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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 11226 | . . . 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 45315 | . . . . . 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 3980 | . . . 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 11258 | . . 3 β’ ((π β§ π₯ β π) β (π₯ + π΄) β β) |
12 | 1red 11240 | . . . 4 β’ ((π β§ π₯ β π) β 1 β β) | |
13 | 0red 11242 | . . . 4 β’ ((π β§ π₯ β π) β 0 β β) | |
14 | 12, 13 | readdcld 11268 | . . 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 12557 | . . . . 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 14137 | . . 3 β’ ((π β§ π¦ β β) β (π¦βπΎ) β β) |
20 | 18 | nn0cnd 12559 | . . . 4 β’ ((π β§ π¦ β β) β πΎ β β) |
21 | nnm1nn0 12538 | . . . . . . 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 14137 | . . . 4 β’ ((π β§ π¦ β β) β (π¦β(πΎ β 1)) β β) |
25 | 20, 24 | mulcld 11259 | . . 3 β’ ((π β§ π¦ β β) β (πΎ Β· (π¦β(πΎ β 1))) β β) |
26 | 1, 4 | dvmptidg 45296 | . . . 4 β’ (π β (π D (π₯ β π β¦ π₯)) = (π₯ β π β¦ 1)) |
27 | 1, 4, 9 | dvmptconst 45294 | . . . 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 25886 | . . 3 β’ (π β (π D (π₯ β π β¦ (π₯ + π΄))) = (π₯ β π β¦ (1 + 0))) |
29 | dvexp 25879 | . . . 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 7422 | . . 3 β’ (π¦ = (π₯ + π΄) β (π¦βπΎ) = ((π₯ + π΄)βπΎ)) | |
32 | oveq1 7422 | . . . 4 β’ (π¦ = (π₯ + π΄) β (π¦β(πΎ β 1)) = ((π₯ + π΄)β(πΎ β 1))) | |
33 | 32 | oveq2d 7431 | . . 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 25898 | . 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 12361 | . . . . . 6 β’ (1 + 0) = 1 | |
36 | 35 | oveq2i 7426 | . . . . 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 12253 | . . . . . . 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 14137 | . . . . . 6 β’ ((π β§ π₯ β π) β ((π₯ + π΄)β(πΎ β 1)) β β) |
42 | 39, 41 | mulcld 11259 | . . . . 5 β’ ((π β§ π₯ β π) β (πΎ Β· ((π₯ + π΄)β(πΎ β 1))) β β) |
43 | 42 | mulridd 11256 | . . . 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 2768 | . . 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 5243 | . 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 2768 | 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 1534 β wcel 2099 β wss 3945 {cpr 4627 β¦ cmpt 5226 βcfv 6543 (class class class)co 7415 βcc 11131 βcr 11132 0cc0 11133 1c1 11134 + caddc 11136 Β· cmul 11138 β cmin 11469 βcn 12237 β0cn0 12497 βcexp 14053 βΎt crest 17396 TopOpenctopn 17397 βfldccnfld 21273 D cdv 25786 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212 |
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 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7866 df-1st 7988 df-2nd 7989 df-supp 8161 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-2o 8482 df-er 8719 df-map 8841 df-pm 8842 df-ixp 8911 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-fsupp 9381 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-icc 13358 df-fz 13512 df-fzo 13655 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17398 df-topn 17399 df-0g 17417 df-gsum 17418 df-topgen 17419 df-pt 17420 df-prds 17423 df-xrs 17478 df-qtop 17483 df-imas 17484 df-xps 17486 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-mulg 19018 df-cntz 19262 df-cmn 19731 df-psmet 21265 df-xmet 21266 df-met 21267 df-bl 21268 df-mopn 21269 df-fbas 21270 df-fg 21271 df-cnfld 21274 df-top 22790 df-topon 22807 df-topsp 22829 df-bases 22843 df-cld 22917 df-ntr 22918 df-cls 22919 df-nei 22996 df-lp 23034 df-perf 23035 df-cn 23125 df-cnp 23126 df-haus 23213 df-tx 23460 df-hmeo 23653 df-fil 23744 df-fm 23836 df-flim 23837 df-flf 23838 df-xms 24220 df-ms 24221 df-tms 24222 df-cncf 24792 df-limc 25789 df-dv 25790 |
This theorem is referenced by: dvnxpaek 45321 |
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