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Theorem modexp 12982
Description: Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.)
Assertion
Ref Expression
modexp (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℝ+) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴𝐶) mod 𝐷) = ((𝐵𝐶) mod 𝐷))

Proof of Theorem modexp
Dummy variables 𝑥 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2l 1085 . 2 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℝ+) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → 𝐶 ∈ ℕ0)
2 id 22 . . 3 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)))
323adant2l 1318 . 2 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℝ+) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)))
4 oveq2 6643 . . . . . 6 (𝑥 = 0 → (𝐴𝑥) = (𝐴↑0))
54oveq1d 6650 . . . . 5 (𝑥 = 0 → ((𝐴𝑥) mod 𝐷) = ((𝐴↑0) mod 𝐷))
6 oveq2 6643 . . . . . 6 (𝑥 = 0 → (𝐵𝑥) = (𝐵↑0))
76oveq1d 6650 . . . . 5 (𝑥 = 0 → ((𝐵𝑥) mod 𝐷) = ((𝐵↑0) mod 𝐷))
85, 7eqeq12d 2635 . . . 4 (𝑥 = 0 → (((𝐴𝑥) mod 𝐷) = ((𝐵𝑥) mod 𝐷) ↔ ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷)))
98imbi2d 330 . . 3 (𝑥 = 0 → ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴𝑥) mod 𝐷) = ((𝐵𝑥) mod 𝐷)) ↔ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))))
10 oveq2 6643 . . . . . 6 (𝑥 = 𝑘 → (𝐴𝑥) = (𝐴𝑘))
1110oveq1d 6650 . . . . 5 (𝑥 = 𝑘 → ((𝐴𝑥) mod 𝐷) = ((𝐴𝑘) mod 𝐷))
12 oveq2 6643 . . . . . 6 (𝑥 = 𝑘 → (𝐵𝑥) = (𝐵𝑘))
1312oveq1d 6650 . . . . 5 (𝑥 = 𝑘 → ((𝐵𝑥) mod 𝐷) = ((𝐵𝑘) mod 𝐷))
1411, 13eqeq12d 2635 . . . 4 (𝑥 = 𝑘 → (((𝐴𝑥) mod 𝐷) = ((𝐵𝑥) mod 𝐷) ↔ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)))
1514imbi2d 330 . . 3 (𝑥 = 𝑘 → ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴𝑥) mod 𝐷) = ((𝐵𝑥) mod 𝐷)) ↔ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷))))
16 oveq2 6643 . . . . . 6 (𝑥 = (𝑘 + 1) → (𝐴𝑥) = (𝐴↑(𝑘 + 1)))
1716oveq1d 6650 . . . . 5 (𝑥 = (𝑘 + 1) → ((𝐴𝑥) mod 𝐷) = ((𝐴↑(𝑘 + 1)) mod 𝐷))
18 oveq2 6643 . . . . . 6 (𝑥 = (𝑘 + 1) → (𝐵𝑥) = (𝐵↑(𝑘 + 1)))
1918oveq1d 6650 . . . . 5 (𝑥 = (𝑘 + 1) → ((𝐵𝑥) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))
2017, 19eqeq12d 2635 . . . 4 (𝑥 = (𝑘 + 1) → (((𝐴𝑥) mod 𝐷) = ((𝐵𝑥) mod 𝐷) ↔ ((𝐴↑(𝑘 + 1)) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷)))
2120imbi2d 330 . . 3 (𝑥 = (𝑘 + 1) → ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴𝑥) mod 𝐷) = ((𝐵𝑥) mod 𝐷)) ↔ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑(𝑘 + 1)) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))))
22 oveq2 6643 . . . . . 6 (𝑥 = 𝐶 → (𝐴𝑥) = (𝐴𝐶))
2322oveq1d 6650 . . . . 5 (𝑥 = 𝐶 → ((𝐴𝑥) mod 𝐷) = ((𝐴𝐶) mod 𝐷))
24 oveq2 6643 . . . . . 6 (𝑥 = 𝐶 → (𝐵𝑥) = (𝐵𝐶))
2524oveq1d 6650 . . . . 5 (𝑥 = 𝐶 → ((𝐵𝑥) mod 𝐷) = ((𝐵𝐶) mod 𝐷))
2623, 25eqeq12d 2635 . . . 4 (𝑥 = 𝐶 → (((𝐴𝑥) mod 𝐷) = ((𝐵𝑥) mod 𝐷) ↔ ((𝐴𝐶) mod 𝐷) = ((𝐵𝐶) mod 𝐷)))
2726imbi2d 330 . . 3 (𝑥 = 𝐶 → ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴𝑥) mod 𝐷) = ((𝐵𝑥) mod 𝐷)) ↔ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴𝐶) mod 𝐷) = ((𝐵𝐶) mod 𝐷))))
28 zcn 11367 . . . . . . 7 (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
29 exp0 12847 . . . . . . 7 (𝐴 ∈ ℂ → (𝐴↑0) = 1)
3028, 29syl 17 . . . . . 6 (𝐴 ∈ ℤ → (𝐴↑0) = 1)
31 zcn 11367 . . . . . . . 8 (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
32 exp0 12847 . . . . . . . 8 (𝐵 ∈ ℂ → (𝐵↑0) = 1)
3331, 32syl 17 . . . . . . 7 (𝐵 ∈ ℤ → (𝐵↑0) = 1)
3433eqcomd 2626 . . . . . 6 (𝐵 ∈ ℤ → 1 = (𝐵↑0))
3530, 34sylan9eq 2674 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴↑0) = (𝐵↑0))
3635oveq1d 6650 . . . 4 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))
37363ad2ant1 1080 . . 3 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))
