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Theorem modexp 14173

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 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ⁺) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑𝐶) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷))

Proof of Theorem modexp Dummy variables 𝑥 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2l 1201	. 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ 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)))
3	2	3adant2l 1180	. 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ⁺) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)))
4		oveq2 7376	. . . . . 6 ⊢ (𝑥 = 0 → (𝐴↑𝑥) = (𝐴↑0))
5	4	oveq1d 7383	. . . . 5 ⊢ (𝑥 = 0 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑0) mod 𝐷))
6		oveq2 7376	. . . . . 6 ⊢ (𝑥 = 0 → (𝐵↑𝑥) = (𝐵↑0))
7	6	oveq1d 7383	. . . . 5 ⊢ (𝑥 = 0 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑0) mod 𝐷))
8	5, 7	eqeq12d 2753	. . . 4 ⊢ (𝑥 = 0 → (((𝐴↑𝑥) mod 𝐷) = ((𝐵↑𝑥) mod 𝐷) ↔ ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷)))
9	8	imbi2d 340	. . 3 ⊢ (𝑥 = 0 → ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑𝑥) mod 𝐷) = ((𝐵↑𝑥) mod 𝐷)) ↔ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))))
10		oveq2 7376	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐴↑𝑥) = (𝐴↑𝑘))
11	10	oveq1d 7383	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑𝑘) mod 𝐷))
12		oveq2 7376	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐵↑𝑥) = (𝐵↑𝑘))
13	12	oveq1d 7383	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷))
14	11, 13	eqeq12d 2753	. . . 4 ⊢ (𝑥 = 𝑘 → (((𝐴↑𝑥) mod 𝐷) = ((𝐵↑𝑥) mod 𝐷) ↔ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)))
15	14	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑘 → ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑𝑥) mod 𝐷) = ((𝐵↑𝑥) mod 𝐷)) ↔ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷))))
16		oveq2 7376	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐴↑𝑥) = (𝐴↑(𝑘 + 1)))
17	16	oveq1d 7383	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑(𝑘 + 1)) mod 𝐷))
18		oveq2 7376	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐵↑𝑥) = (𝐵↑(𝑘 + 1)))
19	18	oveq1d 7383	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))
20	17, 19	eqeq12d 2753	. . . 4 ⊢ (𝑥 = (𝑘 + 1) → (((𝐴↑𝑥) mod 𝐷) = ((𝐵↑𝑥) mod 𝐷) ↔ ((𝐴↑(𝑘 + 1)) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷)))
21	20	imbi2d 340	. . 3 ⊢ (𝑥 = (𝑘 + 1) → ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑𝑥) mod 𝐷) = ((𝐵↑𝑥) 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 7376	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴↑𝑥) = (𝐴↑𝐶))
23	22	oveq1d 7383	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑𝐶) mod 𝐷))
24		oveq2 7376	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵↑𝑥) = (𝐵↑𝐶))
25	24	oveq1d 7383	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷))
26	23, 25	eqeq12d 2753	. . . 4 ⊢ (𝑥 = 𝐶 → (((𝐴↑𝑥) mod 𝐷) = ((𝐵↑𝑥) mod 𝐷) ↔ ((𝐴↑𝐶) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷)))
27	26	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐶 → ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑𝑥) mod 𝐷) = ((𝐵↑𝑥) mod 𝐷)) ↔ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑𝐶) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷))))
28		zcn 12505	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
29		exp0 14000	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴↑0) = 1)
31		zcn 12505	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
32		exp0 14000	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (𝐵↑0) = 1)
33	31, 32	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ ℤ → (𝐵↑0) = 1)
34	33	eqcomd 2743	. . . . . 6 ⊢ (𝐵 ∈ ℤ → 1 = (𝐵↑0))
35	30, 34	sylan9eq 2792	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴↑0) = (𝐵↑0))
36	35	oveq1d 7383	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))
37	36	3ad2ant1 1134	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))
38		simp21l 1292	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → 𝐴 ∈ ℤ)
39		simp1 1137	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → 𝑘 ∈ ℕ₀)
40		zexpcl 14011	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℤ)
41	38, 39, 40	syl2anc 585	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → (𝐴↑𝑘) ∈ ℤ)
42		simp21r 1293	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → 𝐵 ∈ ℤ)
43		zexpcl 14011	. . . . . . . 8 ⊢ ((𝐵 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (𝐵↑𝑘) ∈ ℤ)
44	42, 39, 43	syl2anc 585	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → (𝐵↑𝑘) ∈ ℤ)
45		simp22 1209	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → 𝐷 ∈ ℝ⁺)
46		simp3 1139	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷))
47		simp23 1210	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → (𝐴 mod 𝐷) = (𝐵 mod 𝐷))
48	41, 44, 38, 42, 45, 46, 47	modmul12d 13860	. . . . . 6 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → (((𝐴↑𝑘) · 𝐴) mod 𝐷) = (((𝐵↑𝑘) · 𝐵) mod 𝐷))
49	38	zcnd 12609	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → 𝐴 ∈ ℂ)
50		expp1 14003	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
51	49, 39, 50	syl2anc 585	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
52	51	oveq1d 7383	. . . . . 6 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → ((𝐴↑(𝑘 + 1)) mod 𝐷) = (((𝐴↑𝑘) · 𝐴) mod 𝐷))
53	42	zcnd 12609	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → 𝐵 ∈ ℂ)
54		expp1 14003	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐵↑(𝑘 + 1)) = ((𝐵↑𝑘) · 𝐵))
55	53, 39, 54	syl2anc 585	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → (𝐵↑(𝑘 + 1)) = ((𝐵↑𝑘) · 𝐵))
56	55	oveq1d 7383	. . . . . 6 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → ((𝐵↑(𝑘 + 1)) mod 𝐷) = (((𝐵↑𝑘) · 𝐵) mod 𝐷))
57	48, 52, 56	3eqtr4d 2782	. . . . 5 ⊢ ((𝑘 ∈ ℕ₀ ∧ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ∧ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → ((𝐴↑(𝑘 + 1)) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))
58	57	3exp 1120	. . . 4 ⊢ (𝑘 ∈ ℕ₀ → (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → (((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷) → ((𝐴↑(𝑘 + 1)) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))))
59	58	a2d 29	. . 3 ⊢ (𝑘 ∈ ℕ₀ → ((((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)) → (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑(𝑘 + 1)) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))))
60	9, 15, 21, 27, 37, 59	nn0ind 12599	. 2 ⊢ (𝐶 ∈ ℕ₀ → (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑𝐶) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷)))
61	1, 3, 60	sylc 65	1 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℕ₀ ∧ 𝐷 ∈ ℝ⁺) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑𝐶) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 (class class class)co 7368 ℂcc 11036 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ℕ₀cn0 12413 ℤcz 12500 ℝ⁺crp 12917 mod cmo 13801 ↑cexp 13996

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997

This theorem is referenced by: dvdsmodexp 16199 odzdvds 16735 lgsmod 27302 lgsne0 27314 fmtnoprmfac1lem 47918 sfprmdvdsmersenne 47957 41prothprmlem2 47972

W3C validator