cxpexp - Metamath Proof Explorer

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Theorem cxpexp 26595

Description: Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

**Assertion**
Ref	Expression
cxpexp	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ₀) → (𝐴↑_𝑐𝐵) = (𝐴↑𝐵))

**Proof of Theorem cxpexp**
Step	Hyp	Ref	Expression
1		elnn0 12498	. . . . . 6 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
2		nncn 12244	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
3		nnne0 12270	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0)
4		0cxp 26593	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (0↑_𝑐𝐵) = 0)
5	2, 3, 4	syl2anc 583	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → (0↑_𝑐𝐵) = 0)
6		0exp 14088	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → (0↑𝐵) = 0)
7	5, 6	eqtr4d 2771	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → (0↑_𝑐𝐵) = (0↑𝐵))
8		0cn 11230	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
9		cxpval 26591	. . . . . . . . . . 11 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ) → (0↑_𝑐0) = if(0 = 0, if(0 = 0, 1, 0), (exp‘(0 · (log‘0)))))
10	8, 8, 9	mp2an 691	. . . . . . . . . 10 ⊢ (0↑_𝑐0) = if(0 = 0, if(0 = 0, 1, 0), (exp‘(0 · (log‘0))))
11		eqid 2728	. . . . . . . . . . 11 ⊢ 0 = 0
12	11	iftruei 4531	. . . . . . . . . 10 ⊢ if(0 = 0, if(0 = 0, 1, 0), (exp‘(0 · (log‘0)))) = if(0 = 0, 1, 0)
13	11	iftruei 4531	. . . . . . . . . 10 ⊢ if(0 = 0, 1, 0) = 1
14	10, 12, 13	3eqtri 2760	. . . . . . . . 9 ⊢ (0↑_𝑐0) = 1
15		0exp0e1 14057	. . . . . . . . 9 ⊢ (0↑0) = 1
16	14, 15	eqtr4i 2759	. . . . . . . 8 ⊢ (0↑_𝑐0) = (0↑0)
17		oveq2 7422	. . . . . . . 8 ⊢ (𝐵 = 0 → (0↑_𝑐𝐵) = (0↑_𝑐0))
18		oveq2 7422	. . . . . . . 8 ⊢ (𝐵 = 0 → (0↑𝐵) = (0↑0))
19	16, 17, 18	3eqtr4a 2794	. . . . . . 7 ⊢ (𝐵 = 0 → (0↑_𝑐𝐵) = (0↑𝐵))
20	7, 19	jaoi 856	. . . . . 6 ⊢ ((𝐵 ∈ ℕ ∨ 𝐵 = 0) → (0↑_𝑐𝐵) = (0↑𝐵))
21	1, 20	sylbi 216	. . . . 5 ⊢ (𝐵 ∈ ℕ₀ → (0↑_𝑐𝐵) = (0↑𝐵))
22		oveq1 7421	. . . . . 6 ⊢ (𝐴 = 0 → (𝐴↑_𝑐𝐵) = (0↑_𝑐𝐵))
23		oveq1 7421	. . . . . 6 ⊢ (𝐴 = 0 → (𝐴↑𝐵) = (0↑𝐵))
24	22, 23	eqeq12d 2744	. . . . 5 ⊢ (𝐴 = 0 → ((𝐴↑_𝑐𝐵) = (𝐴↑𝐵) ↔ (0↑_𝑐𝐵) = (0↑𝐵)))
25	21, 24	syl5ibrcom 246	. . . 4 ⊢ (𝐵 ∈ ℕ₀ → (𝐴 = 0 → (𝐴↑_𝑐𝐵) = (𝐴↑𝐵)))
26	25	adantl 481	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ₀) → (𝐴 = 0 → (𝐴↑_𝑐𝐵) = (𝐴↑𝐵)))
27	26	imp 406	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ₀) ∧ 𝐴 = 0) → (𝐴↑_𝑐𝐵) = (𝐴↑𝐵))
28		nn0z 12607	. . . 4 ⊢ (𝐵 ∈ ℕ₀ → 𝐵 ∈ ℤ)
29		cxpexpz 26594	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℤ) → (𝐴↑_𝑐𝐵) = (𝐴↑𝐵))
30	29	3expa 1116	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℤ) → (𝐴↑_𝑐𝐵) = (𝐴↑𝐵))
31	28, 30	sylan2 592	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℕ₀) → (𝐴↑_𝑐𝐵) = (𝐴↑𝐵))
32	31	an32s 651	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ₀) ∧ 𝐴 ≠ 0) → (𝐴↑_𝑐𝐵) = (𝐴↑𝐵))
33	27, 32	pm2.61dane 3025	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ₀) → (𝐴↑_𝑐𝐵) = (𝐴↑𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ifcif 4524 ‘cfv 6542 (class class class)co 7414 ℂcc 11130 0cc0 11132 1c1 11133 · cmul 11137 ℕcn 12236 ℕ₀cn0 12496 ℤcz 12582 ↑cexp 14052 expce 16031 logclog 26481 ↑_𝑐ccxp 26482

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ioo 13354 df-ioc 13355 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-fl 13783 df-mod 13861 df-seq 13993 df-exp 14053 df-fac 14259 df-bc 14288 df-hash 14316 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15441 df-clim 15458 df-rlim 15459 df-sum 15659 df-ef 16037 df-sin 16039 df-cos 16040 df-pi 16042 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-mulg 19017 df-cntz 19261 df-cmn 19730 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-fbas 21269 df-fg 21270 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-nei 22995 df-lp 23033 df-perf 23034 df-cn 23124 df-cnp 23125 df-haus 23212 df-tx 23459 df-hmeo 23652 df-fil 23743 df-fm 23835 df-flim 23836 df-flf 23837 df-xms 24219 df-ms 24220 df-tms 24221 df-cncf 24791 df-limc 25788 df-dv 25789 df-log 26483 df-cxp 26484

This theorem is referenced by: cxp0 26597 cxp1 26598 root1id 26682 dfef2 26896 zetacvg 26940 sgmppw 27123 chpchtsum 27145 logexprlim 27151 dchrabs 27186 bposlem5 27214 bposlem6 27215 ostth2lem3 27561 aks4d1p1p2 41535 aks4d1p1p4 41536 binomcxplemnn0 43780 binomcxplemnotnn0 43787 etransclem46 45662

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