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Mirrors > Home > ILE Home > Th. List > efcllemp | Unicode version |
Description: Lemma for efcl 11370. The series that defines the exponential function converges. The ratio test cvgratgt0 11302 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.) |
Ref | Expression |
---|---|
efcllemp.1 | |
efcllemp.a | |
efcllemp.k | |
efcllemp.ak |
Ref | Expression |
---|---|
efcllemp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 9360 | . 2 | |
2 | eqid 2139 | . 2 | |
3 | halfre 8933 | . . 3 | |
4 | 3 | a1i 9 | . 2 |
5 | halflt1 8937 | . . 3 | |
6 | 5 | a1i 9 | . 2 |
7 | halfgt0 8935 | . . 3 | |
8 | 7 | a1i 9 | . 2 |
9 | efcllemp.k | . . 3 | |
10 | 9 | nnnn0d 9030 | . 2 |
11 | efcllemp.a | . . 3 | |
12 | efcllemp.1 | . . . . 5 | |
13 | 12 | eftvalcn 11363 | . . . 4 |
14 | eftcl 11360 | . . . 4 | |
15 | 13, 14 | eqeltrd 2216 | . . 3 |
16 | 11, 15 | sylan 281 | . 2 |
17 | 11 | adantr 274 | . . . . . 6 |
18 | 17 | abscld 10953 | . . . . 5 |
19 | eluznn0 9393 | . . . . . . 7 | |
20 | 10, 19 | sylan 281 | . . . . . 6 |
21 | nn0p1nn 9016 | . . . . . 6 | |
22 | 20, 21 | syl 14 | . . . . 5 |
23 | 18, 22 | nndivred 8770 | . . . 4 |
24 | 3 | a1i 9 | . . . 4 |
25 | 18, 20 | reexpcld 10441 | . . . . 5 |
26 | 20 | faccld 10482 | . . . . 5 |
27 | 25, 26 | nndivred 8770 | . . . 4 |
28 | 17, 20 | expcld 10424 | . . . . . . 7 |
29 | 28 | absge0d 10956 | . . . . . 6 |
30 | 17, 20 | absexpd 10964 | . . . . . 6 |
31 | 29, 30 | breqtrd 3954 | . . . . 5 |
32 | 26 | nnred 8733 | . . . . 5 |
33 | 26 | nngt0d 8764 | . . . . 5 |
34 | divge0 8631 | . . . . 5 | |
35 | 25, 31, 32, 33, 34 | syl22anc 1217 | . . . 4 |
36 | 2re 8790 | . . . . . . . . . 10 | |
37 | abscl 10823 | . . . . . . . . . 10 | |
38 | remulcl 7748 | . . . . . . . . . 10 | |
39 | 36, 37, 38 | sylancr 410 | . . . . . . . . 9 |
40 | 17, 39 | syl 14 | . . . . . . . 8 |
41 | peano2nn0 9017 | . . . . . . . . . . 11 | |
42 | 10, 41 | syl 14 | . . . . . . . . . 10 |
43 | 42 | nn0red 9031 | . . . . . . . . 9 |
44 | 43 | adantr 274 | . . . . . . . 8 |
45 | 22 | nnred 8733 | . . . . . . . 8 |
46 | 10 | adantr 274 | . . . . . . . . . 10 |
47 | 46 | nn0red 9031 | . . . . . . . . 9 |
48 | efcllemp.ak | . . . . . . . . . 10 | |
49 | 48 | adantr 274 | . . . . . . . . 9 |
50 | 47 | ltp1d 8688 | . . . . . . . . 9 |
51 | 40, 47, 44, 49, 50 | lttrd 7888 | . . . . . . . 8 |
52 | eluzp1p1 9351 | . . . . . . . . . 10 | |
53 | 52 | adantl 275 | . . . . . . . . 9 |
54 | eluzle 9338 | . . . . . . . . 9 | |
55 | 53, 54 | syl 14 | . . . . . . . 8 |
56 | 40, 44, 45, 51, 55 | ltletrd 8185 | . . . . . . 7 |
57 | 18 | recnd 7794 | . . . . . . . 8 |
58 | 2cn 8791 | . . . . . . . 8 | |
59 | mulcom 7749 | . . . . . . . 8 | |
60 | 57, 58, 59 | sylancl 409 | . . . . . . 7 |
61 | 22 | nncnd 8734 | . . . . . . . 8 |
62 | 61 | mulid2d 7784 | . . . . . . 7 |
