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Theorem modexp 14210

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 7398	. . . . . 6 ⊢ (𝑥 = 0 → (𝐴↑𝑥) = (𝐴↑0))
5	4	oveq1d 7405	. . . . 5 ⊢ (𝑥 = 0 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑0) mod 𝐷))
6		oveq2 7398	. . . . . 6 ⊢ (𝑥 = 0 → (𝐵↑𝑥) = (𝐵↑0))
7	6	oveq1d 7405	. . . . 5 ⊢ (𝑥 = 0 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑0) mod 𝐷))
8	5, 7	eqeq12d 2746	. . . 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 7398	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐴↑𝑥) = (𝐴↑𝑘))
11	10	oveq1d 7405	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑𝑘) mod 𝐷))
12		oveq2 7398	. . . . . 6 ⊢ (𝑥 = 𝑘 → (𝐵↑𝑥) = (𝐵↑𝑘))
13	12	oveq1d 7405	. . . . 5 ⊢ (𝑥 = 𝑘 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑𝑘) mod 𝐷))
14	11, 13	eqeq12d 2746	. . . 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 7398	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐴↑𝑥) = (𝐴↑(𝑘 + 1)))
17	16	oveq1d 7405	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑(𝑘 + 1)) mod 𝐷))
18		oveq2 7398	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (𝐵↑𝑥) = (𝐵↑(𝑘 + 1)))
19	18	oveq1d 7405	. . . . 5 ⊢ (𝑥 = (𝑘 + 1) → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑(𝑘 + 1)) mod 𝐷))
20	17, 19	eqeq12d 2746	. . . 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 7398	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴↑𝑥) = (𝐴↑𝐶))
23	22	oveq1d 7405	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴↑𝑥) mod 𝐷) = ((𝐴↑𝐶) mod 𝐷))
24		oveq2 7398	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵↑𝑥) = (𝐵↑𝐶))
25	24	oveq1d 7405	. . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐵↑𝑥) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷))
26	23, 25	eqeq12d 2746	. . . 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 12541	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
29		exp0 14037	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴↑0) = 1)
31		zcn 12541	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ)
32		exp0 14037	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (𝐵↑0) = 1)
33	31, 32	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ ℤ → (𝐵↑0) = 1)
34	33	eqcomd 2736	. . . . . 6 ⊢ (𝐵 ∈ ℤ → 1 = (𝐵↑0))
35	30, 34	sylan9eq 2785	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴↑0) = (𝐵↑0))
36	35	oveq1d 7405	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴↑0) mod 𝐷) = ((𝐵↑0) mod 𝐷))
37	36	3ad2ant1 1133	. . 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 1136	. . . . . . . 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 14048	. . . . . . . 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 14048	. . . . . . . 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 1138	. . . . . . 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 13897	. . . . . 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 12646	. . . . . . . 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 14040	. . . . . . . 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 7405	. . . . . 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 12646	. . . . . . . 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 14040	. . . . . . . 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 7405	. . . . . 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 2775	. . . . 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 1119	. . . 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 12636	. 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 1540 ∈ wcel 2109 (class class class)co 7390 ℂcc 11073 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 ℕ₀cn0 12449 ℤcz 12536 ℝ⁺crp 12958 mod cmo 13838 ↑cexp 14033

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034

This theorem is referenced by: dvdsmodexp 16237 odzdvds 16773 lgsmod 27241 lgsne0 27253 fmtnoprmfac1lem 47569 sfprmdvdsmersenne 47608 41prothprmlem2 47623

W3C validator