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Theorem expival 9257

Description: Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)

**Assertion**
Ref	Expression
expival	⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))

Proof of Theorem expival Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iftrue 3336	. . . . 5 ⊢ (𝑁 = 0 → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) = 1)
2		ax-1cn 6977	. . . . 5 ⊢ 1 ∈ ℂ
3	1, 2	syl6eqel 2128	. . . 4 ⊢ (𝑁 = 0 → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ)
4	3	a1i 9	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝑁 = 0 → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ))
5		elnnz 8255	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
6		elnnuz 8509	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
7	5, 6	bitr3i 175	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 0 < 𝑁) ↔ 𝑁 ∈ (ℤ_≥‘1))
8	7	biimpi 113	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 0 < 𝑁) → 𝑁 ∈ (ℤ_≥‘1))
9	8	adantll 445	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → 𝑁 ∈ (ℤ_≥‘1))
10		cnex 7005	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
11	10	a1i 9	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → ℂ ∈ V)
12		simpl 102	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ (ℤ_≥‘1)) → 𝐴 ∈ ℂ)
13		elnnuz 8509	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℕ ↔ 𝑧 ∈ (ℤ_≥‘1))
14		fvconst2g 5375	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℕ) → ((ℕ × {𝐴})‘𝑧) = 𝐴)
15	13, 14	sylan2br 272	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ (ℤ_≥‘1)) → ((ℕ × {𝐴})‘𝑧) = 𝐴)
16	15	eleq1d 2106	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ (ℤ_≥‘1)) → (((ℕ × {𝐴})‘𝑧) ∈ ℂ ↔ 𝐴 ∈ ℂ))
17	12, 16	mpbird 156	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ (ℤ_≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
18	17	adantlr 446	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 𝑧 ∈ (ℤ_≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
19	18	adantlr 446	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) ∧ 𝑧 ∈ (ℤ_≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
20		mulcl 7008	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑧 · 𝑤) ∈ ℂ)
21	20	adantl 262	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
22	9, 11, 19, 21	iseqcl 9223	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁) ∈ ℂ)
23		iftrue 3336	. . . . . . . . . . . 12 ⊢ (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) = (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁))
24	23	eleq1d 2106	. . . . . . . . . . 11 ⊢ (0 < 𝑁 → (if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ ↔ (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁) ∈ ℂ))
25	24	adantl 262	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → (if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ ↔ (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁) ∈ ℂ))
26	22, 25	mpbird 156	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ)
27	26	ex 108	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
28	27	adantr 261	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
29	28	3adantl3 1062	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
30		simpll2 944	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℤ)
31	30	znegcld 8362	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℤ)
32		simpr 103	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 < 𝑁)
33	30	zred 8360	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℝ)
34		0red 7028	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 ∈ ℝ)
35	33, 34	lenltd 7134	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 ≤ 0 ↔ ¬ 0 < 𝑁))
36	32, 35	mpbird 156	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ≤ 0)
37		simplr 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 𝑁 = 0)
38	37	neneqad 2284	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ≠ 0)
39	38	necomd 2291	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 ≠ 𝑁)
40		0z 8256	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℤ
41		zltlen 8319	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
42	40, 41	mpan2 401	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
43	42	3ad2ant2 926	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
44	43	ad2antrr 457	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
45	36, 39, 44	mpbir2and 851	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 < 0)
46	33	lt0neg1d 7507	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ 0 < -𝑁))
47	45, 46	mpbid 135	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 < -𝑁)
48		elnnz 8255	. . . . . . . . . . . 12 ⊢ (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
49	31, 47, 48	sylanbrc 394	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℕ)
50		elnnuz 8509	. . . . . . . . . . 11 ⊢ (-𝑁 ∈ ℕ ↔ -𝑁 ∈ (ℤ_≥‘1))
51	49, 50	sylib 127	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ (ℤ_≥‘1))
52	10	a1i 9	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ℂ ∈ V)
53	17	3ad2antl1 1066	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ 𝑧 ∈ (ℤ_≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
54	53	adantlr 446	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 𝑧 ∈ (ℤ_≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
55	54	adantlr 446	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) ∧ 𝑧 ∈ (ℤ_≥‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
56	20	adantl 262	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
57	51, 52, 55, 56	iseqcl 9223	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) ∈ ℂ)
58		simpll1 943	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝐴 ∈ ℂ)
59		expivallem 9256	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ -𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0)
60	59	3com23 1110	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ -𝑁 ∈ ℕ ∧ 𝐴 # 0) → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0)
61	60	3expia 1106	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ -𝑁 ∈ ℕ) → (𝐴 # 0 → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0))
62	58, 49, 61	syl2anc 391	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝐴 # 0 → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0))
63	39	neneqd 2226	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 = 𝑁)
