expnegap0 - Intuitionistic Logic Explorer

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Theorem expnegap0 10528

Description: Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)

**Assertion**
Ref	Expression
expnegap0	⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ₀) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))

**Proof of Theorem expnegap0**
Step	Hyp	Ref	Expression
1		elnn0 9178	. . 3 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		nnne0 8947	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
3	2	adantl 277	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0)
4		nncn 8927	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
5	4	adantl 277	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
6	5	negeq0d 8260	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 = 0 ↔ -𝑁 = 0))
7	6	necon3abid 2386	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 ≠ 0 ↔ ¬ -𝑁 = 0))
8	3, 7	mpbid 147	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ¬ -𝑁 = 0)
9	8	iffalsed 3545	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → if(-𝑁 = 0, 1, if(0 < -𝑁, (seq1( · , (ℕ × {𝐴}))‘-𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘--𝑁)))) = if(0 < -𝑁, (seq1( · , (ℕ × {𝐴}))‘-𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘--𝑁))))
10		nnnn0 9183	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
11	10	adantl 277	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
12		nn0nlt0 9202	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → ¬ 𝑁 < 0)
13	11, 12	syl 14	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ¬ 𝑁 < 0)
14	11	nn0red 9230	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
15	14	lt0neg1d 8472	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 < 0 ↔ 0 < -𝑁))
16	13, 15	mtbid 672	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ¬ 0 < -𝑁)
17	16	iffalsed 3545	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → if(0 < -𝑁, (seq1( · , (ℕ × {𝐴}))‘-𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘--𝑁))) = (1 / (seq1( · , (ℕ × {𝐴}))‘--𝑁)))
18	5	negnegd 8259	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → --𝑁 = 𝑁)
19	18	fveq2d 5520	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}))‘--𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
20	19	oveq2d 5891	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (1 / (seq1( · , (ℕ × {𝐴}))‘--𝑁)) = (1 / (seq1( · , (ℕ × {𝐴}))‘𝑁)))
21	9, 17, 20	3eqtrd 2214	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → if(-𝑁 = 0, 1, if(0 < -𝑁, (seq1( · , (ℕ × {𝐴}))‘-𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘--𝑁)))) = (1 / (seq1( · , (ℕ × {𝐴}))‘𝑁)))
22	21	adantlr 477	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑁 ∈ ℕ) → if(-𝑁 = 0, 1, if(0 < -𝑁, (seq1( · , (ℕ × {𝐴}))‘-𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘--𝑁)))) = (1 / (seq1( · , (ℕ × {𝐴}))‘𝑁)))
23		simp1 997	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℂ)
24		simp3 999	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
25	24	nnzd 9374	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
26	25	znegcld 9377	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → -𝑁 ∈ ℤ)
27		simp2 998	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → 𝐴 # 0)
28	27	orcd 733	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → (𝐴 # 0 ∨ 0 ≤ -𝑁))
29		exp3val 10522	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ -𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ -𝑁)) → (𝐴↑-𝑁) = if(-𝑁 = 0, 1, if(0 < -𝑁, (seq1( · , (ℕ × {𝐴}))‘-𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘--𝑁)))))
30	23, 26, 28, 29	syl3anc 1238	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → (𝐴↑-𝑁) = if(-𝑁 = 0, 1, if(0 < -𝑁, (seq1( · , (ℕ × {𝐴}))‘-𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘--𝑁)))))
31	30	3expa 1203	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑁 ∈ ℕ) → (𝐴↑-𝑁) = if(-𝑁 = 0, 1, if(0 < -𝑁, (seq1( · , (ℕ × {𝐴}))‘-𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘--𝑁)))))
32		expnnval 10523	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
33	32	oveq2d 5891	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (1 / (𝐴↑𝑁)) = (1 / (seq1( · , (ℕ × {𝐴}))‘𝑁)))
34	33	adantlr 477	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑁 ∈ ℕ) → (1 / (𝐴↑𝑁)) = (1 / (seq1( · , (ℕ × {𝐴}))‘𝑁)))
35	22, 31, 34	3eqtr4d 2220	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑁 ∈ ℕ) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
36		1div1e1 8661	. . . . . . 7 ⊢ (1 / 1) = 1
37	36	eqcomi 2181	. . . . . 6 ⊢ 1 = (1 / 1)
38		negeq 8150	. . . . . . . . 9 ⊢ (𝑁 = 0 → -𝑁 = -0)
39		neg0 8203	. . . . . . . . 9 ⊢ -0 = 0
40	38, 39	eqtrdi 2226	. . . . . . . 8 ⊢ (𝑁 = 0 → -𝑁 = 0)
41	40	oveq2d 5891	. . . . . . 7 ⊢ (𝑁 = 0 → (𝐴↑-𝑁) = (𝐴↑0))
42		exp0 10524	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
43	41, 42	sylan9eqr 2232	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑-𝑁) = 1)
44		oveq2 5883	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0))
45	44, 42	sylan9eqr 2232	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1)
46	45	oveq2d 5891	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (1 / (𝐴↑𝑁)) = (1 / 1))
47	37, 43, 46	3eqtr4a 2236	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
48	47	adantlr 477	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑁 = 0) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
49	35, 48	jaodan 797	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
50	1, 49	sylan2b 287	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑁 ∈ ℕ₀) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
51	50	3impa 1194	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ₀) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 708 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ifcif 3535 {csn 3593 class class class wbr 4004 × cxp 4625 ‘cfv 5217 (class class class)co 5875 ℂcc 7809 0cc0 7811 1c1 7812 · cmul 7816 < clt 7992 ≤ cle 7993 -cneg 8129 # cap 8538 / cdiv 8629 ℕcn 8919 ℕ₀cn0 9176 ℤcz 9253 seqcseq 10445 ↑cexp 10519

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-n0 9177 df-z 9254 df-uz 9529 df-seqfrec 10446 df-exp 10520

This theorem is referenced by: expineg2 10529 expn1ap0 10530 expnegzap 10554 efexp 11690 pcexp 12309 ex-exp 14482

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