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Theorem expival 8911
 Description: Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
Assertion
Ref Expression
expival ((A 𝑁 (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 3330 . . . . 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 6776 . . . . 5 1
31, 2syl6eqel 2125 . . . 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 𝑁 (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 8031 . . . . . . . . . . . . . 14 (𝑁 ℕ ↔ (𝑁 0 < 𝑁))
6 elnnuz 8285 . . . . . . . . . . . . . 14 (𝑁 ℕ ↔ 𝑁 (ℤ‘1))
75, 6bitr3i 175 . . . . . . . . . . . . 13 ((𝑁 0 < 𝑁) ↔ 𝑁 (ℤ‘1))
87biimpi 113 . . . . . . . . . . . 12 ((𝑁 0 < 𝑁) → 𝑁 (ℤ‘1))
98adantll 445 . . . . . . . . . . 11 (((A 𝑁 ℤ) 0 < 𝑁) → 𝑁 (ℤ‘1))
10 cnex 6803 . . . . . . . . . . . 12 V
1110a1i 9 . . . . . . . . . . 11 (((A 𝑁 ℤ) 0 < 𝑁) → ℂ V)
12 simpl 102 . . . . . . . . . . . . . 14 ((A z (ℤ‘1)) → A ℂ)
13 elnnuz 8285 . . . . . . . . . . . . . . . 16 (z ℕ ↔ z (ℤ‘1))
14 fvconst2g 5318 . . . . . . . . . . . . . . . 16 ((A z ℕ) → ((ℕ × {A})‘z) = A)
1513, 14sylan2br 272 . . . . . . . . . . . . . . 15 ((A z (ℤ‘1)) → ((ℕ × {A})‘z) = A)
1615eleq1d 2103 . . . . . . . . . . . . . 14 ((A z (ℤ‘1)) → (((ℕ × {A})‘z) ℂ ↔ A ℂ))
1712, 16mpbird 156 . . . . . . . . . . . . 13 ((A z (ℤ‘1)) → ((ℕ × {A})‘z) ℂ)
1817adantlr 446 . . . . . . . . . . . 12 (((A 𝑁 ℤ) z (ℤ‘1)) → ((ℕ × {A})‘z) ℂ)
1918adantlr 446 . . . . . . . . . . 11 ((((A 𝑁 ℤ) 0 < 𝑁) z (ℤ‘1)) → ((ℕ × {A})‘z) ℂ)
20 mulcl 6806 . . . . . . . . . . . 12 ((z w ℂ) → (z · w) ℂ)
2120adantl 262 . . . . . . . . . . 11 ((((A 𝑁 ℤ) 0 < 𝑁) (z w ℂ)) → (z · w) ℂ)
229, 11, 19, 21iseqcl 8903 . . . . . . . . . 10 (((A 𝑁 ℤ) 0 < 𝑁) → (seq1( · , (ℕ × {A}), ℂ)‘𝑁) ℂ)
23 iftrue 3330 . . . . . . . . . . . 12 (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {A}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁))) = (seq1( · , (ℕ × {A}), ℂ)‘𝑁))
2423eleq1d 2103 . . . . . . . . . . 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 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 𝑁 ℤ) 0 < 𝑁) → if(0 < 𝑁, (seq1( · , (ℕ × {A}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁))) ℂ)
2726ex 108 . . . . . . . 8 ((A 𝑁 ℤ) → (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {A}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁))) ℂ))
2827adantr 261 . . . . . . 7 (((A 𝑁 ℤ) ¬ 𝑁 = 0) → (0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {A}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁))) ℂ))
29283adantl3 1061 . . . . . 6 (((A 𝑁 (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 943 . . . . . . . . . . . . 13 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → 𝑁 ℤ)
3130znegcld 8138 . . . . . . . . . . . 12 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → -𝑁 ℤ)
32 simpr 103 . . . . . . . . . . . . . . 15 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → ¬ 0 < 𝑁)
