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Theorem lgsneg 27272

Description: The Legendre symbol is either even or odd under negation with respect to the second parameter according to the sign of the first. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Assertion**
Ref	Expression
lgsneg	⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /_L -𝑁) = (if(𝐴 < 0, -1, 1) · (𝐴 /_L 𝑁)))

Proof of Theorem lgsneg Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iftrue 4536	. . . . . . . . 9 ⊢ (𝐴 < 0 → if(𝐴 < 0, -1, 1) = -1)
2	1	adantl 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → if(𝐴 < 0, -1, 1) = -1)
3	2	oveq1d 7439	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → (if(𝐴 < 0, -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) = (-1 · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)))
4		oveq2 7432	. . . . . . . . . 10 ⊢ (if(𝑁 < 0, -1, 1) = -1 → (-1 · if(𝑁 < 0, -1, 1)) = (-1 · -1))
5		neg1mulneg1e1 12461	. . . . . . . . . 10 ⊢ (-1 · -1) = 1
6	4, 5	eqtrdi 2783	. . . . . . . . 9 ⊢ (if(𝑁 < 0, -1, 1) = -1 → (-1 · if(𝑁 < 0, -1, 1)) = 1)
7		oveq2 7432	. . . . . . . . . 10 ⊢ (if(𝑁 < 0, -1, 1) = 1 → (-1 · if(𝑁 < 0, -1, 1)) = (-1 · 1))
8		ax-1cn 11202	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
9	8	mulm1i 11695	. . . . . . . . . 10 ⊢ (-1 · 1) = -1
10	7, 9	eqtrdi 2783	. . . . . . . . 9 ⊢ (if(𝑁 < 0, -1, 1) = 1 → (-1 · if(𝑁 < 0, -1, 1)) = -1)
11	6, 10	ifsb 4543	. . . . . . . 8 ⊢ (-1 · if(𝑁 < 0, -1, 1)) = if(𝑁 < 0, 1, -1)
12		simpr 483	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → 𝐴 < 0)
13	12	biantrud 530	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → (𝑁 < 0 ↔ (𝑁 < 0 ∧ 𝐴 < 0)))
14	13	ifbid 4553	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → if(𝑁 < 0, -1, 1) = if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1))
15	14	oveq2d 7440	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → (-1 · if(𝑁 < 0, -1, 1)) = (-1 · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)))
16		simpl3 1190	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → 𝑁 ≠ 0)
17	16	necomd 2992	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → 0 ≠ 𝑁)
18		simpl2 1189	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → 𝑁 ∈ ℤ)
19	18	zred 12702	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → 𝑁 ∈ ℝ)
20		0re 11252	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
21		ltlen 11351	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
22	19, 20, 21	sylancl 584	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → (𝑁 < 0 ↔ (𝑁 ≤ 0 ∧ 0 ≠ 𝑁)))
23	17, 22	mpbiran2d 706	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → (𝑁 < 0 ↔ 𝑁 ≤ 0))
24	19	le0neg1d 11821	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → (𝑁 ≤ 0 ↔ 0 ≤ -𝑁))
25	19	renegcld 11677	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → -𝑁 ∈ ℝ)
26		lenlt 11328	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ -𝑁 ∈ ℝ) → (0 ≤ -𝑁 ↔ ¬ -𝑁 < 0))
27	20, 25, 26	sylancr 585	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → (0 ≤ -𝑁 ↔ ¬ -𝑁 < 0))
28	23, 24, 27	3bitrd 304	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → (𝑁 < 0 ↔ ¬ -𝑁 < 0))
29	28	ifbid 4553	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → if(𝑁 < 0, 1, -1) = if(¬ -𝑁 < 0, 1, -1))
30		ifnot 4582	. . . . . . . . 9 ⊢ if(¬ -𝑁 < 0, 1, -1) = if(-𝑁 < 0, -1, 1)
31	29, 30	eqtrdi 2783	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → if(𝑁 < 0, 1, -1) = if(-𝑁 < 0, -1, 1))
32	11, 15, 31	3eqtr3a 2791	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → (-1 · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) = if(-𝑁 < 0, -1, 1))
33	12	biantrud 530	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → (-𝑁 < 0 ↔ (-𝑁 < 0 ∧ 𝐴 < 0)))
34	33	ifbid 4553	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → if(-𝑁 < 0, -1, 1) = if((-𝑁 < 0 ∧ 𝐴 < 0), -1, 1))
35	3, 32, 34	3eqtrd 2771	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝐴 < 0) → (if(𝐴 < 0, -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) = if((-𝑁 < 0 ∧ 𝐴 < 0), -1, 1))
36		1t1e1 12410	. . . . . . 7 ⊢ (1 · 1) = 1
37		iffalse 4539	. . . . . . . . 9 ⊢ (¬ 𝐴 < 0 → if(𝐴 < 0, -1, 1) = 1)
