![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 11203 | . . . 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 44652 | . . . . . 6 β’ (π β π β β) |
6 | 5 | adantr 482 | . . . . 5 β’ ((π β§ π₯ β π) β π β β) |
7 | simpr 486 | . . . . 5 β’ ((π β§ π₯ β π) β π₯ β π) | |
8 | 6, 7 | sseldd 3984 | . . . 4 β’ ((π β§ π₯ β π) β π₯ β β) |
9 | dvxpaek.a | . . . . 5 β’ (π β π΄ β β) | |
10 | 9 | adantr 482 | . . . 4 β’ ((π β§ π₯ β π) β π΄ β β) |
11 | 8, 10 | addcld 11233 | . . 3 β’ ((π β§ π₯ β π) β (π₯ + π΄) β β) |
12 | 1red 11215 | . . . 4 β’ ((π β§ π₯ β π) β 1 β β) | |
13 | 0red 11217 | . . . 4 β’ ((π β§ π₯ β π) β 0 β β) | |
14 | 12, 13 | readdcld 11243 | . . 3 β’ ((π β§ π₯ β π) β (1 + 0) β β) |
15 | simpr 486 | . . . 4 β’ ((π β§ π¦ β β) β π¦ β β) | |
16 | dvxpaek.k | . . . . . 6 β’ (π β πΎ β β) | |
17 | 16 | nnnn0d 12532 | . . . . 5 β’ (π β πΎ β β0) |
18 | 17 | adantr 482 | . . . 4 β’ ((π β§ π¦ β β) β πΎ β β0) |
19 | 15, 18 | expcld 14111 | . . 3 β’ ((π β§ π¦ β β) β (π¦βπΎ) β β) |
20 | 18 | nn0cnd 12534 | . . . 4 β’ ((π β§ π¦ β β) β πΎ β β) |
21 | nnm1nn0 12513 | . . . . . . 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 482 | . . . . 5 β’ ((π β§ π¦ β β) β (πΎ β 1) β β0) |
24 | 15, 23 | expcld 14111 | . . . 4 β’ ((π β§ π¦ β β) β (π¦β(πΎ β 1)) β β) |
25 | 20, 24 | mulcld 11234 | . . 3 β’ ((π β§ π¦ β β) β (πΎ Β· (π¦β(πΎ β 1))) β β) |
26 | 1, 4 | dvmptidg 44633 | . . . 4 β’ (π β (π D (π₯ β π β¦ π₯)) = (π₯ β π β¦ 1)) |
27 | 1, 4, 9 | dvmptconst 44631 | . . . 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 25477 | . . 3 β’ (π β (π D (π₯ β π β¦ (π₯ + π΄))) = (π₯ β π β¦ (1 + 0))) |
29 | dvexp 25470 | . . . 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 7416 | . . 3 β’ (π¦ = (π₯ + π΄) β (π¦βπΎ) = ((π₯ + π΄)βπΎ)) | |
32 | oveq1 7416 | . . . 4 β’ (π¦ = (π₯ + π΄) β (π¦β(πΎ β 1)) = ((π₯ + π΄)β(πΎ β 1))) | |
33 | 32 | oveq2d 7425 | . . 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 25489 | . 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 12336 | . . . . . 6 β’ (1 + 0) = 1 | |
36 | 35 | oveq2i 7420 | . . . . 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 12228 | . . . . . . 7 β’ (π β πΎ β β) |
39 | 38 | adantr 482 | . . . . . 6 β’ ((π β§ π₯ β π) β πΎ β β) |
40 | 22 | adantr 482 | . . . . . . 7 β’ ((π β§ π₯ β π) β (πΎ β 1) β β0) |
41 | 11, 40 | expcld 14111 | . . . . . 6 β’ ((π β§ π₯ β π) β ((π₯ + π΄)β(πΎ β 1)) β β) |
42 | 39, 41 | mulcld 11234 | . . . . 5 β’ ((π β§ π₯ β π) β (πΎ Β· ((π₯ + π΄)β(πΎ β 1))) β β) |
43 | 42 | mulridd 11231 | . . . 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 2773 | . . 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 5249 | . 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 2773 | 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 397 = wceq 1542 β wcel 2107 β wss 3949 {cpr 4631 β¦ cmpt 5232 βcfv 6544 (class class class)co 7409 βcc 11108 βcr 11109 0cc0 11110 1c1 11111 + caddc 11113 Β· cmul 11115 β cmin 11444 βcn 12212 β0cn0 12472 βcexp 14027 βΎt crest 17366 TopOpenctopn 17367 βfldccnfld 20944 D cdv 25380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 |
This theorem is referenced by: dvnxpaek 44658 |
Copyright terms: Public domain | W3C validator |