dvexp2 - Metamath Proof Explorer

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Theorem dvexp2 26013

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 12483	. 2 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		dvexp 26012	. . . 4 ⊢ (𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
3		nnne0 12247	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
4	3	neneqd 2962	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ¬ 𝑁 = 0)
5	4	iffalsed 4491	. . . . 5 ⊢ (𝑁 ∈ ℕ → if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))) = (𝑁 · (𝑥↑(𝑁 − 1))))
6	5	mpteq2dv 5194	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))) = (𝑥 ∈ ℂ ↦ (𝑁 · (𝑥↑(𝑁 − 1)))))
7	2, 6	eqtr4d 2800	. . 3 ⊢ (𝑁 ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
8		oveq2 7404	. . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝑥↑𝑁) = (𝑥↑0))
9		exp0 14078	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
10	8, 9	sylan9eq 2817	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑁) = 1)
11	10	mpteq2dva 5193	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ 1))
12		fconstmpt 5709	. . . . . . . 8 ⊢ (ℂ × {1}) = (𝑥 ∈ ℂ ↦ 1)
13	11, 12	eqtr4di 2815	. . . . . . 7 ⊢ (𝑁 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (ℂ × {1}))
14	13	oveq2d 7412	. . . . . 6 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (ℂ D (ℂ × {1})))
15		ax-1cn 11131	. . . . . . 7 ⊢ 1 ∈ ℂ
16		dvconst 25976	. . . . . . 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 2813	. . . . 5 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (ℂ × {0}))
19		fconstmpt 5709	. . . . 5 ⊢ (ℂ × {0}) = (𝑥 ∈ ℂ ↦ 0)
20	18, 19	eqtrdi 2813	. . . 4 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ 0))
21		iftrue 4486	. . . . 5 ⊢ (𝑁 = 0 → if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1)))) = 0)
22	21	mpteq2dv 5194	. . . 4 ⊢ (𝑁 = 0 → (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))) = (𝑥 ∈ ℂ ↦ 0))
23	20, 22	eqtr4d 2800	. . 3 ⊢ (𝑁 = 0 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) = (𝑥 ∈ ℂ ↦ if(𝑁 = 0, 0, (𝑁 · (𝑥↑(𝑁 − 1))))))
24	7, 23	jaoi 868	. 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 219	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

Syntax hints: → wi 4 ∨ wo 858 = wceq 1560 ∈ wcel 2142 ifcif 4480 {csn 4582 ↦ cmpt 5181 × cxp 5645 (class class class)co 7396 ℂcc 11071 0cc0 11073 1c1 11074 · cmul 11078 − cmin 11414 ℕcn 12210 ℕ₀cn0 12481 ↑cexp 14074 D cdv 25922

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-icc 13356 df-fz 13513 df-fzo 13660 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21413 df-xmet 21414 df-met 21415 df-bl 21416 df-mopn 21417 df-fbas 21418 df-fg 21419 df-cnfld 21422 df-top 22951 df-topon 22968 df-topsp 22990 df-bases 23003 df-cld 23076 df-ntr 23077 df-cls 23078 df-nei 23155 df-lp 23193 df-perf 23194 df-cn 23284 df-cnp 23285 df-haus 23372 df-tx 23619 df-hmeo 23812 df-fil 23903 df-fm 23995 df-flim 23996 df-flf 23997 df-xms 24377 df-ms 24378 df-tms 24379 df-cncf 24937 df-limc 25925 df-dv 25926

This theorem is referenced by: dvexp3 26037 dvply1 26345 dvtaylp 26430 pserdvlem2 26488

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