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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvsconst | Structured version Visualization version GIF version |
Description: Derivative of a constant function on the real or complex numbers. The function may return a complex π΄ even if π is β. (Contributed by Steve Rodriguez, 11-Nov-2015.) |
Ref | Expression |
---|---|
dvsconst | β’ ((π β {β, β} β§ π΄ β β) β (π D (π Γ {π΄})) = (π Γ {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconst6g 6791 | . . . 4 β’ (π΄ β β β (β Γ {π΄}):ββΆβ) | |
2 | 1 | anim2i 615 | . . 3 β’ ((π β {β, β} β§ π΄ β β) β (π β {β, β} β§ (β Γ {π΄}):ββΆβ)) |
3 | recnprss 25853 | . . . . . . 7 β’ (π β {β, β} β π β β) | |
4 | c0ex 11246 | . . . . . . . . 9 β’ 0 β V | |
5 | 4 | fconst 6788 | . . . . . . . 8 β’ (β Γ {0}):ββΆ{0} |
6 | 5 | fdmi 6739 | . . . . . . 7 β’ dom (β Γ {0}) = β |
7 | 3, 6 | sseqtrrdi 4033 | . . . . . 6 β’ (π β {β, β} β π β dom (β Γ {0})) |
8 | 7 | adantr 479 | . . . . 5 β’ ((π β {β, β} β§ π΄ β β) β π β dom (β Γ {0})) |
9 | dvconst 25866 | . . . . . . 7 β’ (π΄ β β β (β D (β Γ {π΄})) = (β Γ {0})) | |
10 | 9 | adantl 480 | . . . . . 6 β’ ((π β {β, β} β§ π΄ β β) β (β D (β Γ {π΄})) = (β Γ {0})) |
11 | 10 | dmeqd 5912 | . . . . 5 β’ ((π β {β, β} β§ π΄ β β) β dom (β D (β Γ {π΄})) = dom (β Γ {0})) |
12 | 8, 11 | sseqtrrd 4023 | . . . 4 β’ ((π β {β, β} β§ π΄ β β) β π β dom (β D (β Γ {π΄}))) |
13 | ssid 4004 | . . . 4 β’ β β β | |
14 | 12, 13 | jctil 518 | . . 3 β’ ((π β {β, β} β§ π΄ β β) β (β β β β§ π β dom (β D (β Γ {π΄})))) |
15 | dvres3 25862 | . . 3 β’ (((π β {β, β} β§ (β Γ {π΄}):ββΆβ) β§ (β β β β§ π β dom (β D (β Γ {π΄})))) β (π D ((β Γ {π΄}) βΎ π)) = ((β D (β Γ {π΄})) βΎ π)) | |
16 | 2, 14, 15 | syl2anc 582 | . 2 β’ ((π β {β, β} β§ π΄ β β) β (π D ((β Γ {π΄}) βΎ π)) = ((β D (β Γ {π΄})) βΎ π)) |
17 | xpssres 6027 | . . . . 5 β’ (π β β β ((β Γ {π΄}) βΎ π) = (π Γ {π΄})) | |
18 | 3, 17 | syl 17 | . . . 4 β’ (π β {β, β} β ((β Γ {π΄}) βΎ π) = (π Γ {π΄})) |
19 | 18 | oveq2d 7442 | . . 3 β’ (π β {β, β} β (π D ((β Γ {π΄}) βΎ π)) = (π D (π Γ {π΄}))) |
20 | 19 | adantr 479 | . 2 β’ ((π β {β, β} β§ π΄ β β) β (π D ((β Γ {π΄}) βΎ π)) = (π D (π Γ {π΄}))) |
21 | 10 | reseq1d 5988 | . . 3 β’ ((π β {β, β} β§ π΄ β β) β ((β D (β Γ {π΄})) βΎ π) = ((β Γ {0}) βΎ π)) |
22 | xpssres 6027 | . . . . 5 β’ (π β β β ((β Γ {0}) βΎ π) = (π Γ {0})) | |
23 | 3, 22 | syl 17 | . . . 4 β’ (π β {β, β} β ((β Γ {0}) βΎ π) = (π Γ {0})) |
24 | 23 | adantr 479 | . . 3 β’ ((π β {β, β} β§ π΄ β β) β ((β Γ {0}) βΎ π) = (π Γ {0})) |
25 | 21, 24 | eqtrd 2768 | . 2 β’ ((π β {β, β} β§ π΄ β β) β ((β D (β Γ {π΄})) βΎ π) = (π Γ {0})) |
26 | 16, 20, 25 | 3eqtr3d 2776 | 1 β’ ((π β {β, β} β§ π΄ β β) β (π D (π Γ {π΄})) = (π Γ {0})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3949 {csn 4632 {cpr 4634 Γ cxp 5680 dom cdm 5682 βΎ cres 5684 βΆwf 6549 (class class class)co 7426 βcc 11144 βcr 11145 0cc0 11146 D cdv 25812 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fi 9442 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-icc 13371 df-fz 13525 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-rest 17411 df-topn 17412 df-topgen 17432 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-cncf 24818 df-limc 25815 df-dv 25816 |
This theorem is referenced by: dvconstbi 43802 |
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