Theorem ef0lem 8572

Description: The series defining the exponential function converges in the (trivial) case of a zero argument.

Hypothesis

Ref	Expression
ef0lem.1	|- F = {<.j, y>. | (j e. NN0 /\ y = ((A^j) / (!` j)))}

Assertion

Ref	Expression
ef0lem	|- (A = 0 -> ( + seq0 F) ~~> 1)

Distinct variable group:   y,j,A

Proof of Theorem ef0lem

Step	Hyp	Ref	Expression
1		oprex 4907	. . 3 |- ( + seq0 F) e. _V
2		0z 7355	. . 3 |- 0 e. ZZ
3	1, 2	climconsti 8354	. 2 |- ((1 e. CC /\ A.m e. (ZZ>=` 0)(( + seq0 F)` m) = 1) -> ( + seq0 F) ~~> 1)
4		ax1cn 6422	. 2 |- 1 e. CC
5		ef0lem.1	. . . . . . . . . . 11 |- F = {<.j, y>. | (j e. NN0 /\ y = ((A^j) / (!` j)))}
6		oprex 4907	. . . . . . . . . . 11 |- ((A^(m + 1)) / (!` (m + 1))) e. _V
7		opreq2 4890	. . . . . . . . . . . 12 |- (j = (m + 1) -> (A^j) = (A^(m + 1)))
8		fveq2 4681	. . . . . . . . . . . 12 |- (j = (m + 1) -> (!` j) = (!` (m + 1)))
9	7, 8	opreq12d 4900	. . . . . . . . . . 11 |- (j = (m + 1) -> ((A^j) / (!` j)) = ((A^(m + 1)) / (!` (m + 1))))
10	5, 6, 9	ser0p1i 7810	. . . . . . . . . 10 |- (m e. NN0 -> (( + seq0 F)` (m + 1)) = ((( + seq0 F)` m) + ((A^(m + 1)) / (!` (m + 1)))))
11	10	ad2antrr 440	. . . . . . . . 9 |- (((m e. NN0 /\ A = 0) /\ (( + seq0 F)` m) = 1) -> (( + seq0 F)` (m + 1)) = ((( + seq0 F)` m) + ((A^(m + 1)) / (!` (m + 1)))))
12		opreq1 4889	. . . . . . . . . 10 |- ((( + seq0 F)` m) = 1 -> ((( + seq0 F)` m) + ((A^(m + 1)) / (!` (m + 1)))) = (1 + ((A^(m + 1)) / (!` (m + 1)))))
13		opreq1 4889	. . . . . . . . . . . . . . 15 |- (A = 0 -> (A^(m + 1)) = (0^(m + 1)))
14		nn0p1nn 7384	. . . . . . . . . . . . . . . 16 |- (m e. NN0 -> (m + 1) e. NN)
15		0exp 7832	. . . . . . . . . . . . . . . 16 |- ((m + 1) e. NN -> (0^(m + 1)) = 0)
16	14, 15	syl 12	. . . . . . . . . . . . . . 15 |- (m e. NN0 -> (0^(m + 1)) = 0)
17	13, 16	sylan9eqr 1951	. . . . . . . . . . . . . 14 |- ((m e. NN0 /\ A = 0) -> (A^(m + 1)) = 0)
18	17	opreq1d 4897	. . . . . . . . . . . . 13 |- ((m e. NN0 /\ A = 0) -> ((A^(m + 1)) / (!` (m + 1))) = (0 / (!` (m + 1))))
19		peano2nn0 7333	. . . . . . . . . . . . . . . 16 |- (m e. NN0 -> (m + 1) e. NN0)
20		faccl 8192	. . . . . . . . . . . . . . . . 17 |- ((m + 1) e. NN0 -> (!` (m + 1)) e. NN)
21		nncn 7113	. . . . . . . . . . . . . . . . 17 |- ((!` (m + 1)) e. NN -> (!` (m + 1)) e. CC)
22	20, 21	syl 12	. . . . . . . . . . . . . . . 16 |- ((m + 1) e. NN0 -> (!` (m + 1)) e. CC)
