efcllem - Metamath Proof Explorer

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Theorem efcllem 14808

Description: Lemma for efcl 14813. The series that defines the exponential function converges, in the case where its argument is nonzero. The ratio test cvgrat 14615 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)

**Hypothesis**
Ref	Expression
eftval.1

**Assertion**
Ref	Expression
efcllem

Distinct variable group:

Allowed substitution hint:

(

)

Proof of Theorem efcllem Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 11722	. 2
2		eqid 2622	. 2
3		halfre 11246	. . 3
4	3	a1i 11	. 2
5		halflt1 11250	. . 3
6	5	a1i 11	. 2
7		2re 11090	. . . 4
8		abscl 14018	. . . 4
9		remulcl 10021	. . . 4 $/\$
10	7, 8, 9	sylancr 695	. . 3
11		absge0 14027	. . . 4
12		0le2 11111	. . . . 5
13		mulge0 10546	. . . . 5 $/\$ $/\$ $/\$
14	7, 12, 13	mpanl12 718	. . . 4 $/\$
15	8, 11, 14	syl2anc 693	. . 3
16		flge0nn0 12621	. . 3 $/\$
17	10, 15, 16	syl2anc 693	. 2
18		eftval.1	. . . . 5
19	18	eftval 14807	. . . 4
20	19	adantl 482	. . 3 $/\$
21		eftcl 14804	. . 3 $/\$
22	20, 21	eqeltrd 2701	. 2 $/\$
23	8	adantr 481	. . . . 5 $/\$
24		eluznn0 11757	. . . . . . 7 $/\$
25	17, 24	sylan 488	. . . . . 6 $/\$
26		nn0p1nn 11332	. . . . . 6
27	25, 26	syl 17	. . . . 5 $/\$
28	23, 27	nndivred 11069	. . . 4 $/\$
29	3	a1i 11	. . . 4 $/\$
30	23, 25	reexpcld 13025	. . . . 5 $/\$
31		faccl 13070	. . . . . 6
32	25, 31	syl 17	. . . . 5 $/\$
33	30, 32	nndivred 11069	. . . 4 $/\$
34		expcl 12878	. . . . . . . 8 $/\$
35	25, 34	syldan 487	. . . . . . 7 $/\$
36	35	absge0d 14183	. . . . . 6 $/\$
37		absexp 14044	. . . . . . 7 $/\$
38	25, 37	syldan 487	. . . . . 6 $/\$
39	36, 38	breqtrd 4679	. . . . 5 $/\$
40	32	nnred 11035	. . . . 5 $/\$
41	32	nngt0d 11064	. . . . 5 $/\$
42		divge0 10892	. . . . 5 $/\$ $/\$ $/\$
43	30, 39, 40, 41, 42	syl22anc 1327	. . . 4 $/\$
44	10	adantr 481	. . . . . . . 8 $/\$
45		peano2nn0 11333	. . . . . . . . . . 11
46	17, 45	syl 17	. . . . . . . . . 10
47	46	nn0red 11352	. . . . . . . . 9
48	47	adantr 481	. . . . . . . 8 $/\$
49	27	nnred 11035	. . . . . . . 8 $/\$
50		flltp1 12601	. . . . . . . . 9
51	44, 50	syl 17	. . . . . . . 8 $/\$
52		eluzp1p1 11713	. . . . . . . . . 10
53	52	adantl 482	. . . . . . . . 9 $/\$
54		eluzle 11700	. . . . . . . . 9
55	53, 54	syl 17	. . . . . . . 8 $/\$
56	44, 48, 49, 51, 55	ltletrd 10197	. . . . . . 7 $/\$
57	23	recnd 10068	. . . . . . . 8 $/\$
58		2cn 11091	. . . . . . . 8
59		mulcom 10022	. . . . . . . 8 $/\$
60	57, 58, 59	sylancl 694	. . . . . . 7 $/\$
61	27	nncnd 11036	. . . . . . . 8 $/\$
62	61	mulid2d 10058	. . . . . . 7 $/\$
63	56, 60, 62	3brtr4d 4685	. . . . . 6 $/\$
64		2pos 11112	. . . . . . . . 9
65	7, 64	pm3.2i 471	. . . . . . . 8 $/\$
