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Theorem lgsmod 27234

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 13853	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 mod 𝑁) ∈ ℕ₀)
2	1	3adant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (𝐴 mod 𝑁) ∈ ℕ₀)
3	2	nn0zd 12555	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → (𝐴 mod 𝑁) ∈ ℤ)
4	3	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (𝐴 mod 𝑁) ∈ ℤ)
5		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ)
6	5	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∈ ℙ)
7		simpl3 1194	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → ¬ 2 ∥ 𝑁)
8		breq1 5110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 2 → (𝑛 ∥ 𝑁 ↔ 2 ∥ 𝑁))
9	8	notbid 318	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 2 → (¬ 𝑛 ∥ 𝑁 ↔ ¬ 2 ∥ 𝑁))
10	7, 9	syl5ibrcom 247	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → (𝑛 = 2 → ¬ 𝑛 ∥ 𝑁))
11	10	necon2ad 2940	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → (𝑛 ∥ 𝑁 → 𝑛 ≠ 2))
12	11	imp 406	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ≠ 2)
13		eldifsn 4750	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℙ ∖ {2}) ↔ (𝑛 ∈ ℙ ∧ 𝑛 ≠ 2))
14	6, 12, 13	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∈ (ℙ ∖ {2}))
15		oddprm 16781	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℙ ∖ {2}) → ((𝑛 − 1) / 2) ∈ ℕ)
16	14, 15	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝑛 − 1) / 2) ∈ ℕ)
17	16	nnnn0d 12503	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝑛 − 1) / 2) ∈ ℕ₀)
18		zexpcl 14041	. . . . . . . . . . . . 13 ⊢ (((𝐴 mod 𝑁) ∈ ℤ ∧ ((𝑛 − 1) / 2) ∈ ℕ₀) → ((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) ∈ ℤ)
19	4, 17, 18	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) ∈ ℤ)
20	19	zred 12638	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁)↑((𝑛 − 1) / 2)) ∈ ℝ)
21		simpll1 1213	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝐴 ∈ ℤ)
22		zexpcl 14041	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ ((𝑛 − 1) / 2) ∈ ℕ₀) → (𝐴↑((𝑛 − 1) / 2)) ∈ ℤ)
23	21, 17, 22	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (𝐴↑((𝑛 − 1) / 2)) ∈ ℤ)
24	23	zred 12638	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → (𝐴↑((𝑛 − 1) / 2)) ∈ ℝ)
25		1red 11175	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 1 ∈ ℝ)
26		prmnn 16644	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ)
27	26	ad2antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∈ ℕ)
28	27	nnrpd 12993	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∈ ℝ⁺)
29		prmz 16645	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℤ)
30	29	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∈ ℤ)
31		simp2 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → 𝑁 ∈ ℕ)
32	31	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑁 ∈ ℕ)
33	32	nnzd 12556	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑁 ∈ ℤ)
34	4, 21	zsubcld 12643	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) − 𝐴) ∈ ℤ)
35		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∥ 𝑁)
36	21	zred 12638	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝐴 ∈ ℝ)
37	32	nnrpd 12993	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑁 ∈ ℝ⁺)
38		modabs2 13867	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ⁺) → ((𝐴 mod 𝑁) mod 𝑁) = (𝐴 mod 𝑁))
39	36, 37, 38	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) mod 𝑁) = (𝐴 mod 𝑁))
40		moddvds 16233	. . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑁 ∥ ((𝐴 mod 𝑁) − 𝐴))
43	30, 33, 34, 35, 42	dvdstrd 16265	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → 𝑛 ∥ ((𝐴 mod 𝑁) − 𝐴))
44		moddvds 16233	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) mod 𝑛) = (𝐴 mod 𝑛))
47		modexp 14203	. . . . . . . . . . . 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 1383	. . . . . . . . . . 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 13870	. . . . . . . . . . 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 1383	. . . . . . . . . 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 7402	. . . . . . . . 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 27226	. . . . . . . . . 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 584	. . . . . . . . 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 27226	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (ℙ ∖ {2})) → (𝐴 /_L 𝑛) = ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))
55	21, 14, 54	syl2anc 584	. . . . . . . . 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 2774	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) /_L 𝑛) = (𝐴 /_L 𝑛))
57	56	oveq1d 7402	. . . . . . 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 726	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (𝐴 mod 𝑁) ∈ ℤ)
59	29	ad2antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → 𝑛 ∈ ℤ)
60		lgscl 27222	. . . . . . . . . . . 12 ⊢ (((𝐴 mod 𝑁) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝐴 mod 𝑁) /_L 𝑛) ∈ ℤ)
61	58, 59, 60	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) /_L 𝑛) ∈ ℤ)
62	61	zcnd 12639	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → ((𝐴 mod 𝑁) /_L 𝑛) ∈ ℂ)
63	62	exp0d 14105	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (((𝐴 mod 𝑁) /_L 𝑛)↑0) = 1)
64		simpll1 1213	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → 𝐴 ∈ ℤ)
65		lgscl 27222	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝐴 /_L 𝑛) ∈ ℤ)
66	64, 59, 65	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (𝐴 /_L 𝑛) ∈ ℤ)
67	66	zcnd 12639	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (𝐴 /_L 𝑛) ∈ ℂ)
68	67	exp0d 14105	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → ((𝐴 /_L 𝑛)↑0) = 1)
69	63, 68	eqtr4d 2767	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (((𝐴 mod 𝑁) /_L 𝑛)↑0) = ((𝐴 /_L 𝑛)↑0))
70	31	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → 𝑁 ∈ ℕ)
71		pceq0 16842	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℙ ∧ 𝑁 ∈ ℕ) → ((𝑛 pCnt 𝑁) = 0 ↔ ¬ 𝑛 ∥ 𝑁))
72	5, 70, 71	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑁) = 0 ↔ ¬ 𝑛 ∥ 𝑁))
73	72	biimpar 477	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝑛 ∈ ℙ) ∧ ¬ 𝑛 ∥ 𝑁) → (𝑛 pCnt 𝑁) = 0)
74	73	oveq2d 7403	. . . . . . . 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 7403	. . . . . . . 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 2774	. . . . . . 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 812	. . . . . 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 5201	. . . 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 13974	. . 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 6860	. 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 2729	. . . 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 27230	. . 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 584	. 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 2729	. . . 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 27230	. . 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 1132	. 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 2774	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3911 ifcif 4488 {csn 4589 class class class wbr 5107 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 − cmin 11405 / cdiv 11835 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 ℤcz 12529 ℝ⁺crp 12951 mod cmo 13831 seqcseq 13966 ↑cexp 14026 ∥ cdvds 16222 ℙcprime 16641 pCnt cpc 16807 /_L clgs 27205

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 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-gcd 16465 df-prm 16642 df-phi 16736 df-pc 16808 df-lgs 27206

This theorem is referenced by: lgsmodeq 27253 lgsqr 27262 lgsdchrval 27265

W3C validator