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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climexp | Structured version Unicode version |
Description: The limit of natural powers, is the natural power of the limit. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
climexp.1 |
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climexp.2 |
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climexp.3 |
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climexp.4 |
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climexp.5 |
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climexp.6 |
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climexp.7 |
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climexp.8 |
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climexp.9 |
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climexp.10 |
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Ref | Expression |
---|---|
climexp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climexp.4 |
. . . 4
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2 | climexp.5 |
. . . 4
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3 | climexp.8 |
. . . . . 6
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4 | eqid 2451 |
. . . . . . 7
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5 | 4 | expcn 20573 |
. . . . . 6
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6 | 3, 5 | syl 16 |
. . . . 5
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7 | 4 | cncfcn1 20611 |
. . . . 5
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8 | 6, 7 | syl6eleqr 2550 |
. . . 4
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9 | climexp.6 |
. . . 4
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10 | climexp.7 |
. . . 4
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11 | climcl 13088 |
. . . . 5
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12 | 10, 11 | syl 16 |
. . . 4
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13 | 1, 2, 8, 9, 10, 12 | climcncf 20601 |
. . 3
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14 | eqidd 2452 |
. . . 4
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15 | simpr 461 |
. . . . 5
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16 | 15 | oveq1d 6208 |
. . . 4
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17 | 12, 3 | expcld 12118 |
. . . 4
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18 | 14, 16, 12, 17 | fvmptd 5881 |
. . 3
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19 | 13, 18 | breqtrd 4417 |
. 2
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20 | climexp.9 |
. . 3
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21 | cnex 9467 |
. . . . 5
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22 | 21 | mptex 6050 |
. . . 4
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23 | fvex 5802 |
. . . . . 6
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24 | 1, 23 | eqeltri 2535 |
. . . . 5
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25 | fex 6052 |
. . . . 5
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26 | 9, 24, 25 | sylancl 662 |
. . . 4
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27 | coexg 6631 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 22, 26, 27 | sylancr 663 |
. . 3
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29 | eqidd 2452 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | simpr 461 |
. . . . . 6
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31 | 30 | oveq1d 6208 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 9 | ffvelrnda 5945 |
. . . . 5
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33 | 3 | adantr 465 |
. . . . . 6
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34 | 32, 33 | expcld 12118 |
. . . . 5
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35 | 29, 31, 32, 34 | fvmptd 5881 |
. . . 4
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36 | fvco3 5870 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
37 | 9, 36 | sylan 471 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | climexp.1 |
. . . . . . 7
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39 | nfv 1674 |
. . . . . . 7
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40 | 38, 39 | nfan 1863 |
. . . . . 6
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41 | climexp.3 |
. . . . . . . 8
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42 | nfcv 2613 |
. . . . . . . 8
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43 | 41, 42 | nffv 5799 |
. . . . . . 7
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44 | climexp.2 |
. . . . . . . . 9
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45 | 44, 42 | nffv 5799 |
. . . . . . . 8
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46 | nfcv 2613 |
. . . . . . . 8
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47 | nfcv 2613 |
. . . . . . . 8
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48 | 45, 46, 47 | nfov 6216 |
. . . . . . 7
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49 | 43, 48 | nfeq 2623 |
. . . . . 6
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50 | 40, 49 | nfim 1855 |
. . . . 5
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51 | eleq1 2523 |
. . . . . . 7
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52 | 51 | anbi2d 703 |
. . . . . 6
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53 | fveq2 5792 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
54 | fveq2 5792 |
. . . . . . . 8
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55 | 54 | oveq1d 6208 |
. . . . . . 7
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56 | 53, 55 | eqeq12d 2473 |
. . . . . 6
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57 | 52, 56 | imbi12d 320 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
58 | climexp.10 |
. . . . 5
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59 | 50, 57, 58 | chvar 1966 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
60 | 35, 37, 59 | 3eqtr4rd 2503 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
61 | 1, 20, 28, 2, 60 | climeq 13156 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
62 | 19, 61 | mpbird 232 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 ax-inf2 7951 ax-cnex 9442 ax-resscn 9443 ax-1cn 9444 ax-icn 9445 ax-addcl 9446 ax-addrcl 9447 ax-mulcl 9448 ax-mulrcl 9449 ax-mulcom 9450 ax-addass 9451 ax-mulass 9452 ax-distr 9453 ax-i2m1 9454 ax-1ne0 9455 ax-1rid 9456 ax-rnegex 9457 ax-rrecex 9458 ax-cnre 9459 ax-pre-lttri 9460 ax-pre-lttrn 9461 ax-pre-ltadd 9462 ax-pre-mulgt0 9463 ax-pre-sup 9464 ax-mulf 9466 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-pss 3445 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-tp 3983 df-op 3985 df-uni 4193 df-int 4230 df-iun 4274 df-iin 4275 df-br 4394 df-opab 4452 df-mpt 4453 df-tr 4487 df-eprel 4733 df-id 4737 df-po 4742 df-so 4743 df-fr 4780 df-se 4781 df-we 4782 df-ord 4823 df-on 4824 df-lim 4825 df-suc 4826 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-isom 5528 df-riota 6154 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-of 6423 df-om 6580 df-1st 6680 df-2nd 6681 df-supp 6794 df-recs 6935 df-rdg 6969 df-1o 7023 df-2o 7024 df-oadd 7027 df-er 7204 df-map 7319 df-ixp 7367 df-en 7414 df-dom 7415 df-sdom 7416 df-fin 7417 df-fsupp 7725 df-fi 7765 df-sup 7795 df-oi 7828 df-card 8213 df-cda 8441 df-pnf 9524 df-mnf 9525 df-xr 9526 df-ltxr 9527 df-le 9528 df-sub 9701 df-neg 9702 df-div 10098 df-nn 10427 df-2 10484 df-3 10485 df-4 10486 df-5 10487 df-6 10488 df-7 10489 df-8 10490 df-9 10491 df-10 10492 df-n0 10684 df-z 10751 df-dec 10860 df-uz 10966 df-q 11058 df-rp 11096 df-xneg 11193 df-xadd 11194 df-xmul 11195 df-icc 11411 df-fz 11548 df-fzo 11659 df-seq 11917 df-exp 11976 df-hash 12214 df-cj 12699 df-re 12700 df-im 12701 df-sqr 12835 df-abs 12836 df-clim 13077 df-struct 14287 df-ndx 14288 df-slot 14289 df-base 14290 df-sets 14291 df-ress 14292 df-plusg 14362 df-mulr 14363 df-starv 14364 df-sca 14365 df-vsca 14366 df-ip 14367 df-tset 14368 df-ple 14369 df-ds 14371 df-unif 14372 df-hom 14373 df-cco 14374 df-rest 14472 df-topn 14473 df-0g 14491 df-gsum 14492 df-topgen 14493 df-pt 14494 df-prds 14497 df-xrs 14551 df-qtop 14556 df-imas 14557 df-xps 14559 df-mre 14635 df-mrc 14636 df-acs 14638 df-mnd 15526 df-submnd 15576 df-mulg 15659 df-cntz 15946 df-cmn 16392 df-psmet 17927 df-xmet 17928 df-met 17929 df-bl 17930 df-mopn 17931 df-cnfld 17937 df-top 18628 df-bases 18630 df-topon 18631 df-topsp 18632 df-cn 18956 df-cnp 18957 df-tx 19260 df-hmeo 19453 df-xms 20020 df-ms 20021 df-tms 20022 df-cncf 20579 |
This theorem is referenced by: stirlinglem8 30017 |
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