66	65	a1i 11	. . . . . . 7 $/\$ $/\$
67		1red 10055	. . . . . . 7 $/\$
68	27	nngt0d 11064	. . . . . . . 8 $/\$
69	49, 68	jca 554	. . . . . . 7 $/\$ $/\$
70		lt2mul2div 10901	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
71	23, 66, 67, 69, 70	syl22anc 1327	. . . . . 6 $/\$
72	63, 71	mpbid 222	. . . . 5 $/\$
73		ltle 10126	. . . . . 6 $/\$
74	28, 3, 73	sylancl 694	. . . . 5 $/\$
75	72, 74	mpd 15	. . . 4 $/\$
76	28, 29, 33, 43, 75	lemul2ad 10964	. . 3 $/\$
77		peano2nn0 11333	. . . . . . 7
78	25, 77	syl 17	. . . . . 6 $/\$
79	18	eftval 14807	. . . . . 6
80	78, 79	syl 17	. . . . 5 $/\$
81	80	fveq2d 6195	. . . 4 $/\$
82		absexp 14044	. . . . . . . 8 $/\$
83	78, 82	syldan 487	. . . . . . 7 $/\$
84	57, 25	expp1d 13009	. . . . . . 7 $/\$
85	83, 84	eqtrd 2656	. . . . . 6 $/\$
86		faccl 13070	. . . . . . . . . 10
87	78, 86	syl 17	. . . . . . . . 9 $/\$
88	87	nnred 11035	. . . . . . . 8 $/\$
89	87	nnnn0d 11351	. . . . . . . . 9 $/\$
90	89	nn0ge0d 11354	. . . . . . . 8 $/\$
91	88, 90	absidd 14161	. . . . . . 7 $/\$
92		facp1 13065	. . . . . . . 8
93	25, 92	syl 17	. . . . . . 7 $/\$
94	91, 93	eqtrd 2656	. . . . . 6 $/\$
95	85, 94	oveq12d 6668	. . . . 5 $/\$
96		expcl 12878	. . . . . . 7 $/\$
97	78, 96	syldan 487	. . . . . 6 $/\$
98	87	nncnd 11036	. . . . . 6 $/\$
99	87	nnne0d 11065	. . . . . 6 $/\$
100	97, 98, 99	absdivd 14194	. . . . 5 $/\$
101	30	recnd 10068	. . . . . 6 $/\$
102	32	nncnd 11036	. . . . . 6 $/\$
103	32	nnne0d 11065	. . . . . 6 $/\$
104	27	nnne0d 11065	. . . . . 6 $/\$
105	101, 102, 57, 61, 103, 104	divmuldivd 10842	. . . . 5 $/\$
106	95, 100, 105	3eqtr4d 2666	. . . 4 $/\$
107	81, 106	eqtrd 2656	. . 3 $/\$
108		halfcn 11247	. . . . 5
109	25, 22	syldan 487	. . . . . . 7 $/\$
110	109	abscld 14175	. . . . . 6 $/\$
111	110	recnd 10068	. . . . 5 $/\$
112		mulcom 10022	. . . . 5 $/\$
113	108, 111, 112	sylancr 695	. . . 4 $/\$
114	25, 19	syl 17	. . . . . . 7 $/\$
115	114	fveq2d 6195	. . . . . 6 $/\$
116		eftabs 14806	. . . . . . 7 $/\$
117	25, 116	syldan 487	. . . . . 6 $/\$
118	115, 117	eqtrd 2656	. . . . 5 $/\$
119	118	oveq1d 6665	. . . 4 $/\$
120	113, 119	eqtrd 2656	. . 3 $/\$
121	76, 107, 120	3brtr4d 4685	. 2 $/\$
122	1, 2, 4, 6, 17, 22, 121	cvgrat 14615	1

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990 class class class wbr 4653

cmpt 4729

cdm 5114

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cle 10075

cdiv 10684

cn 11020

c2 11070

cn0 11292

cuz 11687

cfl 12591

cseq 12801

cexp 12860

cfa 13060

cabs 13974

cli 14215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417

This theorem is referenced by: eff 14812 efcvg 14815 reefcl 14817 efaddlem 14823 eftlcvg 14836 effsumlt 14841 eflegeo 14851 eirrlem 14932 expfac 39889

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