expnegap0 - Intuitionistic Logic Explorer

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Theorem expnegap0 10142

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 8831	. . 3 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2		nnne0 8606	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0)
3	2	adantl 273	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0)
4		nncn 8586	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
5	4	adantl 273	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
6	5	negeq0d 7936	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 = 0 ↔ -𝑁 = 0))
7	6	necon3abid 2306	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 ≠ 0 ↔ ¬ -𝑁 = 0))
8	3, 7	mpbid 146	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ¬ -𝑁 = 0)
9	8	iffalsed 3431	. . . . . . 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 8836	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
11	10	adantl 273	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ₀)
12		nn0nlt0 8855	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → ¬ 𝑁 < 0)
13	11, 12	syl 14	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ¬ 𝑁 < 0)
14	11	nn0red 8883	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
15	14	lt0neg1d 8144	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑁 < 0 ↔ 0 < -𝑁))
16	13, 15	mtbid 638	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ¬ 0 < -𝑁)
17	16	iffalsed 3431	. . . . . . 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 7935	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → --𝑁 = 𝑁)
19	18	fveq2d 5357	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}))‘--𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
20	19	oveq2d 5722	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (1 / (seq1( · , (ℕ × {𝐴}))‘--𝑁)) = (1 / (seq1( · , (ℕ × {𝐴}))‘𝑁)))
21	9, 17, 20	3eqtrd 2136	. . . . . 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 464	. . . . 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 949	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℂ)
24		simp3 951	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
25	24	nnzd 9024	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
26	25	znegcld 9027	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → -𝑁 ∈ ℤ)
27		simp2 950	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → 𝐴 # 0)
28	27	orcd 693	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → (𝐴 # 0 ∨ 0 ≤ -𝑁))
29		exp3val 10136	. . . . . . 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 1184	. . . . . 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 1149	. . . . 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 10137	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
33	32	oveq2d 5722	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (1 / (𝐴↑𝑁)) = (1 / (seq1( · , (ℕ × {𝐴}))‘𝑁)))
34	33	adantlr 464	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑁 ∈ ℕ) → (1 / (𝐴↑𝑁)) = (1 / (seq1( · , (ℕ × {𝐴}))‘𝑁)))
35	22, 31, 34	3eqtr4d 2142	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑁 ∈ ℕ) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
36		1div1e1 8325	. . . . . . 7 ⊢ (1 / 1) = 1
37	36	eqcomi 2104	. . . . . 6 ⊢ 1 = (1 / 1)
38		negeq 7826	. . . . . . . . 9 ⊢ (𝑁 = 0 → -𝑁 = -0)
39		neg0 7879	. . . . . . . . 9 ⊢ -0 = 0
40	38, 39	syl6eq 2148	. . . . . . . 8 ⊢ (𝑁 = 0 → -𝑁 = 0)
41	40	oveq2d 5722	. . . . . . 7 ⊢ (𝑁 = 0 → (𝐴↑-𝑁) = (𝐴↑0))
42		exp0 10138	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
43	41, 42	sylan9eqr 2154	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑-𝑁) = 1)
44		oveq2 5714	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0))
45	44, 42	sylan9eqr 2154	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑𝑁) = 1)
46	45	oveq2d 5722	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (1 / (𝐴↑𝑁)) = (1 / 1))
47	37, 43, 46	3eqtr4a 2158	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 = 0) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
48	47	adantlr 464	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑁 = 0) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
49	35, 48	jaodan 752	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
50	1, 49	sylan2b 283	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑁 ∈ ℕ₀) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
51	50	3impa 1144	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 103 ∨ wo 670 ∧ w3a 930 = wceq 1299 ∈ wcel 1448 ≠ wne 2267 ifcif 3421 {csn 3474 class class class wbr 3875 × cxp 4475 ‘cfv 5059 (class class class)co 5706 ℂcc 7498 0cc0 7500 1c1 7501 · cmul 7505 < clt 7672 ≤ cle 7673 -cneg 7805 # cap 8209 / cdiv 8293 ℕcn 8578 ℕ₀cn0 8829 ℤcz 8906 seqcseq 10059 ↑cexp 10133

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613

This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-if 3422 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-ilim 4229 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-frec 6218 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-n0 8830 df-z 8907 df-uz 9177 df-seqfrec 10060 df-exp 10134

This theorem is referenced by: expineg2 10143 expn1ap0 10144 expnegzap 10168 efexp 11186 ex-exp 12542

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