23	19, 22	syl 12	. . . . . . . . . . . . . . 15 |- (m e. NN0 -> (!` (m + 1)) e. CC)
24		facne0 8193	. . . . . . . . . . . . . . . 16 |- ((m + 1) e. NN0 -> (!` (m + 1)) =/= 0)
25	19, 24	syl 12	. . . . . . . . . . . . . . 15 |- (m e. NN0 -> (!` (m + 1)) =/= 0)
26		div0 6943	. . . . . . . . . . . . . . 15 |- (((!` (m + 1)) e. CC /\ (!` (m + 1)) =/= 0) -> (0 / (!` (m + 1))) = 0)
27	23, 25, 26	syl11anc 524	. . . . . . . . . . . . . 14 |- (m e. NN0 -> (0 / (!` (m + 1))) = 0)
28	27	adantr 425	. . . . . . . . . . . . 13 |- ((m e. NN0 /\ A = 0) -> (0 / (!` (m + 1))) = 0)
29	18, 28	eqtrd 1925	. . . . . . . . . . . 12 |- ((m e. NN0 /\ A = 0) -> ((A^(m + 1)) / (!` (m + 1))) = 0)
30	29	opreq2d 4898	. . . . . . . . . . 11 |- ((m e. NN0 /\ A = 0) -> (1 + ((A^(m + 1)) / (!` (m + 1)))) = (1 + 0))
31	4	addid1i 6483	. . . . . . . . . . 11 |- (1 + 0) = 1
32	30, 31	syl6eq 1944	. . . . . . . . . 10 |- ((m e. NN0 /\ A = 0) -> (1 + ((A^(m + 1)) / (!` (m + 1)))) = 1)
33	12, 32	sylan9eqr 1951	. . . . . . . . 9 |- (((m e. NN0 /\ A = 0) /\ (( + seq0 F)` m) = 1) -> ((( + seq0 F)` m) + ((A^(m + 1)) / (!` (m + 1)))) = 1)
34	11, 33	eqtrd 1925	. . . . . . . 8 |- (((m e. NN0 /\ A = 0) /\ (( + seq0 F)` m) = 1) -> (( + seq0 F)` (m + 1)) = 1)
35	34	exp31 407	. . . . . . 7 |- (m e. NN0 -> (A = 0 -> ((( + seq0 F)` m) = 1 -> (( + seq0 F)` (m + 1)) = 1)))
36	35	a2d 16	. . . . . 6 |- (m e. NN0 -> ((A = 0 -> (( + seq0 F)` m) = 1) -> (A = 0 -> (( + seq0 F)` (m + 1)) = 1)))
37		addex 6470	. . . . . . . . 9 |- + e. _V
38		nn0ex 7314	. . . . . . . . . 10 |- NN0 e. _V
39	38, 5	fopabex2 4541	. . . . . . . . 9 |- F e. _V
40	37, 39	seq00 7793	. . . . . . . 8 |- (( + seq0 F)` 0) = (F` 0)
41	40	a1i 8	. . . . . . 7 |- (A = 0 -> (( + seq0 F)` 0) = (F` 0))
42		0nn0 7322	. . . . . . . . 9 |- 0 e. NN0
43	42	a1i 8	. . . . . . . 8 |- (A = 0 -> 0 e. NN0)
44		opreq2 4890	. . . . . . . . . 10 |- (j = 0 -> (A^j) = (A^0))
45		fveq2 4681	. . . . . . . . . 10 |- (j = 0 -> (!` j) = (!` 0))
46	44, 45	opreq12d 4900	. . . . . . . . 9 |- (j = 0 -> ((A^j) / (!` j)) = ((A^0) / (!` 0)))
47		oprex 4907	. . . . . . . . 9 |- ((A^0) / (!` 0)) e. _V
48	46, 5, 47	fvopab4 4743	. . . . . . . 8 |- (0 e. NN0 -> (F` 0) = ((A^0) / (!` 0)))
49	43, 48	syl 12	. . . . . . 7 |- (A = 0 -> (F` 0) = ((A^0) / (!` 0)))
50		opreq1 4889	. . . . . . . . . 10 |- (A = 0 -> (A^0) = (0^0))
51		0cn 6481	. . . . . . . . . . 11 |- 0 e. CC
52		exp0 7814	. . . . . . . . . . 11 |- (0 e. CC -> (0^0) = 1)
