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Theorem lgsmod 27465

Description: The Legendre (Jacobi) symbol is preserved under reduction mod 𝑛 when 𝑛 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Assertion**
Ref	Expression
lgsmod	⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → ((𝐴 mod 𝑁) /_L 𝑁) = (𝐴 /_L 𝑁))

Proof of Theorem lgsmod Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zmodcl 13926	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 mod 𝑁) ∈ ℕ₀)
2	1	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (𝐴 mod 𝑁) ∈ ℕ₀)
3	2	nn0zd 12617	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (𝐴 mod 𝑁) ∈ ℤ)
4	3	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (𝐴 mod 𝑁) ∈ ℤ)
5		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ)
6	5	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∈ ℙ)
7		simpl3 1212	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → ¬ 2 ∥ 𝑁)
8		breq1 5113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 2 → (𝑛 ∥ 𝑁 ↔ 2 ∥ 𝑁))
9	8	notbid 321	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 2 → (¬ 𝑛 ∥ 𝑁 ↔ ¬ 2 ∥ 𝑁))
10	7, 9	syl5ibrcom 250	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → (𝑛 = 2 → ¬ 𝑛 ∥ 𝑁))
11	10	necon2ad 2973	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → (𝑛 ∥ 𝑁 → 𝑛 ≠ 2))
12	11	imp 411	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ≠ 2)
13		eldifsn 4754	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℙ ∖ {2}) ↔ (𝑛 ∈ ℙ ∧ 𝑛 ≠ 2))
14	6, 12, 13	sylanbrc 594	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∈ (ℙ ∖ {2}))
15		oddprm 16871	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℙ ∖ {2}) → ((𝑛 − 1) / 2) ∈ ℕ)
16	14, 15	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝑛 − 1) / 2) ∈ ℕ)
17	16	nnnn0d 12566	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝑛 − 1) / 2) ∈ ℕ₀)
18		zexpcl 14114	. . . . . . . . . . . . 13 ⊢ (((𝐴 mod 𝑁) ∈ ℤ ∧ ((𝑛 − 1) / 2) ∈ ℕ₀) → ((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) ∈ ℤ)
19	4, 17, 18	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) ∈ ℤ)
20	19	zred 12701	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) ∈ ℝ)
21		simpll1 1231	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝐴 ∈ ℤ)
22		zexpcl 14114	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ ((𝑛 − 1) / 2) ∈ ℕ₀) → (𝐴↑((𝑛 − 1) / 2)) ∈ ℤ)
23	21, 17, 22	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (𝐴↑((𝑛 − 1) / 2)) ∈ ℤ)
24	23	zred 12701	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (𝐴↑((𝑛 − 1) / 2)) ∈ ℝ)
25		1red 11210	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 1 ∈ ℝ)
26		prmnn 16733	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ)
27	26	ad2antlr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∈ ℕ)
28	27	nnrpd 13059	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∈ ℝ⁺)
29		prmz 16734	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℤ)
30	29	ad2antlr 739	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∈ ℤ)
31		simp2 1155	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → 𝑁 ∈ ℕ)
32	31	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑁 ∈ ℕ)
33	32	nnzd 12618	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑁 ∈ ℤ)
34	4, 21	zsubcld 12706	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) − 𝐴) ∈ ℤ)
35		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∥ 𝑁)
36	21	zred 12701	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝐴 ∈ ℝ)
37	32	nnrpd 13059	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑁 ∈ ℝ⁺)
38		modabs2 13940	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → ((𝐴 mod 𝑁) mod 𝑁) = (𝐴 mod 𝑁))
39	36, 37, 38	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) mod 𝑁) = (𝐴 mod 𝑁))
40		moddvds 16322	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 mod 𝑁) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (((𝐴 mod 𝑁) mod 𝑁) = (𝐴 mod 𝑁) ↔ 𝑁 ∥ ((𝐴 mod 𝑁) − 𝐴)))
41	32, 4, 21, 40	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (((𝐴 mod 𝑁) mod 𝑁) = (𝐴 mod 𝑁) ↔ 𝑁 ∥ ((𝐴 mod 𝑁) − 𝐴)))
42	39, 41	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑁 ∥ ((𝐴 mod 𝑁) − 𝐴))
43	30, 33, 34, 35, 42	dvdstrd 16354	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∥ ((𝐴 mod 𝑁) − 𝐴))
44		moddvds 16322	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴 mod 𝑁) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (((𝐴 mod 𝑁) mod 𝑛) = (𝐴 mod 𝑛) ↔ 𝑛 ∥ ((𝐴 mod 𝑁) − 𝐴)))
45	27, 4, 21, 44	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (((𝐴 mod 𝑁) mod 𝑛) = (𝐴 mod 𝑛) ↔ 𝑛 ∥ ((𝐴 mod 𝑁) − 𝐴)))
46	43, 45	mpbird 260	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) mod 𝑛) = (𝐴 mod 𝑛))
47		modexp 14276	. . . . . . . . . . . 12 ⊢ ((((𝐴 mod 𝑁) ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (((𝑛 − 1) / 2) ∈ ℕ₀ ∧ 𝑛 ∈ ℝ⁺) ∧ ((𝐴 mod 𝑁) mod 𝑛) = (𝐴 mod 𝑛)) → (((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) mod 𝑛) = ((𝐴↑((𝑛 − 1) / 2)) mod 𝑛))
48	4, 21, 17, 28, 46, 47	syl221anc 1408	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) mod 𝑛) = ((𝐴↑((𝑛 − 1) / 2)) mod 𝑛))
49		modadd1 13943	. . . . . . . . . . 11 ⊢ (((((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) ∈ ℝ ∧ (𝐴↑((𝑛 − 1) / 2)) ∈ ℝ) ∧ (1 ∈ ℝ ∧ 𝑛 ∈ ℝ⁺) ∧ (((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) mod 𝑛) = ((𝐴↑((𝑛 − 1) / 2)) mod 𝑛)) → ((((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) + 1) mod 𝑛) = (((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛))
