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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)) → (𝐴𝑁) = 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.
StepHypRef 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
31, 2syl6eqel 2128 . . . 4 (𝑁 = 0 → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))) ∈ ℂ)
43a1i 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))
75, 6bitr3i 175 . . . . . . . . . . . . 13 ((𝑁 ∈ ℤ ∧ 0 < 𝑁) ↔ 𝑁 ∈ (ℤ‘1))
87biimpi 113 . . . . . . . . . . . 12 ((𝑁 ∈ ℤ ∧ 0 < 𝑁) → 𝑁 ∈ (ℤ‘1))
98adantll 445 . . . . . . . . . . 11 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → 𝑁 ∈ (ℤ‘1))
10 cnex 7005 . . . . . . . . . . . 12 ℂ ∈ V
1110a1i 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)
1513, 14sylan2br 272 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ 𝑧 ∈ (ℤ‘1)) → ((ℕ × {𝐴})‘𝑧) = 𝐴)
1615eleq1d 2106 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℂ ∧ 𝑧 ∈ (ℤ‘1)) → (((ℕ × {𝐴})‘𝑧) ∈ ℂ ↔ 𝐴 ∈ ℂ))
1712, 16mpbird 156 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑧 ∈ (ℤ‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
1817adantlr 446 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 𝑧 ∈ (ℤ‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
1918adantlr 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)
2120adantl 262 . . . . . . . . . . 11 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
229, 11, 19, 21iseqcl 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))
2423eleq1d 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))
2524adantl 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))
2622, 25mpbird 156 . . . . . . . . 9 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ 0 < 𝑁) → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ)
2726ex 108 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))) ∈ ℂ))
2827adantr 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))
29283adantl3 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)
3130znegcld 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)
3330zred 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)
3533, 34lenltd 7134 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 ≤ 0 ↔ ¬ 0 < 𝑁))
3632, 35mpbird 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)
3837neneqad 2284 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ≠ 0)
3938necomd 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)))
4240, 41mpan2 401 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℤ → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
43423ad2ant2 926 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
4443ad2antrr 457 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
4536, 39, 44mpbir2and 851 . . . . . . . . . . . . 13 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 < 0)
4633lt0neg1d 7507 . . . . . . . . . . . . 13 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ 0 < -𝑁))
4745, 46mpbid 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))
4931, 47, 48sylanbrc 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))
5149, 50sylib 127 . . . . . . . . . 10 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ (ℤ‘1))
5210a1i 9 . . . . . . . . . 10 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ℂ ∈ V)
53173ad2antl1 1066 . . . . . . . . . . . 12 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ 𝑧 ∈ (ℤ‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
5453adantlr 446 . . . . . . . . . . 11 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 𝑧 ∈ (ℤ‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
5554adantlr 446 . . . . . . . . . 10 (((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) ∧ 𝑧 ∈ (ℤ‘1)) → ((ℕ × {𝐴})‘𝑧) ∈ ℂ)
5620adantl 262 . . . . . . . . . 10 (((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) ∧ (𝑧 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → (𝑧 · 𝑤) ∈ ℂ)
5751, 52, 55, 56iseqcl 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)
60593com23 1110 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ -𝑁 ∈ ℕ ∧ 𝐴 # 0) → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0)
61603expia 1106 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ -𝑁 ∈ ℕ) → (𝐴 # 0 → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0))
6258, 49, 61syl2anc 391 . . . . . . . . . 10 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝐴 # 0 → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0))
6339neneqd 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))
6532, 63, 64sylanbrc 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)))
6740, 66mpan 400 . . . . . . . . . . . . . 14 (𝑁 ∈ ℤ → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
68673ad2ant2 926 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
6968ad2antrr 457 . . . . . . . . . . . 12 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
7065, 69mtbird 598 . . . . . . . . . . 11 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 ≤ 𝑁)
7170pm2.21d 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))
7362, 71, 72mpjaod 638 . . . . . . . . 9 ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁) # 0)
7457, 73recclapd 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)))
7675eleq1d 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))
7776adantl 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))
7874, 77mpbird 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)
7978ex 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)
8140, 80mpan 400 . . . . . . . . . 10 (𝑁 ∈ ℤ → DECID 0 < 𝑁)
82 df-dc 743 . . . . . . . . . 10 (DECID 0 < 𝑁 ↔ (0 < 𝑁 ∨ ¬ 0 < 𝑁))
8381, 82sylib 127 . . . . . . . . 9 (𝑁 ∈ ℤ → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
8483adantl 262 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
8584adantr 261 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
86853adantl3 1062 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → (0 < 𝑁 ∨ ¬ 0 < 𝑁))
8729, 79, 86mpjaod 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))))
8988eleq1d 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))
9089adantl 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))
9187, 90mpbird 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)
9291ex 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)
9440, 93mpan2 401 . . . . 5 (𝑁 ∈ ℤ → DECID 𝑁 = 0)
95 df-dc 743 . . . . 5 (DECID 𝑁 = 0 ↔ (𝑁 = 0 ∨ ¬ 𝑁 = 0))
9694, 95sylib 127 . . . 4 (𝑁 ∈ ℤ → (𝑁 = 0 ∨ ¬ 𝑁 = 0))
97963ad2ant2 926 . . 3 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝑁 = 0 ∨ ¬ 𝑁 = 0))
984, 92, 97mpjaod 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})
10099xpeq2d 4369 . . . . . . 7 (𝑥 = 𝐴 → (ℕ × {𝑥}) = (ℕ × {𝐴}))
101 iseqeq3 9216 . . . . . . 7 ((ℕ × {𝑥}) = (ℕ × {𝐴}) → seq1( · , (ℕ × {𝑥}), ℂ) = seq1( · , (ℕ × {𝐴}), ℂ))
102100, 101syl 14 . . . . . 6 (𝑥 = 𝐴 → seq1( · , (ℕ × {𝑥}), ℂ) = seq1( · , (ℕ × {𝐴}), ℂ))
103102fveq1d 5180 . . . . 5 (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦) = (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑦))
104102fveq1d 5180 . . . . . 6 (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦) = (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦))
105104oveq2d 5528 . . . . 5 (𝑥 = 𝐴 → (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦)))
106103, 105ifeq12d 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))))
107106ifeq2d 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)
112111fveq2d 5182 . . . . . 6 (𝑦 = 𝑁 → (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦) = (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁))
113112oveq2d 5528 . . . . 5 (𝑦 = 𝑁 → (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))
114109, 110, 113ifbieq12d 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))))
115108, 114ifbieq2d 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)))))
117107, 115, 116ovmpt2g 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) → (𝐴𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))
11898, 117syld3an3 1180 1 ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝐴𝑁) = 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
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