64		ioran 669	. . . . . . . . . . . . 13 ⊢ (¬ (0 < 𝑁 ∨ 0 = 𝑁) ↔ (¬ 0 < 𝑁 ∧ ¬ 0 = 𝑁))
65	32, 63, 64	sylanbrc 394	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ (0 < 𝑁 ∨ 0 = 𝑁))
66		zleloe 8292	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
67	40, 66	mpan 400	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
68	67	3ad2ant2 926	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
69	68	ad2antrr 457	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
70	65, 69	mtbird 598	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 ≤ 𝑁)
71	70	pm2.21d 549	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (0 ≤ 𝑁 → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0))
72		simpll3 945	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝐴 # 0 ∨ 0 ≤ 𝑁))
73	62, 71, 72	mpjaod 638	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0)
74	57, 73	recclapd 7757	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)) ∈ ℂ)
75		iffalse 3339	. . . . . . . . . 10 ⊢ (¬ 0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) = (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))
76	75	eleq1d 2106	. . . . . . . . 9 ⊢ (¬ 0 < 𝑁 → (if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ ↔ (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)) ∈ ℂ))
77	76	adantl 262	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ ↔ (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)) ∈ ℂ))
78	74, 77	mpbird 156	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ)
79	78	ex 108	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → (¬ 0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
80		zdclt 8318	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 0 < 𝑁)
81	40, 80	mpan 400	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → DECID 0 < 𝑁)
82		df-dc 743	. . . . . . . . . 10 ⊢ (DECID 0 < 𝑁 ↔ (0 < 𝑁 ∨ ¬ 0 < 𝑁))
83	81, 82	sylib 127	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
84	83	adantl 262	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
85	84	adantr 261	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
86	85	3adantl3 1062	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
87	29, 79, 86	mpjaod 638	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ)
88		iffalse 3339	. . . . . . 7 ⊢ (¬ 𝑁 = 0 → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) = if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))))
89	88	eleq1d 2106	. . . . . 6 ⊢ (¬ 𝑁 = 0 → (if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ ↔ if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
90	89	adantl 262	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → (if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ ↔ if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
91	87, 90	mpbird 156	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ)
92	91	ex 108	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (¬ 𝑁 = 0 → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ))
93		zdceq 8316	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
94	40, 93	mpan2 401	. . . . 5 ⊢ (𝑁 ∈ ℤ → DECID 𝑁 = 0)
95		df-dc 743	. . . . 5 ⊢ (DECID 𝑁 = 0 ↔ (𝑁 = 0 ∨ ¬ 𝑁 = 0))
96	94, 95	sylib 127	. . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ∨ ¬ 𝑁 = 0))
97	96	3ad2ant2 926	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝑁 = 0 ∨ ¬ 𝑁 = 0))
98	4, 92, 97	mpjaod 638	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ)
99		sneq 3386	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
100	99	xpeq2d 4369	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (ℕ × {𝑥}) = (ℕ × {𝐴}))
101		iseqeq3 9216	. . . . . . 7 ⊢ ((ℕ × {𝑥}) = (ℕ × {𝐴}) → seq1( · , (ℕ × {𝑥}), ℂ) = seq1( · , (ℕ × {𝐴}), ℂ))
102	100, 101	syl 14	. . . . . 6 ⊢ (𝑥 = 𝐴 → seq1( · , (ℕ × {𝑥}), ℂ) = seq1( · , (ℕ × {𝐴}), ℂ))
103	102	fveq1d 5180	. . . . 5 ⊢ (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦) = (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦))
104	102	fveq1d 5180	. . . . . 6 ⊢ (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦) = (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦))
105	104	oveq2d 5528	. . . . 5 ⊢ (𝑥 = 𝐴 → (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦)))
106	103, 105	ifeq12d 3347	. . . 4 ⊢ (𝑥 = 𝐴 → if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦))) = if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦))))
107	106	ifeq2d 3346	. . 3 ⊢ (𝑥 = 𝐴 → if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)))) = if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦)))))
108		eqeq1 2046	. . . 4 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
109		breq2 3768	. . . . 5 ⊢ (𝑦 = 𝑁 → (0 < 𝑦 ↔ 0 < 𝑁))
110		fveq2 5178	. . . . 5 ⊢ (𝑦 = 𝑁 → (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦) = (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁))
111		negeq 7204	. . . . . . 7 ⊢ (𝑦 = 𝑁 → -𝑦 = -𝑁)
112	111	fveq2d 5182	. . . . . 6 ⊢ (𝑦 = 𝑁 → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦) = (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))
113	112	oveq2d 5528	. . . . 5 ⊢ (𝑦 = 𝑁 → (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))
114	109, 110, 113	ifbieq12d 3354	. . . 4 ⊢ (𝑦 = 𝑁 → if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦))) = if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))))
115	108, 114	ifbieq2d 3352	. . 3 ⊢ (𝑦 = 𝑁 → if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦)))) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))
116		df-iexp 9255	. . 3 ⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)))))
117	107, 115, 116	ovmpt2g 5635	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))
118	98, 117	syld3an3 1180	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 629 DECID wdc 742 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 ≠ wne 2204 Vcvv 2557 ifcif 3331 {csn 3375 class class class wbr 3764 × cxp 4343 ‘cfv 4902 (class class class)co 5512 ℂcc 6887 0cc0 6889 1c1 6890 · cmul 6894 < clt 7060 ≤ cle 7061 -cneg 7183 # cap 7572 / cdiv 7651 ℕcn 7914 ℤcz 8245 ℤ_≥cuz 8473 seqcseq 9211 ↑cexp 9254

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-frec 5978 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-n0 8182 df-z 8246 df-uz 8474 df-iseq 9212 df-iexp 9255

This theorem is referenced by: expinnval 9258 exp0 9259 expnegap0 9263

W3C validator