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Mirrors > Home > ILE Home > Th. List > eflegeo | Unicode version |
Description: The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.) |
Ref | Expression |
---|---|
eflegeo.1 | |
eflegeo.2 | |
eflegeo.3 |
Ref | Expression |
---|---|
eflegeo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 9492 | . . 3 | |
2 | 0zd 9195 | . . 3 | |
3 | eflegeo.1 | . . . . 5 | |
4 | 3 | recnd 7919 | . . . 4 |
5 | eqid 2164 | . . . . 5 | |
6 | 5 | eftvalcn 11588 | . . . 4 |
7 | 4, 6 | sylan 281 | . . 3 |
8 | reeftcl 11586 | . . . 4 | |
9 | 3, 8 | sylan 281 | . . 3 |
10 | simpr 109 | . . . 4 | |
11 | 3 | adantr 274 | . . . . 5 |
12 | 11, 10 | reexpcld 10595 | . . . 4 |
13 | oveq2 5845 | . . . . 5 | |
14 | eqid 2164 | . . . . 5 | |
15 | 13, 14 | fvmptg 5557 | . . . 4 |
16 | 10, 12, 15 | syl2anc 409 | . . 3 |
17 | reexpcl 10463 | . . . 4 | |
18 | 3, 17 | sylan 281 | . . 3 |
19 | faccl 10638 | . . . . . . 7 | |
20 | 19 | adantl 275 | . . . . . 6 |
21 | 20 | nnred 8862 | . . . . 5 |
22 | eflegeo.2 | . . . . . . 7 | |
23 | 22 | adantr 274 | . . . . . 6 |
24 | 11, 10, 23 | expge0d 10596 | . . . . 5 |
25 | 20 | nnge1d 8892 | . . . . 5 |
26 | 18, 21, 24, 25 | lemulge12d 8825 | . . . 4 |
27 | 20 | nngt0d 8893 | . . . . 5 |
28 | ledivmul 8764 | . . . . 5 | |
29 | 18, 18, 21, 27, 28 | syl112anc 1231 | . . . 4 |
30 | 26, 29 | mpbird 166 | . . 3 |
31 | 5 | efcllem 11590 | . . . 4 |
32 | 4, 31 | syl 14 | . . 3 |
33 | seqex 10373 | . . . 4 | |
34 | eflegeo.3 | . . . . . 6 | |
35 | 1red 7906 | . . . . . . 7 | |
36 | difrp 9620 | . . . . . . 7 | |
37 | 3, 35, 36 | syl2anc 409 | . . . . . 6 |
38 | 34, 37 | mpbid 146 | . . . . 5 |
39 | 38 | rpreccld 9635 | . . . 4 |
40 | 3, 22 | absidd 11099 | . . . . . 6 |
41 | 40, 34 | eqbrtrd 3999 | . . . . 5 |
42 | 4, 41, 16 | geolim 11442 | . . . 4 |
43 | breldmg 4805 | . . . 4 | |
44 | 33, 39, 42, 43 | mp3an2i 1331 | . . 3 |
45 | 1, 2, 7, 9, 16, 18, 30, 32, 44 | isumle 11426 | . 2 |
46 | efval 11592 | . . 3 | |
47 | 4, 46 | syl 14 | . 2 |
48 | expcl 10464 | . . . . 5 | |
49 | 4, 48 | sylan 281 | . . . 4 |
50 | 1, 2, 16, 49, 42 | isumclim 11352 | . . 3 |
51 | 50 | eqcomd 2170 | . 2 |
52 | 45, 47, 51 | 3brtr4d 4009 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cvv 2722 class class class wbr 3977 cmpt 4038 cdm 4599 cfv 5183 (class class class)co 5837 cc 7743 cr 7744 cc0 7745 c1 7746 caddc 7748 cmul 7750 clt 7925 cle 7926 cmin 8061 cdiv 8560 cn 8849 cn0 9106 crp 9581 cseq 10371 cexp 10445 cfa 10628 cabs 10929 cli 11209 csu 11284 ce 11573 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-mulrcl 7844 ax-addcom 7845 ax-mulcom 7846 ax-addass 7847 ax-mulass 7848 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-1rid 7852 ax-0id 7853 ax-rnegex 7854 ax-precex 7855 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-apti 7860 ax-pre-ltadd 7861 ax-pre-mulgt0 7862 ax-pre-mulext 7863 ax-arch 7864 ax-caucvg 7865 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-if 3517 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-po 4269 df-iso 4270 df-iord 4339 df-on 4341 df-ilim 4342 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-isom 5192 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-irdg 6330 df-frec 6351 df-1o 6376 df-oadd 6380 df-er 6493 df-en 6699 df-dom 6700 df-fin 6701 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-reap 8465 df-ap 8472 df-div 8561 df-inn 8850 df-2 8908 df-3 8909 df-4 8910 df-n0 9107 df-z 9184 df-uz 9459 df-q 9550 df-rp 9582 df-ico 9822 df-fz 9937 df-fzo 10069 df-seqfrec 10372 df-exp 10446 df-fac 10629 df-ihash 10679 df-cj 10774 df-re 10775 df-im 10776 df-rsqrt 10930 df-abs 10931 df-clim 11210 df-sumdc 11285 df-ef 11579 |
This theorem is referenced by: (None) |
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