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Theorem modexp 14274

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 1198	. 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 1177	. 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 7439	. . . . . 6 ⊢ (𝑥 = 0 → (𝐴↑𝑥) = (𝐴↑0))
5	4	oveq1d 7446	. . . . 5 ⊢ (𝑥 = 0 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑0) mod 𝐷))
6		oveq2 7439	. . . . . 6 ⊢ (𝑥 = 0 → (𝐵↑𝑥) = (𝐵↑0))
7	6	oveq1d 7446	. . . . 5 ⊢ (𝑥 = 0 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑0) mod 𝐷))
8	5, 7	eqeq12d 2751	. . . 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 7439	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐴↑𝑥) = (𝐴↑𝑘))
11	10	oveq1d 7446	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑𝑘) mod 𝐷))
12		oveq2 7439	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐵↑𝑥) = (𝐵↑𝑘))
13	12	oveq1d 7446	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷))
14	11, 13	eqeq12d 2751	. . . 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 7439	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐴↑𝑥) = (𝐴↑(𝑘 + 1)))
17	16	oveq1d 7446	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑(𝑘 + 1)) mod 𝐷))
18		oveq2 7439	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐵↑𝑥) = (𝐵↑(𝑘 + 1)))
19	18	oveq1d 7446	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))
20	17, 19	eqeq12d 2751	. . . 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 7439	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴↑𝑥) = (𝐴↑𝐶))
23	22	oveq1d 7446	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑𝐶) mod 𝐷))
24		oveq2 7439	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵↑𝑥) = (𝐵↑𝐶))
25	24	oveq1d 7446	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷))
26	23, 25	eqeq12d 2751	. . . 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 12616	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
29		exp0 14103	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴↑0) = 1)
31		zcn 12616	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
32		exp0 14103	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (𝐵↑0) = 1)
33	31, 32	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ ℤ → (𝐵↑0) = 1)
34	33	eqcomd 2741	. . . . . 6 ⊢ (𝐵 ∈ ℤ → 1 = (𝐵↑0))
35	30, 34	sylan9eq 2795	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴↑0) = (𝐵↑0))
36	35	oveq1d 7446	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))
37	36	3ad2ant1 1132	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))
38		simp21l 1289	. . . . . . . 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 1135	. . . . . . . 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 14114	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℤ)
41	38, 39, 40	syl2anc 584	. . . . . . 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 1290	. . . . . . . 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 14114	. . . . . . . 8 ⊢ ((𝐵 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (𝐵↑𝑘) ∈ ℤ)
44	42, 39, 43	syl2anc 584	. . . . . . 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 1206	. . . . . . 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 1137	. . . . . . 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 1207	. . . . . . 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 13963	. . . . . 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 12721	. . . . . . . 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 14106	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
51	49, 39, 50	syl2anc 584	. . . . . . 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 7446	. . . . . 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 12721	. . . . . . . 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 14106	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐵↑(𝑘 + 1)) = ((𝐵↑𝑘) · 𝐵))
55	53, 39, 54	syl2anc 584	. . . . . . 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 7446	. . . . . 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 2785	. . . . 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 1118	. . . 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 12711	. 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 1086 = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ℕ₀cn0 12524 ℤcz 12611 ℝ⁺crp 13032 mod cmo 13906 ↑cexp 14099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100

This theorem is referenced by: dvdsmodexp 16295 odzdvds 16829 lgsmod 27382 lgsne0 27394 fmtnoprmfac1lem 47489 sfprmdvdsmersenne 47528 41prothprmlem2 47543

W3C validator