50	20, 24, 25, 28, 48, 49	syl221anc 1408	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) + 1) mod 𝑛) = (((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛))
51	50	oveq1d 7427	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (((((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1) = ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))
52		lgsval3 27457	. . . . . . . . . 10 ⊢ (((𝐴 mod 𝑁) ∈ ℤ ∧ 𝑛 ∈ (ℙ ∖ {2})) → ((𝐴 mod 𝑁) /_L 𝑛) = (((((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))
53	4, 14, 52	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) /_L 𝑛) = (((((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))
54		lgsval3 27457	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (ℙ ∖ {2})) → (𝐴 /_L 𝑛) = ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))
55	21, 14, 54	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (𝐴 /_L 𝑛) = ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))
56	51, 53, 55	3eqtr4d 2808	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) /_L 𝑛) = (𝐴 /_L 𝑛))
57	56	oveq1d 7427	. . . . . . 7 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)) = ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)))
58	3	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (𝐴 mod 𝑁) ∈ ℤ)
59	29	ad2antlr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → 𝑛 ∈ ℤ)
60		lgscl 27453	. . . . . . . . . . . 12 ⊢ (((𝐴 mod 𝑁) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝐴 mod 𝑁) /_L 𝑛) ∈ ℤ)
61	58, 59, 60	syl2anc 595	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) /_L 𝑛) ∈ ℤ)
62	61	zcnd 12702	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) /_L 𝑛) ∈ ℂ)
63	62	exp0d 14178	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (((𝐴 mod 𝑁) /_L 𝑛)↑0) = 1)
64		simpll1 1231	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → 𝐴 ∈ ℤ)
65		lgscl 27453	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝐴 /_L 𝑛) ∈ ℤ)
66	64, 59, 65	syl2anc 595	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (𝐴 /_L 𝑛) ∈ ℤ)
67	66	zcnd 12702	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (𝐴 /_L 𝑛) ∈ ℂ)
68	67	exp0d 14178	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → ((𝐴 /_L 𝑛)↑0) = 1)
69	63, 68	eqtr4d 2801	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (((𝐴 mod 𝑁) /_L 𝑛)↑0) = ((𝐴 /_L 𝑛)↑0))
70	31	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → 𝑁 ∈ ℕ)
71		pceq0 16932	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((𝑛 pCnt 𝑁) = 0 ↔ ¬ 𝑛 ∥ 𝑁))
72	5, 70, 71	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑁) = 0 ↔ ¬ 𝑛 ∥ 𝑁))
73	72	biimpar 482	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (𝑛 pCnt 𝑁) = 0)
74	73	oveq2d 7428	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)) = (((𝐴 mod 𝑁) /_L 𝑛)↑0))
75	73	oveq2d 7428	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)) = ((𝐴 /_L 𝑛)↑0))
76	69, 74, 75	3eqtr4d 2808	. . . . . . 7 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)) = ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)))
77	57, 76	pm2.61dan 824	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)) = ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)))
78	77	ifeq1da 4520	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → if(𝑛 ∈ ℙ, (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1) = if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))
79	78	mpteq2dv 5206	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))
80	79	seqeq3d 14047	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))))
81	80	fveq1d 6885	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘𝑁) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘𝑁))
82		eqid 2763	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))
83	82	lgsval4a 27461	. . 3 ⊢ (((𝐴 mod 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) /_L 𝑁) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘𝑁))
84	3, 31, 83	syl2anc 595	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → ((𝐴 mod 𝑁) /_L 𝑁) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (((𝐴 mod 𝑁) /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘𝑁))
85		eqid 2763	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))
86	85	lgsval4a 27461	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 /_L 𝑁) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘𝑁))
87	86	3adant3 1150	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (𝐴 /_L 𝑁) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘𝑁))
88	81, 84, 87	3eqtr4d 2808	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → ((𝐴 mod 𝑁) /_L 𝑁) = (𝐴 /_L 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3903 ifcif 4488 {csn 4590 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6538 (class class class)co 7412 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 − cmin 11442 / cdiv 11872 ℕcn 12234 2c2 12296 ℕ₀cn0 12505 ℤcz 12592 ℝ⁺crp 13017 mod cmo 13904 seqcseq 14039 ↑cexp 14099 ∥ cdvds 16311 ℙcprime 16730 pCnt cpc 16897 /_L clgs 27436

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-oadd 8458 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-inf 9404 df-dju 9888 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-xnn0 12579 df-z 12593 df-uz 12864 df-q 12974 df-rp 13018 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-dvds 16312 df-gcd 16554 df-prm 16731 df-phi 16826 df-pc 16898 df-lgs 27437

This theorem is referenced by: lgsmodeq 27484 lgsqr 27493 lgsdchrval 27496

W3C validator