63 | 56, 60, 62 | 3brtr4d 3960 | . . . . . 6 |
64 | 2rp 9446 | . . . . . . . 8 | |
65 | 64 | a1i 9 | . . . . . . 7 |
66 | 1red 7781 | . . . . . . 7 | |
67 | 22 | nnrpd 9482 | . . . . . . 7 |
68 | 18, 65, 66, 67 | lt2mul2divd 9552 | . . . . . 6 |
69 | 63, 68 | mpbid 146 | . . . . 5 |
70 | ltle 7851 | . . . . . 6 | |
71 | 23, 3, 70 | sylancl 409 | . . . . 5 |
72 | 69, 71 | mpd 13 | . . . 4 |
73 | 23, 24, 27, 35, 72 | lemul2ad 8698 | . . 3 |
74 | peano2nn0 9017 | . . . . . . 7 | |
75 | 20, 74 | syl 14 | . . . . . 6 |
76 | 12 | eftvalcn 11363 | . . . . . 6 |
77 | 11, 75, 76 | syl2an2r 584 | . . . . 5 |
78 | 77 | fveq2d 5425 | . . . 4 |
79 | 17, 75 | absexpd 10964 | . . . . . . 7 |
80 | 57, 20 | expp1d 10425 | . . . . . . 7 |
81 | 79, 80 | eqtrd 2172 | . . . . . 6 |
82 | 75 | faccld 10482 | . . . . . . . . 9 |
83 | 82 | nnred 8733 | . . . . . . . 8 |
84 | 82 | nnnn0d 9030 | . . . . . . . . 9 |
85 | 84 | nn0ge0d 9033 | . . . . . . . 8 |
86 | 83, 85 | absidd 10939 | . . . . . . 7 |
87 | facp1 10476 | . . . . . . . 8 | |
88 | 20, 87 | syl 14 | . . . . . . 7 |
89 | 86, 88 | eqtrd 2172 | . . . . . 6 |
90 | 81, 89 | oveq12d 5792 | . . . . 5 |
91 | 17, 75 | expcld 10424 | . . . . . 6 |
92 | 82 | nncnd 8734 | . . . . . 6 |
93 | 82 | nnap0d 8766 | . . . . . 6 # |
94 | 91, 92, 93 | absdivapd 10967 | . . . . 5 |
95 | 25 | recnd 7794 | . . . . . 6 |
96 | 26 | nncnd 8734 | . . . . . 6 |
97 | 26 | nnap0d 8766 | . . . . . 6 # |
98 | 22 | nnap0d 8766 | . . . . . 6 # |
99 | 95, 96, 57, 61, 97, 98 | divmuldivapd 8592 | . . . . 5 |
100 | 90, 94, 99 | 3eqtr4d 2182 | . . . 4 |
101 | 78, 100 | eqtrd 2172 | . . 3 |
102 | halfcn 8934 | . . . . 5 | |
103 | 11, 20, 15 | syl2an2r 584 | . . . . . . 7 |
104 | 103 | abscld 10953 | . . . . . 6 |
105 | 104 | recnd 7794 | . . . . 5 |
106 | mulcom 7749 | . . . . 5 | |
107 | 102, 105, 106 | sylancr 410 | . . . 4 |
108 | 11, 20, 13 | syl2an2r 584 | . . . . . . 7 |
109 | 108 | fveq2d 5425 | . . . . . 6 |
110 | eftabs 11362 | . . . . . . 7 | |
111 | 11, 20, 110 | syl2an2r 584 | . . . . . 6 |
112 | 109, 111 | eqtrd 2172 | . . . . 5 |
113 | 112 | oveq1d 5789 | . . . 4 |
114 | 107, 113 | eqtrd 2172 | . . 3 |
115 | 73, 101, 114 | 3brtr4d 3960 | . 2 |
116 | 1, 2, 4, 6, 8, 10, 16, 115 | cvgratgt0 11302 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 class class class wbr 3929 cmpt 3989 cdm 4539 cfv 5123 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 caddc 7623 cmul 7625 clt 7800 cle 7801 cdiv 8432 cn 8720 c2 8771 cn0 8977 cuz 9326 crp 9441 cseq 10218 cexp 10292 cfa 10471 cabs 10769 cli 11047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-ico 9677 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-fac 10472 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 |
This theorem is referenced by: efcllem 11365 |
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