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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvxpaek | Structured version Visualization version GIF version |
Description: Derivative of the polynomial (π₯ + π΄)βπΎ. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
dvxpaek.s | β’ (π β π β {β, β}) |
dvxpaek.x | β’ (π β π β ((TopOpenββfld) βΎt π)) |
dvxpaek.a | β’ (π β π΄ β β) |
dvxpaek.k | β’ (π β πΎ β β) |
Ref | Expression |
---|---|
dvxpaek | β’ (π β (π D (π₯ β π β¦ ((π₯ + π΄)βπΎ))) = (π₯ β π β¦ (πΎ Β· ((π₯ + π΄)β(πΎ β 1))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvxpaek.s | . . 3 β’ (π β π β {β, β}) | |
2 | cnelprrecn 11200 | . . . 4 β’ β β {β, β} | |
3 | 2 | a1i 11 | . . 3 β’ (π β β β {β, β}) |
4 | dvxpaek.x | . . . . . . 7 β’ (π β π β ((TopOpenββfld) βΎt π)) | |
5 | 1, 4 | dvdmsscn 45162 | . . . . . 6 β’ (π β π β β) |
6 | 5 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β π) β π β β) |
7 | simpr 484 | . . . . 5 β’ ((π β§ π₯ β π) β π₯ β π) | |
8 | 6, 7 | sseldd 3976 | . . . 4 β’ ((π β§ π₯ β π) β π₯ β β) |
9 | dvxpaek.a | . . . . 5 β’ (π β π΄ β β) | |
10 | 9 | adantr 480 | . . . 4 β’ ((π β§ π₯ β π) β π΄ β β) |
11 | 8, 10 | addcld 11231 | . . 3 β’ ((π β§ π₯ β π) β (π₯ + π΄) β β) |
12 | 1red 11213 | . . . 4 β’ ((π β§ π₯ β π) β 1 β β) | |
13 | 0red 11215 | . . . 4 β’ ((π β§ π₯ β π) β 0 β β) | |
14 | 12, 13 | readdcld 11241 | . . 3 β’ ((π β§ π₯ β π) β (1 + 0) β β) |
15 | simpr 484 | . . . 4 β’ ((π β§ π¦ β β) β π¦ β β) | |
16 | dvxpaek.k | . . . . . 6 β’ (π β πΎ β β) | |
17 | 16 | nnnn0d 12530 | . . . . 5 β’ (π β πΎ β β0) |
18 | 17 | adantr 480 | . . . 4 β’ ((π β§ π¦ β β) β πΎ β β0) |
19 | 15, 18 | expcld 14109 | . . 3 β’ ((π β§ π¦ β β) β (π¦βπΎ) β β) |
20 | 18 | nn0cnd 12532 | . . . 4 β’ ((π β§ π¦ β β) β πΎ β β) |
21 | nnm1nn0 12511 | . . . . . . 7 β’ (πΎ β β β (πΎ β 1) β β0) | |
22 | 16, 21 | syl 17 | . . . . . 6 β’ (π β (πΎ β 1) β β0) |
23 | 22 | adantr 480 | . . . . 5 β’ ((π β§ π¦ β β) β (πΎ β 1) β β0) |
24 | 15, 23 | expcld 14109 | . . . 4 β’ ((π β§ π¦ β β) β (π¦β(πΎ β 1)) β β) |
25 | 20, 24 | mulcld 11232 | . . 3 β’ ((π β§ π¦ β β) β (πΎ Β· (π¦β(πΎ β 1))) β β) |
26 | 1, 4 | dvmptidg 45143 | . . . 4 β’ (π β (π D (π₯ β π β¦ π₯)) = (π₯ β π β¦ 1)) |
27 | 1, 4, 9 | dvmptconst 45141 | . . . 4 β’ (π β (π D (π₯ β π β¦ π΄)) = (π₯ β π β¦ 0)) |
28 | 1, 8, 12, 26, 10, 13, 27 | dvmptadd 25816 | . . 3 β’ (π β (π D (π₯ β π β¦ (π₯ + π΄))) = (π₯ β π β¦ (1 + 0))) |
29 | dvexp 25809 | . . . 4 β’ (πΎ β β β (β D (π¦ β β β¦ (π¦βπΎ))) = (π¦ β β β¦ (πΎ Β· (π¦β(πΎ β 1))))) | |
30 | 16, 29 | syl 17 | . . 3 β’ (π β (β D (π¦ β β β¦ (π¦βπΎ))) = (π¦ β β β¦ (πΎ Β· (π¦β(πΎ β 1))))) |
31 | oveq1 7409 | . . 3 β’ (π¦ = (π₯ + π΄) β (π¦βπΎ) = ((π₯ + π΄)βπΎ)) | |
32 | oveq1 7409 | . . . 4 β’ (π¦ = (π₯ + π΄) β (π¦β(πΎ β 1)) = ((π₯ + π΄)β(πΎ β 1))) | |
33 | 32 | oveq2d 7418 | . . 3 β’ (π¦ = (π₯ + π΄) β (πΎ Β· (π¦β(πΎ β 1))) = (πΎ Β· ((π₯ + π΄)β(πΎ β 1)))) |