3330zred 8136 . . . . . . . . . . . . . . . 16 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → 𝑁 ℝ)
34 0red 6826 . . . . . . . . . . . . . . . 16 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → 0 ℝ)
3533, 34lenltd 6931 . . . . . . . . . . . . . . 15 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → (𝑁 ≤ 0 ↔ ¬ 0 < 𝑁))
3632, 35mpbird 156 . . . . . . . . . . . . . 14 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → 𝑁 ≤ 0)
37 simplr 482 . . . . . . . . . . . . . . . 16 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → ¬ 𝑁 = 0)
3837neneqad 2278 . . . . . . . . . . . . . . 15 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → 𝑁 ≠ 0)
3938necomd 2285 . . . . . . . . . . . . . 14 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → 0 ≠ 𝑁)
40 0z 8032 . . . . . . . . . . . . . . . . 17 0
41 zltlen 8095 . . . . . . . . . . . . . . . . 17 ((𝑁 0 ℤ) → (𝑁 < 0 ↔ (𝑁 ≤ 0 0 ≠ 𝑁)))
4240, 41mpan2 401 . . . . . . . . . . . . . . . 16 (𝑁 ℤ → (𝑁 < 0 ↔ (𝑁 ≤ 0 0 ≠ 𝑁)))
43423ad2ant2 925 . . . . . . . . . . . . . . 15 ((A 𝑁 (A # 0 0 ≤ 𝑁)) → (𝑁 < 0 ↔ (𝑁 ≤ 0 0 ≠ 𝑁)))
4443ad2antrr 457 . . . . . . . . . . . . . 14 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → (𝑁 < 0 ↔ (𝑁 ≤ 0 0 ≠ 𝑁)))
4536, 39, 44mpbir2and 850 . . . . . . . . . . . . 13 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → 𝑁 < 0)
4633lt0neg1d 7302 . . . . . . . . . . . . 13 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → (𝑁 < 0 ↔ 0 < -𝑁))
4745, 46mpbid 135 . . . . . . . . . . . 12 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → 0 < -𝑁)
48 elnnz 8031 . . . . . . . . . . . 12 (-𝑁 ℕ ↔ (-𝑁 0 < -𝑁))
4931, 47, 48sylanbrc 394 . . . . . . . . . . 11 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → -𝑁 ℕ)
50 elnnuz 8285 . . . . . . . . . . 11 (-𝑁 ℕ ↔ -𝑁 (ℤ‘1))
5149, 50sylib 127 . . . . . . . . . 10 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → -𝑁 (ℤ‘1))
5210a1i 9 . . . . . . . . . 10 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → ℂ V)
53173ad2antl1 1065 . . . . . . . . . . . 12 (((A 𝑁 (A # 0 0 ≤ 𝑁)) z (ℤ‘1)) → ((ℕ × {A})‘z) ℂ)
5453adantlr 446 . . . . . . . . . . 11 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) z (ℤ‘1)) → ((ℕ × {A})‘z) ℂ)
5554adantlr 446 . . . . . . . . . 10 (((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) z (ℤ‘1)) → ((ℕ × {A})‘z) ℂ)
5620adantl 262 . . . . . . . . . 10 (((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) (z w ℂ)) → (z · w) ℂ)
5751, 52, 55, 56iseqcl 8903 . . . . . . . . 9 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → (seq1( · , (ℕ × {A}), ℂ)‘-𝑁) ℂ)
58 simpll1 942 . . . . . . . . . . 11 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → A ℂ)
59 expivallem 8910 . . . . . . . . . . . . 13 ((A A # 0 -𝑁 ℕ) → (seq1( · , (ℕ × {A}), ℂ)‘-𝑁) # 0)
60593com23 1109 . . . . . . . . . . . 12 ((A -𝑁 A # 0) → (seq1( · , (ℕ × {A}), ℂ)‘-𝑁) # 0)
61603expia 1105 . . . . . . . . . . 11 ((A -𝑁 ℕ) → (A # 0 → (seq1( · , (ℕ × {A}), ℂ)‘-𝑁) # 0))
