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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem19 | Structured version Visualization version GIF version |
Description: The π-th derivative of π» is 0 if π is large enough. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem19.s | β’ (π β π β {β, β}) |
etransclem19.x | β’ (π β π β ((TopOpenββfld) βΎt π)) |
etransclem19.p | β’ (π β π β β) |
etransclem19.1 | β’ π» = (π β (0...π) β¦ (π₯ β π β¦ ((π₯ β π)βif(π = 0, (π β 1), π)))) |
etransclem19.J | β’ (π β π½ β (0...π)) |
etransclem19.n | β’ (π β π β β€) |
etransclem19.7 | β’ (π β if(π½ = 0, (π β 1), π) < π) |
Ref | Expression |
---|---|
etransclem19 | β’ (π β ((π Dπ (π»βπ½))βπ) = (π₯ β π β¦ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | etransclem19.s | . . 3 β’ (π β π β {β, β}) | |
2 | etransclem19.x | . . 3 β’ (π β π β ((TopOpenββfld) βΎt π)) | |
3 | etransclem19.p | . . 3 β’ (π β π β β) | |
4 | etransclem19.1 | . . 3 β’ π» = (π β (0...π) β¦ (π₯ β π β¦ ((π₯ β π)βif(π = 0, (π β 1), π)))) | |
5 | etransclem19.J | . . 3 β’ (π β π½ β (0...π)) | |
6 | etransclem19.n | . . . 4 β’ (π β π β β€) | |
7 | 0red 11222 | . . . . 5 β’ (π β 0 β β) | |
8 | 6 | zred 12671 | . . . . 5 β’ (π β π β β) |
9 | nnm1nn0 12518 | . . . . . . . . 9 β’ (π β β β (π β 1) β β0) | |
10 | 3, 9 | syl 17 | . . . . . . . 8 β’ (π β (π β 1) β β0) |
11 | 10 | nn0red 12538 | . . . . . . 7 β’ (π β (π β 1) β β) |
12 | 3 | nnred 12232 | . . . . . . 7 β’ (π β π β β) |
13 | 11, 12 | ifcld 4574 | . . . . . 6 β’ (π β if(π½ = 0, (π β 1), π) β β) |
14 | 10 | nn0ge0d 12540 | . . . . . . . . 9 β’ (π β 0 β€ (π β 1)) |
15 | 14 | adantr 480 | . . . . . . . 8 β’ ((π β§ π½ = 0) β 0 β€ (π β 1)) |
16 | iftrue 4534 | . . . . . . . . . 10 β’ (π½ = 0 β if(π½ = 0, (π β 1), π) = (π β 1)) | |
17 | 16 | eqcomd 2737 | . . . . . . . . 9 β’ (π½ = 0 β (π β 1) = if(π½ = 0, (π β 1), π)) |
18 | 17 | adantl 481 | . . . . . . . 8 β’ ((π β§ π½ = 0) β (π β 1) = if(π½ = 0, (π β 1), π)) |
19 | 15, 18 | breqtrd 5174 | . . . . . . 7 β’ ((π β§ π½ = 0) β 0 β€ if(π½ = 0, (π β 1), π)) |
20 | 3 | nnnn0d 12537 | . . . . . . . . . 10 β’ (π β π β β0) |
21 | 20 | nn0ge0d 12540 | . . . . . . . . 9 β’ (π β 0 β€ π) |
22 | 21 | adantr 480 | . . . . . . . 8 β’ ((π β§ Β¬ π½ = 0) β 0 β€ π) |
23 | iffalse 4537 | . . . . . . . . . 10 β’ (Β¬ π½ = 0 β if(π½ = 0, (π β 1), π) = π) | |
24 | 23 | eqcomd 2737 | . . . . . . . . 9 β’ (Β¬ π½ = 0 β π = if(π½ = 0, (π β 1), π)) |
25 | 24 | adantl 481 | . . . . . . . 8 β’ ((π β§ Β¬ π½ = 0) β π = if(π½ = 0, (π β 1), π)) |
26 | 22, 25 | breqtrd 5174 | . . . . . . 7 β’ ((π β§ Β¬ π½ = 0) β 0 β€ if(π½ = 0, (π β 1), π)) |
27 | 19, 26 | pm2.61dan 810 | . . . . . 6 β’ (π β 0 β€ if(π½ = 0, (π β 1), π)) |
28 | etransclem19.7 | . . . . . 6 β’ (π β if(π½ = 0, (π β 1), π) < π) | |
29 | 7, 13, 8, 27, 28 | lelttrd 11377 | . . . . 5 β’ (π β 0 < π) |
30 | 7, 8, 29 | ltled 11367 | . . . 4 β’ (π β 0 β€ π) |
31 | elnn0z 12576 | . . . 4 β’ (π β β0 β (π β β€ β§ 0 β€ π)) | |
32 | 6, 30, 31 | sylanbrc 582 | . . 3 β’ (π β π β β0) |
33 | 1, 2, 3, 4, 5, 32 | etransclem17 45266 | . 2 β’ (π β ((π Dπ (π»βπ½))βπ) = (π₯ β π β¦ if(if(π½ = 0, (π β 1), π) < π, 0, (((!βif(π½ = 0, (π β 1), π)) / (!β(if(π½ = 0, (π β 1), π) β π))) Β· ((π₯ β π½)β(if(π½ = 0, (π β 1), π) β π)))))) |
34 | 28 | iftrued 4536 | . . 3 β’ (π β if(if(π½ = 0, (π β 1), π) < π, 0, (((!βif(π½ = 0, (π β 1), π)) / (!β(if(π½ = 0, (π β 1), π) β π))) Β· ((π₯ β π½)β(if(π½ = 0, (π β 1), π) β π)))) = 0) |
35 | 34 | mpteq2dv 5250 | . 2 β’ (π β (π₯ β π β¦ if(if(π½ = 0, (π β 1), π) < π, 0, (((!βif(π½ = 0, (π β 1), π)) / (!β(if(π½ = 0, (π β 1), π) β π))) Β· ((π₯ β π½)β(if(π½ = 0, (π β 1), π) β π))))) = (π₯ β π β¦ 0)) |
36 | 33, 35 | eqtrd 2771 | 1 β’ (π β ((π Dπ (π»βπ½))βπ) = (π₯ β π β¦ 0)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 ifcif 4528 {cpr 4630 class class class wbr 5148 β¦ cmpt 5231 βcfv 6543 (class class class)co 7412 βcc 11112 βcr 11113 0cc0 11114 1c1 11115 Β· cmul 11119 < clt 11253 β€ cle 11254 β cmin 11449 / cdiv 11876 βcn 12217 β0cn0 12477 β€cz 12563 ...cfz 13489 βcexp 14032 !cfa 14238 βΎt crest 17371 TopOpenctopn 17372 βfldccnfld 21145 Dπ cdvn 25614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-icc 13336 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-fac 14239 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-limc 25616 df-dv 25617 df-dvn 25618 |
This theorem is referenced by: etransclem32 45281 |
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