34 | 1, 3, 11, 14, 19, 25, 28, 30, 31, 33 | dvmptco 25828 | . 2 β’ (π β (π D (π₯ β π β¦ ((π₯ + π΄)βπΎ))) = (π₯ β π β¦ ((πΎ Β· ((π₯ + π΄)β(πΎ β 1))) Β· (1 + 0)))) |
35 | 1p0e1 12334 | . . . . . 6 β’ (1 + 0) = 1 | |
36 | 35 | oveq2i 7413 | . . . . 5 β’ ((πΎ Β· ((π₯ + π΄)β(πΎ β 1))) Β· (1 + 0)) = ((πΎ Β· ((π₯ + π΄)β(πΎ β 1))) Β· 1) |
37 | 36 | a1i 11 | . . . 4 β’ ((π β§ π₯ β π) β ((πΎ Β· ((π₯ + π΄)β(πΎ β 1))) Β· (1 + 0)) = ((πΎ Β· ((π₯ + π΄)β(πΎ β 1))) Β· 1)) |
38 | 16 | nncnd 12226 | . . . . . . 7 β’ (π β πΎ β β) |
39 | 38 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β π) β πΎ β β) |
40 | 22 | adantr 480 | . . . . . . 7 β’ ((π β§ π₯ β π) β (πΎ β 1) β β0) |
41 | 11, 40 | expcld 14109 | . . . . . 6 β’ ((π β§ π₯ β π) β ((π₯ + π΄)β(πΎ β 1)) β β) |
42 | 39, 41 | mulcld 11232 | . . . . 5 β’ ((π β§ π₯ β π) β (πΎ Β· ((π₯ + π΄)β(πΎ β 1))) β β) |
43 | 42 | mulridd 11229 | . . . 4 β’ ((π β§ π₯ β π) β ((πΎ Β· ((π₯ + π΄)β(πΎ β 1))) Β· 1) = (πΎ Β· ((π₯ + π΄)β(πΎ β 1)))) |
44 | 37, 43 | eqtrd 2764 | . . 3 β’ ((π β§ π₯ β π) β ((πΎ Β· ((π₯ + π΄)β(πΎ β 1))) Β· (1 + 0)) = (πΎ Β· ((π₯ + π΄)β(πΎ β 1)))) |
45 | 44 | mpteq2dva 5239 | . 2 β’ (π β (π₯ β π β¦ ((πΎ Β· ((π₯ + π΄)β(πΎ β 1))) Β· (1 + 0))) = (π₯ β π β¦ (πΎ Β· ((π₯ + π΄)β(πΎ β 1))))) |
46 | 34, 45 | eqtrd 2764 | 1 β’ (π β (π D (π₯ β π β¦ ((π₯ + π΄)βπΎ))) = (π₯ β π β¦ (πΎ Β· ((π₯ + π΄)β(πΎ β 1))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wss 3941 {cpr 4623 β¦ cmpt 5222 βcfv 6534 (class class class)co 7402 βcc 11105 βcr 11106 0cc0 11107 1c1 11108 + caddc 11110 Β· cmul 11112 β cmin 11442 βcn 12210 β0cn0 12470 βcexp 14025 βΎt crest 17367 TopOpenctopn 17368 βfldccnfld 21230 D cdv 25716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-q 12931 df-rp 12973 df-xneg 13090 df-xadd 13091 df-xmul 13092 df-icc 13329 df-fz 13483 df-fzo 13626 df-seq 13965 df-exp 14026 df-hash 14289 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-starv 17213 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-unif 17221 df-hom 17222 df-cco 17223 df-rest 17369 df-topn 17370 df-0g 17388 df-gsum 17389 df-topgen 17390 df-pt 17391 df-prds 17394 df-xrs 17449 df-qtop 17454 df-imas 17455 df-xps 17457 df-mre 17531 df-mrc 17532 df-acs 17534 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-mulg 18988 df-cntz 19225 df-cmn 19694 df-psmet 21222 df-xmet 21223 df-met 21224 df-bl 21225 df-mopn 21226 df-fbas 21227 df-fg 21228 df-cnfld 21231 df-top 22720 df-topon 22737 df-topsp 22759 df-bases 22773 df-cld 22847 df-ntr 22848 df-cls 22849 df-nei 22926 df-lp 22964 df-perf 22965 df-cn 23055 df-cnp 23056 df-haus 23143 df-tx 23390 df-hmeo 23583 df-fil 23674 df-fm 23766 df-flim 23767 df-flf 23768 df-xms 24150 df-ms 24151 df-tms 24152 df-cncf 24722 df-limc 25719 df-dv 25720 |
This theorem is referenced by: dvnxpaek 45168 |
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