6258, 49, 61syl2anc 391 . . . . . . . . . 10 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → (A # 0 → (seq1( · , (ℕ × {A}), ℂ)‘-𝑁) # 0))
6339neneqd 2221 . . . . . . . . . . . . 13 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → ¬ 0 = 𝑁)
64 ioran 668 . . . . . . . . . . . . 13 (¬ (0 < 𝑁 0 = 𝑁) ↔ (¬ 0 < 𝑁 ¬ 0 = 𝑁))
6532, 63, 64sylanbrc 394 . . . . . . . . . . . 12 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → ¬ (0 < 𝑁 0 = 𝑁))
66 zleloe 8068 . . . . . . . . . . . . . . 15 ((0 𝑁 ℤ) → (0 ≤ 𝑁 ↔ (0 < 𝑁 0 = 𝑁)))
6740, 66mpan 400 . . . . . . . . . . . . . 14 (𝑁 ℤ → (0 ≤ 𝑁 ↔ (0 < 𝑁 0 = 𝑁)))
68673ad2ant2 925 . . . . . . . . . . . . 13 ((A 𝑁 (A # 0 0 ≤ 𝑁)) → (0 ≤ 𝑁 ↔ (0 < 𝑁 0 = 𝑁)))
6968ad2antrr 457 . . . . . . . . . . . 12 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → (0 ≤ 𝑁 ↔ (0 < 𝑁 0 = 𝑁)))
7065, 69mtbird 597 . . . . . . . . . . 11 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → ¬ 0 ≤ 𝑁)
7170pm2.21d 549 . . . . . . . . . 10 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → (0 ≤ 𝑁 → (seq1( · , (ℕ × {A}), ℂ)‘-𝑁) # 0))
72 simpll3 944 . . . . . . . . . 10 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → (A # 0 0 ≤ 𝑁))
7362, 71, 72mpjaod 637 . . . . . . . . 9 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → (seq1( · , (ℕ × {A}), ℂ)‘-𝑁) # 0)
7457, 73recclapd 7539 . . . . . . . 8 ((((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) ¬ 0 < 𝑁) → (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁)) ℂ)
75 iffalse 3333 . . . . . . . . . 10 (¬ 0 < 𝑁 → if(0 < 𝑁, (seq1( · , (ℕ × {A}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁))) = (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁)))
7675eleq1d 2103 . . . . . . . . 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 𝑁 (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 𝑁 (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 𝑁 (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 8094 . . . . . . . . . . 11 ((0 𝑁 ℤ) → DECID 0 < 𝑁)
8140, 80mpan 400 . . . . . . . . . 10 (𝑁 ℤ → DECID 0 < 𝑁)
82 df-dc 742 . . . . . . . . . 10 (DECID 0 < 𝑁 ↔ (0 < 𝑁 ¬ 0 < 𝑁))
8381, 82sylib 127 . . . . . . . . 9 (𝑁 ℤ → (0 < 𝑁 ¬ 0 < 𝑁))
8483adantl 262 . . . . . . . 8 ((A 𝑁 ℤ) → (0 < 𝑁 ¬ 0 < 𝑁))
8584adantr 261 . . . . . . 7 (((A 𝑁 ℤ) ¬ 𝑁 = 0) → (0 < 𝑁 ¬ 0 < 𝑁))
86853adantl3 1061 . . . . . 6 (((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) → (0 < 𝑁 ¬ 0 < 𝑁))
8729, 79, 86mpjaod 637 . . . . 5 (((A 𝑁 (A # 0 0 ≤ 𝑁)) ¬ 𝑁 = 0) → if(0 < 𝑁, (seq1( · , (ℕ × {A}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁))) ℂ)
88 iffalse 3333 . . . . . . 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 2103 . . . . . 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 𝑁 (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 𝑁 (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 𝑁 (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 8092 . . . . . 6 ((𝑁 0 ℤ) → DECID 𝑁 = 0)