53	51, 52	ax-mp 7	. . . . . . . . . 10 |- (0^0) = 1
54	50, 53	syl6eq 1944	. . . . . . . . 9 |- (A = 0 -> (A^0) = 1)
55	54	opreq1d 4897	. . . . . . . 8 |- (A = 0 -> ((A^0) / (!` 0)) = (1 / (!` 0)))
56		fac0 8186	. . . . . . . . . 10 |- (!` 0) = 1
57	56	opreq2i 4893	. . . . . . . . 9 |- (1 / (!` 0)) = (1 / 1)
58	4	div1i 6948	. . . . . . . . 9 |- (1 / 1) = 1
59	57, 58	eqtri 1908	. . . . . . . 8 |- (1 / (!` 0)) = 1
60	55, 59	syl6eq 1944	. . . . . . 7 |- (A = 0 -> ((A^0) / (!` 0)) = 1)
61	41, 49, 60	3eqtrd 1929	. . . . . 6 |- (A = 0 -> (( + seq0 F)` 0) = 1)
62		fveq2 4681	. . . . . . . 8 |- (k = 0 -> (( + seq0 F)` k) = (( + seq0 F)` 0))
63	62	eqeq1d 1892	. . . . . . 7 |- (k = 0 -> ((( + seq0 F)` k) = 1 <-> (( + seq0 F)` 0) = 1))
64	63	imbi2d 674	. . . . . 6 |- (k = 0 -> ((A = 0 -> (( + seq0 F)` k) = 1) <-> (A = 0 -> (( + seq0 F)` 0) = 1)))
65		fveq2 4681	. . . . . . . 8 |- (k = m -> (( + seq0 F)` k) = (( + seq0 F)` m))
66	65	eqeq1d 1892	. . . . . . 7 |- (k = m -> ((( + seq0 F)` k) = 1 <-> (( + seq0 F)` m) = 1))
67	66	imbi2d 674	. . . . . 6 |- (k = m -> ((A = 0 -> (( + seq0 F)` k) = 1) <-> (A = 0 -> (( + seq0 F)` m) = 1)))
68		fveq2 4681	. . . . . . . 8 |- (k = (m + 1) -> (( + seq0 F)` k) = (( + seq0 F)` (m + 1)))
69	68	eqeq1d 1892	. . . . . . 7 |- (k = (m + 1) -> ((( + seq0 F)` k) = 1 <-> (( + seq0 F)` (m + 1)) = 1))
70	69	imbi2d 674	. . . . . 6 |- (k = (m + 1) -> ((A = 0 -> (( + seq0 F)` k) = 1) <-> (A = 0 -> (( + seq0 F)` (m + 1)) = 1)))
71	36, 61, 64, 67, 70, 67	nn0indALT 7425	. . . . 5 |- (m e. NN0 -> (A = 0 -> (( + seq0 F)` m) = 1))
72	71	impcom 378	. . . 4 |- ((A = 0 /\ m e. NN0) -> (( + seq0 F)` m) = 1)
73		elnn0uz 7610	. . . 4 |- (m e. NN0 <-> m e. (ZZ>=` 0))
74	72, 73	sylan2br 502	. . 3 |- ((A = 0 /\ m e. (ZZ>=` 0)) -> (( + seq0 F)` m) = 1)
75	74	r19.21aiva 2176	. 2 |- (A = 0 -> A.m e. (ZZ>=` 0)(( + seq0 F)` m) = 1)
76	3, 4, 75	sylancr 526	1 |- (A = 0 -> ( + seq0 F) ~~> 1)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   + caddc 6389   / cdiv 6447  NNcn 6449  NN0cn0 6450  ZZ>=cuz 7586   seq0 cseq0 7775  ^cexp 7811  !cfa 8183   ~~> cli 8234

This theorem is referenced by:  ef0 8597

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  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-uz 7587  df-seq1 7721  df-shft 7754  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

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