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Mirrors > Home > MPE Home > Th. List > dfef2 | Structured version Visualization version GIF version |
Description: The limit of the sequence (1 + π΄ / π)βπ as π goes to +β is (expβπ΄). This is another common definition of e. (Contributed by Mario Carneiro, 1-Mar-2015.) |
Ref | Expression |
---|---|
dfef2.1 | β’ (π β πΉ β π) |
dfef2.2 | β’ (π β π΄ β β) |
dfef2.3 | β’ ((π β§ π β β) β (πΉβπ) = ((1 + (π΄ / π))βπ)) |
Ref | Expression |
---|---|
dfef2 | β’ (π β πΉ β (expβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfef2.2 | . . 3 β’ (π β π΄ β β) | |
2 | ax-1cn 11197 | . . . . . . . 8 β’ 1 β β | |
3 | simpl 482 | . . . . . . . . 9 β’ ((π΄ β β β§ π₯ β β) β π΄ β β) | |
4 | nncn 12251 | . . . . . . . . . 10 β’ (π₯ β β β π₯ β β) | |
5 | 4 | adantl 481 | . . . . . . . . 9 β’ ((π΄ β β β§ π₯ β β) β π₯ β β) |
6 | nnne0 12277 | . . . . . . . . . 10 β’ (π₯ β β β π₯ β 0) | |
7 | 6 | adantl 481 | . . . . . . . . 9 β’ ((π΄ β β β§ π₯ β β) β π₯ β 0) |
8 | 3, 5, 7 | divcld 12021 | . . . . . . . 8 β’ ((π΄ β β β§ π₯ β β) β (π΄ / π₯) β β) |
9 | addcl 11221 | . . . . . . . 8 β’ ((1 β β β§ (π΄ / π₯) β β) β (1 + (π΄ / π₯)) β β) | |
10 | 2, 8, 9 | sylancr 586 | . . . . . . 7 β’ ((π΄ β β β§ π₯ β β) β (1 + (π΄ / π₯)) β β) |
11 | nnnn0 12510 | . . . . . . . 8 β’ (π₯ β β β π₯ β β0) | |
12 | 11 | adantl 481 | . . . . . . 7 β’ ((π΄ β β β§ π₯ β β) β π₯ β β0) |
13 | cxpexp 26615 | . . . . . . 7 β’ (((1 + (π΄ / π₯)) β β β§ π₯ β β0) β ((1 + (π΄ / π₯))βππ₯) = ((1 + (π΄ / π₯))βπ₯)) | |
14 | 10, 12, 13 | syl2anc 583 | . . . . . 6 β’ ((π΄ β β β§ π₯ β β) β ((1 + (π΄ / π₯))βππ₯) = ((1 + (π΄ / π₯))βπ₯)) |
15 | 14 | mpteq2dva 5248 | . . . . 5 β’ (π΄ β β β (π₯ β β β¦ ((1 + (π΄ / π₯))βππ₯)) = (π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯))) |
16 | nnrp 13018 | . . . . . . . 8 β’ (π₯ β β β π₯ β β+) | |
17 | 16 | ssriv 3984 | . . . . . . 7 β’ β β β+ |
18 | 17 | a1i 11 | . . . . . 6 β’ (π΄ β β β β β β+) |
19 | eqid 2728 | . . . . . . 7 β’ (0(ballβ(abs β β ))(1 / ((absβπ΄) + 1))) = (0(ballβ(abs β β ))(1 / ((absβπ΄) + 1))) | |
20 | 19 | efrlim 26914 | . . . . . 6 β’ (π΄ β β β (π₯ β β+ β¦ ((1 + (π΄ / π₯))βππ₯)) βπ (expβπ΄)) |
21 | 18, 20 | rlimres2 15538 | . . . . 5 β’ (π΄ β β β (π₯ β β β¦ ((1 + (π΄ / π₯))βππ₯)) βπ (expβπ΄)) |
22 | 15, 21 | eqbrtrrd 5172 | . . . 4 β’ (π΄ β β β (π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯)) βπ (expβπ΄)) |
23 | nnuz 12896 | . . . . 5 β’ β = (β€β₯β1) | |
24 | 1zzd 12624 | . . . . 5 β’ (π΄ β β β 1 β β€) | |
25 | 10, 12 | expcld 14143 | . . . . . 6 β’ ((π΄ β β β§ π₯ β β) β ((1 + (π΄ / π₯))βπ₯) β β) |
26 | 25 | fmpttd 7125 | . . . . 5 β’ (π΄ β β β (π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯)):ββΆβ) |
