_k e. (0...m)((B^k) / (!` k)) e. CC) -> (sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) e. CC)
37	29, 35, 36	syl11anc 524	. . . . . . 7 |- (m e. NN0 -> (sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) e. CC)
38	24, 37	eqeltrd 1971	. . . . . 6 |- (m e. NN0 -> (H` m) e. CC)
39	18, 38	jca 310	. . . . 5 |- (m e. NN0 -> ((F` m) e. CC /\ (H` m) e. CC))
40	6, 39	syl 12	. . . 4 |- (m e. NN -> ((F` m) e. CC /\ (H` m) e. CC))
41	40	rgen 2159	. . 3 |- A.m e. NN ((F` m) e. CC /\ (H` m) e. CC)
42	5, 41	pm3.2i 307	. 2 |- ((exp` (A + B)) e. CC /\ A.m e. NN ((F` m) e. CC /\ (H` m) e. CC))
43		eqid 1884	. . . . . . 7 |- (((|_` ((abs` A) + 1)) x. (|_` ((abs` B) + 1)))^2) = (((|_` ((abs` A) + 1)) x. (|_` ((abs` B) + 1)))^2)
44		eqid 1884	. . . . . . 7 |- (2 x. ((2^(3^2)) x. ((4 x. (((|_` ((abs` A) + 1)) x. (|_` ((abs` B) + 1)))^2))^((4 x. (((|_` ((abs` A) + 1)) x. (|_` ((abs` B) + 1)))^2)) + 3)))) = (2 x. ((2^(3^2)) x. ((4 x. (((|_` ((abs` A) + 1)) x. (|_` ((abs` B) + 1)))^2))^((4 x. (((|_` ((abs` A) + 1)) x. (|_` ((abs` B) + 1)))^2)) + 3))))
45	1, 2, 43, 44	efaddlem25 8624	. . . . . 6 |- ((x e. RR /\ 0 < x) -> E.f e. NN A.m e. NN (f <_ m -> (abs` ((sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) - sum_j e. (0...m)(((A + B)^j) / (!` j)))) < x))
46	24, 11	opreq12d 4900	. . . . . . . . . . . 12 |- (m e. NN0 -> ((H` m) - (F` m)) = ((sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) - sum_j e. (0...m)(((A + B)^j) / (!` j))))
47	6, 46	syl 12	. . . . . . . . . . 11 |- (m e. NN -> ((H` m) - (F` m)) = ((sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) - sum_j e. (0...m)(((A + B)^j) / (!` j))))
48	47	fveq2d 4685	. . . . . . . . . 10 |- (m e. NN -> (abs` ((H` m) - (F` m))) = (abs` ((sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) - sum_j e. (0...m)(((A + B)^j) / (!` j)))))
49	48	breq1d 3348	. . . . . . . . 9 |- (m e. NN -> ((abs` ((H` m) - (F` m))) < x <-> (abs` ((sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) - sum_j e. (0...m)(((A + B)^j) / (!` j)))) < x))
50	49	imbi2d 674	. . . . . . . 8 |- (m e. NN -> ((f <_ m -> (abs` ((H` m) - (F` m))) < x) <-> (f <_ m -> (abs` ((sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) - sum_j e. (0...m)(((A + B)^j) / (!` j)))) < x)))
51	50	ralbiia 2133	. . . . . . 7 |- (A.m e. NN (f <_ m -> (abs` ((H` m) - (F` m))) < x) <-> A.m e. NN (f <_ m -> (abs` ((sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) - sum_j e. (0...m)(((A + B)^j) / (!` j)))) < x))
52	51	rexbii 2128	. . . . . 6 |- (E.f e. NN A.m e. NN (f <_ m -> (abs` ((H` m) - (F` m))) < x) <-> E.f e. NN A.m e. NN (f <_ m -> (abs` ((sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) - sum_j e. (0...m)(((A + B)^j) / (!` j)))) < x))
53	45, 52	sylibr 217	. . . . 5 |- ((x e. RR /\ 0 < x) -> E.f e. NN A.m e. NN (f <_ m -> (abs` ((H` m) - (F` m))) < x))
54	53	ex 402	. . . 4 |- (x e. RR -> (0 < x -> E.f e. NN A.m e. NN (f <_ m -> (abs` ((H` m) - (F` m))) < x)))
55	54	rgen 2159	. . 3 |- A.x e. RR (0 < x -> E.f e. NN A.m e. NN (f <_ m -> (abs` ((H` m) - (F` m))) < x))
56	9	efcvgfsum 8577	. . . 4 |- ((A + B) e. CC -> F ~~> (exp` (A + B)))
57	3, 56	ax-mp 7	. . 3 |- F ~~> (exp` (A + B))
58	55, 57	pm3.2i 307	. 2 |- (A.x e. RR (0 < x -> E.f e. NN A.m e. NN (f <_ m -> (abs` ((H` m) - (F` m))) < x)) /\ F ~~> (exp` (A + B)))
59		nn0ex 7314	. . . 4 |- NN0 e. _V
60	59, 22	fopabex2 4541	. . 3 |- H e. _V
61	60	2climnn 8362	. 2 |- ((((exp` (A + B)) e. CC /\ A.m e. NN ((F` m) e. CC /\ (H` m) e. CC)) /\ (A.x e. RR (0 < x -> E.f e. NN A.m e. NN (f <_ m -> (abs` ((H` m) - (F` m))) < x)) /\ F ~~> (exp` (A + B)))) -> H ~~> (exp` (A + B)))
62	42, 58, 61	mp2an 761	1 |- H ~~> (exp` (A + B))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   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   - cmin 6445   / cdiv 6447   <_ cle 6448  NNcn 6449  NN0cn0 6450   < clt 6653  2c2 7145  3c3 7146  4c4 7147  |_cfl 7462  ...cfz 7637  ^cexp 7811  abscabs 8000  !cfa 8183   ~~> cli 8234  sum_csu 8239  expce 8555

This theorem is referenced by:  efaddlem28 8627

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-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-fl 7463  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-bc 8209  df-clim 8235  df-sum 8240  df-ef 8560

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