9440, 93mpan2 401 . . . . 5 (𝑁 ℤ → DECID 𝑁 = 0)
95 df-dc 742 . . . . 5 (DECID 𝑁 = 0 ↔ (𝑁 = 0 ¬ 𝑁 = 0))
9694, 95sylib 127 . . . 4 (𝑁 ℤ → (𝑁 = 0 ¬ 𝑁 = 0))
97963ad2ant2 925 . . 3 ((A 𝑁 (A # 0 0 ≤ 𝑁)) → (𝑁 = 0 ¬ 𝑁 = 0))
984, 92, 97mpjaod 637 . 2 ((A 𝑁 (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 3378 . . . . . . . 8 (x = A → {x} = {A})
10099xpeq2d 4312 . . . . . . 7 (x = A → (ℕ × {x}) = (ℕ × {A}))
101 iseqeq3 8896 . . . . . . 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 5123 . . . . 5 (x = A → (seq1( · , (ℕ × {x}), ℂ)‘y) = (seq1( · , (ℕ × {A}), ℂ)‘y))
104102fveq1d 5123 . . . . . 6 (x = A → (seq1( · , (ℕ × {x}), ℂ)‘-y) = (seq1( · , (ℕ × {A}), ℂ)‘-y))
105104oveq2d 5471 . . . . 5 (x = A → (1 / (seq1( · , (ℕ × {x}), ℂ)‘-y)) = (1 / (seq1( · , (ℕ × {A}), ℂ)‘-y)))
106103, 105ifeq12d 3341 . . . 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 3340 . . 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 2043 . . . 4 (y = 𝑁 → (y = 0 ↔ 𝑁 = 0))
109 breq2 3759 . . . . 5 (y = 𝑁 → (0 < y ↔ 0 < 𝑁))
110 fveq2 5121 . . . . 5 (y = 𝑁 → (seq1( · , (ℕ × {A}), ℂ)‘y) = (seq1( · , (ℕ × {A}), ℂ)‘𝑁))
111 negeq 7001 . . . . . . 7 (y = 𝑁 → -y = -𝑁)
112111fveq2d 5125 . . . . . 6 (y = 𝑁 → (seq1( · , (ℕ × {A}), ℂ)‘-y) = (seq1( · , (ℕ × {A}), ℂ)‘-𝑁))
113112oveq2d 5471 . . . . 5 (y = 𝑁 → (1 / (seq1( · , (ℕ × {A}), ℂ)‘-y)) = (1 / (seq1( · , (ℕ × {A}), ℂ)‘-𝑁)))
114109, 110, 113ifbieq12d 3348 . . . 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 3346 . . 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 8909 . . 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 5577 . 2 ((A 𝑁 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 1179 1 ((A 𝑁 (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 628  DECID wdc 741   ∧ w3a 884   = wceq 1242   ∈ wcel 1390   ≠ wne 2201  Vcvv 2551  ifcif 3325  {csn 3367   class class class wbr 3755   × cxp 4286  ‘cfv 4845  (class class class)co 5455  ℂcc 6709  0cc0 6711  1c1 6712   · cmul 6716   < clt 6857   ≤ cle 6858  -cneg 6980   # cap 7365   / cdiv 7433  ℕcn 7695  ℤcz 8021  ℤ≥cuz 8249  seqcseq 8892  ↑cexp 8908 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-mulrcl 6782  ax-addcom 6783  ax-mulcom 6784  ax-addass 6785  ax-mulass 6786  ax-distr 6787  ax-i2m1 6788  ax-1rid 6790  ax-0id 6791  ax-rnegex 6792  ax-precex 6793  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-lttrn 6797  ax-pre-apti 6798  ax-pre-ltadd 6799  ax-pre-mulgt0 6800  ax-pre-mulext 6801 This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-if 3326  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-frec 5918  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-sub 6981  df-neg 6982  df-reap 7359  df-ap 7366  df-div 7434  df-inn 7696  df-n0 7958  df-z 8022  df-uz 8250  df-iseq 8893  df-iexp 8909 This theorem is referenced by:  expinnval  8912  exp0  8913  expnegap0  8917
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