dvexp2 - Metamath Proof Explorer

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Theorem dvexp2 26283

Description: Derivative of an exponential, possibly zero power. (Contributed by Stefan O'Rear, 13-Nov-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

**Assertion**
Ref	Expression
dvexp2	⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))

Distinct variable group: 𝑥,𝑁

**Proof of Theorem dvexp2**
Step	Hyp	Ref	Expression
1		elnn0 12615	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		dvexp 26282	. . . 4 ⊢ (𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
3		nnne0 12379	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
4	3	neneqd 2960	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = 0)
5	4	iffalsed 4492	. . . . 5 ⊢ (𝑁 ∈ ℕ → if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
6	5	mpteq2dv 5198	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
7	2, 6	eqtr4d 2798	. . 3 ⊢ (𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
8		oveq2 7432	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝑥↑𝑁) = (𝑥↑0))
9		exp0 14215	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
10	8, 9	sylan9eq 2815	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑁) = 1)
11	10	mpteq2dva 5197	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ 1))
12		fconstmpt 5713	. . . . . . . 8 ⊢ (ℂ × {1}) = (𝑥 ∈ ℂ ↦ 1)
13	11, 12	eqtr4di 2813	. . . . . . 7 ⊢ (𝑁 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (ℂ × {1}))
14	13	oveq2d 7440	. . . . . 6 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (ℂ D (ℂ × {1})))
15		ax-1cn 11265	. . . . . . 7 ⊢ 1 ∈ ℂ
16		dvconst 26246	. . . . . . 7 ⊢ (1 ∈ ℂ → (ℂ D (ℂ × {1})) = (ℂ × {0}))
17	15, 16	ax-mp 5	. . . . . 6 ⊢ (ℂ D (ℂ × {1})) = (ℂ × {0})
18	14, 17	eqtrdi 2811	. . . . 5 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (ℂ × {0}))
19		fconstmpt 5713	. . . . 5 ⊢ (ℂ × {0}) = (𝑥 ∈ ℂ ↦ 0)
20	18, 19	eqtrdi 2811	. . . 4 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ 0))
21		iftrue 4487	. . . . 5 ⊢ (𝑁 = 0 → if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))) = 0)
22	21	mpteq2dv 5198	. . . 4 ⊢ (𝑁 = 0 → (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))) = (𝑥 ∈ ℂ ↦ 0))
23	20, 22	eqtr4d 2798	. . 3 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
24	7, 23	jaoi 871	. 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
25	1, 24	sylbi 220	1 ⊢ (𝑁 ∈ ℕ₀ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ifcif 4481 {csn 4584 ↦ cmpt 5185 × cxp 5649 (class class class)co 7424 ℂcc 11205 0cc0 11207 1c1 11208 · cmul 11212 − cmin 11548 ℕcn 12342 ℕ₀cn0 12613 ↑cexp 14211 D cdv 26192

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 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 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-er 8725 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-icc 13490 df-fz 13647 df-fzo 13796 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-mulg 19287 df-cntz 19540 df-cmn 20005 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-lp 23463 df-perf 23464 df-cn 23554 df-cnp 23555 df-haus 23642 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-limc 26195 df-dv 26196

This theorem is used by: dvexp3 26307 dvply1 26614 dvtaylp 26706 pserdvlem2 26764

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