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Theorem modexp 14277

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 1200	. 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 1179	. 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 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 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 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 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 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 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 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 12618	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
29		exp0 14106	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴↑0) = 1)
31		zcn 12618	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
32		exp0 14106	. . . . . . . 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 2797	. . . . 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 1134	. . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐷 ∈ ℝ⁺ ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))
38		simp21l 1291	. . . . . . . 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 14117	. . . . . . . 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 1292	. . . . . . . 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 14117	. . . . . . . 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 1208	. . . . . . 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 1209	. . . . . . 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 13966	. . . . . 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 12723	. . . . . . . 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 14109	. . . . . . . 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 12723	. . . . . . . 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 14109	. . . . . . . 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 2787	. . . . 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 12713	. 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 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℕ₀cn0 12526 ℤcz 12613 ℝ⁺crp 13034 mod cmo 13909 ↑cexp 14102

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103

This theorem is referenced by: dvdsmodexp 16298 odzdvds 16833 lgsmod 27367 lgsne0 27379 fmtnoprmfac1lem 47551 sfprmdvdsmersenne 47590 41prothprmlem2 47605

W3C validator