27 | 23, 24, 26 | rlimclim 15523 | . . . 4 β’ (π΄ β β β ((π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯)) βπ (expβπ΄) β (π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯)) β (expβπ΄))) |
28 | 22, 27 | mpbid 231 | . . 3 β’ (π΄ β β β (π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯)) β (expβπ΄)) |
29 | 1, 28 | syl 17 | . 2 β’ (π β (π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯)) β (expβπ΄)) |
30 | nnex 12249 | . . . . 5 β’ β β V | |
31 | 30 | mptex 7235 | . . . 4 β’ (π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯)) β V |
32 | 31 | a1i 11 | . . 3 β’ (π β (π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯)) β V) |
33 | dfef2.1 | . . 3 β’ (π β πΉ β π) | |
34 | 1zzd 12624 | . . 3 β’ (π β 1 β β€) | |
35 | oveq2 7428 | . . . . . . . 8 β’ (π₯ = π β (π΄ / π₯) = (π΄ / π)) | |
36 | 35 | oveq2d 7436 | . . . . . . 7 β’ (π₯ = π β (1 + (π΄ / π₯)) = (1 + (π΄ / π))) |
37 | id 22 | . . . . . . 7 β’ (π₯ = π β π₯ = π) | |
38 | 36, 37 | oveq12d 7438 | . . . . . 6 β’ (π₯ = π β ((1 + (π΄ / π₯))βπ₯) = ((1 + (π΄ / π))βπ)) |
39 | eqid 2728 | . . . . . 6 β’ (π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯)) = (π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯)) | |
40 | ovex 7453 | . . . . . 6 β’ ((1 + (π΄ / π))βπ) β V | |
41 | 38, 39, 40 | fvmpt 7005 | . . . . 5 β’ (π β β β ((π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯))βπ) = ((1 + (π΄ / π))βπ)) |
42 | 41 | adantl 481 | . . . 4 β’ ((π β§ π β β) β ((π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯))βπ) = ((1 + (π΄ / π))βπ)) |
43 | dfef2.3 | . . . 4 β’ ((π β§ π β β) β (πΉβπ) = ((1 + (π΄ / π))βπ)) | |
44 | 42, 43 | eqtr4d 2771 | . . 3 β’ ((π β§ π β β) β ((π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯))βπ) = (πΉβπ)) |
45 | 23, 32, 33, 34, 44 | climeq 15544 | . 2 β’ (π β ((π₯ β β β¦ ((1 + (π΄ / π₯))βπ₯)) β (expβπ΄) β πΉ β (expβπ΄))) |
46 | 29, 45 | mpbid 231 | 1 β’ (π β πΉ β (expβπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2937 Vcvv 3471 β wss 3947 class class class wbr 5148 β¦ cmpt 5231 β ccom 5682 βcfv 6548 (class class class)co 7420 βcc 11137 0cc0 11139 1c1 11140 + caddc 11142 β cmin 11475 / cdiv 11902 βcn 12243 β0cn0 12503 β+crp 13007 βcexp 14059 abscabs 15214 β cli 15461 βπ crli 15462 expce 16038 ballcbl 21266 βπccxp 26502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13361 df-ioc 13362 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-fac 14266 df-bc 14295 df-hash 14323 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 df-sin 16046 df-cos 16047 df-tan 16048 df-pi 16049 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-mulg 19024 df-cntz 19268 df-cmn 19737 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-cmp 23304 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-limc 25808 df-dv 25809 df-log 26503 df-cxp 26504 |
This theorem is referenced by: (None) |
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