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Theorem modexp 13595

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 1196	. 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 1175	. 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 7143	. . . . . 6 ⊢ (𝑥 = 0 → (𝐴↑𝑥) = (𝐴↑0))
5	4	oveq1d 7150	. . . . 5 ⊢ (𝑥 = 0 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑0) mod 𝐷))
6		oveq2 7143	. . . . . 6 ⊢ (𝑥 = 0 → (𝐵↑𝑥) = (𝐵↑0))
7	6	oveq1d 7150	. . . . 5 ⊢ (𝑥 = 0 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑0) mod 𝐷))
8	5, 7	eqeq12d 2814	. . . 4 ⊢ (𝑥 = 0 → (((𝐴↑𝑥) mod 𝐷) = ((𝐵↑𝑥) mod 𝐷) ↔ ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷)))
9	8	imbi2d 344	. . 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 7143	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐴↑𝑥) = (𝐴↑𝑘))
11	10	oveq1d 7150	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑𝑘) mod 𝐷))
12		oveq2 7143	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐵↑𝑥) = (𝐵↑𝑘))
13	12	oveq1d 7150	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷))
14	11, 13	eqeq12d 2814	. . . 4 ⊢ (𝑥 = 𝑘 → (((𝐴↑𝑥) mod 𝐷) = ((𝐵↑𝑥) mod 𝐷) ↔ ((𝐴↑𝑘) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷)))
15	14	imbi2d 344	. . 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 7143	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐴↑𝑥) = (𝐴↑(𝑘 + 1)))
17	16	oveq1d 7150	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑(𝑘 + 1)) mod 𝐷))
18		oveq2 7143	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐵↑𝑥) = (𝐵↑(𝑘 + 1)))
19	18	oveq1d 7150	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))
20	17, 19	eqeq12d 2814	. . . 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 344	. . 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 7143	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴↑𝑥) = (𝐴↑𝐶))
23	22	oveq1d 7150	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑𝐶) mod 𝐷))
24		oveq2 7143	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵↑𝑥) = (𝐵↑𝐶))
25	24	oveq1d 7150	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷))
26	23, 25	eqeq12d 2814	. . . 4 ⊢ (𝑥 = 𝐶 → (((𝐴↑𝑥) mod 𝐷) = ((𝐵↑𝑥) mod 𝐷) ↔ ((𝐴↑𝐶) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷)))
27	26	imbi2d 344	. . 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 11974	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
29		exp0 13429	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴↑0) = 1)
31		zcn 11974	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
32		exp0 13429	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (𝐵↑0) = 1)
33	31, 32	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ ℤ → (𝐵↑0) = 1)
34	33	eqcomd 2804	. . . . . 6 ⊢ (𝐵 ∈ ℤ → 1 = (𝐵↑0))
35	30, 34	sylan9eq 2853	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴↑0) = (𝐵↑0))
36	35	oveq1d 7150	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))
37	36	3ad2ant1 1130	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))
38		simp21l 1287	. . . . . . . 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 1133	. . . . . . . 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 13440	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℤ)
41	38, 39, 40	syl2anc 587	. . . . . . 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 1288	. . . . . . . 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 13440	. . . . . . . 8 ⊢ ((𝐵 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (𝐵↑𝑘) ∈ ℤ)
44	42, 39, 43	syl2anc 587	. . . . . . 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 1204	. . . . . . 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 1135	. . . . . . 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 1205	. . . . . . 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 13288	. . . . . 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 12076	. . . . . . . 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 13432	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
51	49, 39, 50	syl2anc 587	. . . . . . 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 7150	. . . . . 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 12076	. . . . . . . 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 13432	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐵↑(𝑘 + 1)) = ((𝐵↑𝑘) · 𝐵))
55	53, 39, 54	syl2anc 587	. . . . . . 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 7150	. . . . . 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 2843	. . . . 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 1116	. . . 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 12065	. 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ℕ₀cn0 11885 ℤcz 11969 ℝ⁺crp 12377 mod cmo 13232 ↑cexp 13425

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426

This theorem is referenced by: dvdsmodexp 15607 odzdvds 16122 lgsmod 25907 lgsne0 25919 fmtnoprmfac1lem 44081 sfprmdvdsmersenne 44121 41prothprmlem2 44136

W3C validator