38 simp21l 1176 . . . . . . . 8 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → 𝐴 ∈ ℤ)
39 simp1 1059 . . . . . . . 8 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → 𝑘 ∈ ℕ0)
40 zexpcl 12858 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℤ)
4138, 39, 40syl2anc 692 . . . . . . 7 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → (𝐴𝑘) ∈ ℤ)
42 simp21r 1177 . . . . . . . 8 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → 𝐵 ∈ ℤ)
43 zexpcl 12858 . . . . . . . 8 ((𝐵 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → (𝐵𝑘) ∈ ℤ)
4442, 39, 43syl2anc 692 . . . . . . 7 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → (𝐵𝑘) ∈ ℤ)
45 simp22 1093 . . . . . . 7 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → 𝐷 ∈ ℝ+)
46 simp3 1061 . . . . . . 7 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷))
47 simp23 1094 . . . . . . 7 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → (𝐴 mod 𝐷) = (𝐵 mod 𝐷))
4841, 44, 38, 42, 45, 46, 47modmul12d 12707 . . . . . 6 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → (((𝐴𝑘) · 𝐴) mod 𝐷) = (((𝐵𝑘) · 𝐵) mod 𝐷))
4938zcnd 11468 . . . . . . . 8 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → 𝐴 ∈ ℂ)
50 expp1 12850 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑(𝑘 + 1)) = ((𝐴𝑘) · 𝐴))
5149, 39, 50syl2anc 692 . . . . . . 7 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → (𝐴↑(𝑘 + 1)) = ((𝐴𝑘) · 𝐴))
5251oveq1d 6650 . . . . . 6 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → ((𝐴↑(𝑘 + 1)) mod 𝐷) = (((𝐴𝑘) · 𝐴) mod 𝐷))
5342zcnd 11468 . . . . . . . 8 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → 𝐵 ∈ ℂ)
54 expp1 12850 . . . . . . . 8 ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐵↑(𝑘 + 1)) = ((𝐵𝑘) · 𝐵))
5553, 39, 54syl2anc 692 . . . . . . 7 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → (𝐵↑(𝑘 + 1)) = ((𝐵𝑘) · 𝐵))
5655oveq1d 6650 . . . . . 6 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → ((𝐵↑(𝑘 + 1)) mod 𝐷) = (((𝐵𝑘) · 𝐵) mod 𝐷))
5748, 52, 563eqtr4d 2664 . . . . 5 ((𝑘 ∈ ℕ0 ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → ((𝐴↑(𝑘 + 1)) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))
58573exp 1262 . . . 4 (𝑘 ∈ ℕ0 → (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → (((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷) → ((𝐴↑(𝑘 + 1)) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))))
5958a2d 29 . . 3 (𝑘 ∈ ℕ0 → ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴𝑘) mod 𝐷) = ((𝐵𝑘) mod 𝐷)) → (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑(𝑘 + 1)) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))))
609, 15, 21, 27, 37, 59nn0ind 11457 . 2 (𝐶 ∈ ℕ0 → (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ+ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴𝐶) mod 𝐷) = ((𝐵𝐶) mod 𝐷)))
611, 3, 60sylc 65 1 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ0𝐷 ∈ ℝ+) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴𝐶) mod 𝐷) = ((𝐵𝐶) mod 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  (class class class)co 6635  cc 9919  0cc0 9921  1c1 9922   + caddc 9924   · cmul 9926  0cn0 11277  cz 11362  +crp 11817   mod cmo 12651  cexp 12843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-sup 8333  df-inf 8334  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-rp 11818  df-fl 12576  df-mod 12652  df-seq 12785  df-exp 12844
This theorem is referenced by:  fermltl  15470  odzdvds  15481  lgslem4  25006  lgsmod  25029  lgsne0  25041  fmtnoprmfac1lem  41241  sfprmdvdsmersenne  41285  41prothprmlem2  41300
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