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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 11154 | . . . 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 44279 | . . . . . 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 3949 | . . . 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 11184 | . . 3 β’ ((π β§ π₯ β π) β (π₯ + π΄) β β) |
12 | 1red 11166 | . . . 4 β’ ((π β§ π₯ β π) β 1 β β) | |
13 | 0red 11168 | . . . 4 β’ ((π β§ π₯ β π) β 0 β β) | |
14 | 12, 13 | readdcld 11194 | . . 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 12483 | . . . . 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 14062 | . . 3 β’ ((π β§ π¦ β β) β (π¦βπΎ) β β) |
20 | 18 | nn0cnd 12485 | . . . 4 β’ ((π β§ π¦ β β) β πΎ β β) |
21 | nnm1nn0 12464 | . . . . . . 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 14062 | . . . 4 β’ ((π β§ π¦ β β) β (π¦β(πΎ β 1)) β β) |
25 | 20, 24 | mulcld 11185 | . . 3 β’ ((π β§ π¦ β β) β (πΎ Β· (π¦β(πΎ β 1))) β β) |
26 | 1, 4 | dvmptidg 44260 | . . . 4 β’ (π β (π D (π₯ β π β¦ π₯)) = (π₯ β π β¦ 1)) |
27 | 1, 4, 9 | dvmptconst 44258 | . . . 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 25362 | . . 3 β’ (π β (π D (π₯ β π β¦ (π₯ + π΄))) = (π₯ β π β¦ (1 + 0))) |
29 | dvexp 25355 | . . . 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 7370 | . . 3 β’ (π¦ = (π₯ + π΄) β (π¦βπΎ) = ((π₯ + π΄)βπΎ)) | |
32 | oveq1 7370 | . . . 4 β’ (π¦ = (π₯ + π΄) β (π¦β(πΎ β 1)) = ((π₯ + π΄)β(πΎ β 1))) | |
33 | 32 | oveq2d 7379 | . . 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 25374 | . 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 12287 | . . . . . 6 β’ (1 + 0) = 1 | |
36 | 35 | oveq2i 7374 | . . . . 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 12179 | . . . . . . 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 14062 | . . . . . 6 β’ ((π β§ π₯ β π) β ((π₯ + π΄)β(πΎ β 1)) β β) |
42 | 39, 41 | mulcld 11185 | . . . . 5 β’ ((π β§ π₯ β π) β (πΎ Β· ((π₯ + π΄)β(πΎ β 1))) β β) |
43 | 42 | mulridd 11182 | . . . 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 2772 | . . 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 5211 | . 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 2772 | 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 3914 {cpr 4594 β¦ cmpt 5194 βcfv 6502 (class class class)co 7363 βcc 11059 βcr 11060 0cc0 11061 1c1 11062 + caddc 11064 Β· cmul 11066 β cmin 11395 βcn 12163 β0cn0 12423 βcexp 13978 βΎt crest 17317 TopOpenctopn 17318 βfldccnfld 20834 D cdv 25265 |
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 2703 ax-rep 5248 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-resscn 11118 ax-1cn 11119 ax-icn 11120 ax-addcl 11121 ax-addrcl 11122 ax-mulcl 11123 ax-mulrcl 11124 ax-mulcom 11125 ax-addass 11126 ax-mulass 11127 ax-distr 11128 ax-i2m1 11129 ax-1ne0 11130 ax-1rid 11131 ax-rnegex 11132 ax-rrecex 11133 ax-cnre 11134 ax-pre-lttri 11135 ax-pre-lttrn 11136 ax-pre-ltadd 11137 ax-pre-mulgt0 11138 ax-pre-sup 11139 ax-addf 11140 ax-mulf 11141 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4872 df-int 4914 df-iun 4962 df-iin 4963 df-br 5112 df-opab 5174 df-mpt 5195 df-tr 5229 df-id 5537 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5594 df-se 5595 df-we 5596 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-pred 6259 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7319 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7623 df-om 7809 df-1st 7927 df-2nd 7928 df-supp 8099 df-frecs 8218 df-wrecs 8249 df-recs 8323 df-rdg 8362 df-1o 8418 df-2o 8419 df-er 8656 df-map 8775 df-pm 8776 df-ixp 8844 df-en 8892 df-dom 8893 df-sdom 8894 df-fin 8895 df-fsupp 9314 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9456 df-card 9885 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 df-sub 11397 df-neg 11398 df-div 11823 df-nn 12164 df-2 12226 df-3 12227 df-4 12228 df-5 12229 df-6 12230 df-7 12231 df-8 12232 df-9 12233 df-n0 12424 df-z 12510 df-dec 12629 df-uz 12774 df-q 12884 df-rp 12926 df-xneg 13043 df-xadd 13044 df-xmul 13045 df-icc 13282 df-fz 13436 df-fzo 13579 df-seq 13918 df-exp 13979 df-hash 14242 df-cj 14997 df-re 14998 df-im 14999 df-sqrt 15133 df-abs 15134 df-struct 17031 df-sets 17048 df-slot 17066 df-ndx 17078 df-base 17096 df-ress 17125 df-plusg 17161 df-mulr 17162 df-starv 17163 df-sca 17164 df-vsca 17165 df-ip 17166 df-tset 17167 df-ple 17168 df-ds 17170 df-unif 17171 df-hom 17172 df-cco 17173 df-rest 17319 df-topn 17320 df-0g 17338 df-gsum 17339 df-topgen 17340 df-pt 17341 df-prds 17344 df-xrs 17399 df-qtop 17404 df-imas 17405 df-xps 17407 df-mre 17481 df-mrc 17482 df-acs 17484 df-mgm 18512 df-sgrp 18561 df-mnd 18572 df-submnd 18617 df-mulg 18888 df-cntz 19112 df-cmn 19579 df-psmet 20826 df-xmet 20827 df-met 20828 df-bl 20829 df-mopn 20830 df-fbas 20831 df-fg 20832 df-cnfld 20835 df-top 22281 df-topon 22298 df-topsp 22320 df-bases 22334 df-cld 22408 df-ntr 22409 df-cls 22410 df-nei 22487 df-lp 22525 df-perf 22526 df-cn 22616 df-cnp 22617 df-haus 22704 df-tx 22951 df-hmeo 23144 df-fil 23235 df-fm 23327 df-flim 23328 df-flf 23329 df-xms 23711 df-ms 23712 df-tms 23713 df-cncf 24279 df-limc 25268 df-dv 25269 |
This theorem is referenced by: dvnxpaek 44285 |
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