38	37	adantl 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ ¬ 𝐴 < 0) → if(𝐴 < 0, -1, 1) = 1)
39		simpr 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ ¬ 𝐴 < 0) → ¬ 𝐴 < 0)
40	39	intnand 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ ¬ 𝐴 < 0) → ¬ (𝑁 < 0 ∧ 𝐴 < 0))
41	40	iffalsed 4541	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ ¬ 𝐴 < 0) → if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) = 1)
42	38, 41	oveq12d 7442	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ ¬ 𝐴 < 0) → (if(𝐴 < 0, -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) = (1 · 1))
43	39	intnand 487	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ ¬ 𝐴 < 0) → ¬ (-𝑁 < 0 ∧ 𝐴 < 0))
44	43	iffalsed 4541	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ ¬ 𝐴 < 0) → if((-𝑁 < 0 ∧ 𝐴 < 0), -1, 1) = 1)
45	36, 42, 44	3eqtr4a 2793	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ ¬ 𝐴 < 0) → (if(𝐴 < 0, -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) = if((-𝑁 < 0 ∧ 𝐴 < 0), -1, 1))
46	35, 45	pm2.61dan 811	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (if(𝐴 < 0, -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) = if((-𝑁 < 0 ∧ 𝐴 < 0), -1, 1))
47	46	eqcomd 2733	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → if((-𝑁 < 0 ∧ 𝐴 < 0), -1, 1) = (if(𝐴 < 0, -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)))
48		simpr 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ)
49		simpl2 1189	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℙ) → 𝑁 ∈ ℤ)
50		zq 12974	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℚ)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℙ) → 𝑁 ∈ ℚ)
52		pcneg 16848	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℙ ∧ 𝑁 ∈ ℚ) → (𝑛 pCnt -𝑁) = (𝑛 pCnt 𝑁))
53	48, 51, 52	syl2anc 582	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt -𝑁) = (𝑛 pCnt 𝑁))
54	53	oveq2d 7440	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑛 ∈ ℙ) → ((𝐴 /_L 𝑛)↑(𝑛 pCnt -𝑁)) = ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)))
55	54	ifeq1da 4561	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt -𝑁)), 1) = if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))
56	55	mpteq2dv 5252	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt -𝑁)), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))
57	56	seqeq3d 14012	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt -𝑁)), 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))))
58		zcn 12599	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
59	58	3ad2ant2 1131	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℂ)
60	59	absnegd 15434	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘-𝑁) = (abs‘𝑁))
61	57, 60	fveq12d 6907	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt -𝑁)), 1)))‘(abs‘-𝑁)) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)))
62	47, 61	oveq12d 7442	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (if((-𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt -𝑁)), 1)))‘(abs‘-𝑁))) = ((if(𝐴 < 0, -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁))))
63		neg1cn 12362	. . . . . 6 ⊢ -1 ∈ ℂ
64	63, 8	ifcli 4577	. . . . 5 ⊢ if(𝐴 < 0, -1, 1) ∈ ℂ
65	64	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → if(𝐴 < 0, -1, 1) ∈ ℂ)
66	63, 8	ifcli 4577	. . . . 5 ⊢ if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) ∈ ℂ
67	66	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) ∈ ℂ)
68		nnabscl 15310	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
69	68	3adant1 1127	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ)
70		nnuz 12901	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
71	69, 70	eleqtrdi 2838	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ (ℤ_≥‘1))
72		eqid 2727	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))
73	72	lgsfcl3 27269	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)):ℕ⟶ℤ)
74		elfznn 13568	. . . . . . 7 ⊢ (𝑥 ∈ (1...(abs‘𝑁)) → 𝑥 ∈ ℕ)
75		ffvelcdm 7094	. . . . . . 7 ⊢ (((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)):ℕ⟶ℤ ∧ 𝑥 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))‘𝑥) ∈ ℤ)
76	73, 74, 75	syl2an 594	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑥 ∈ (1...(abs‘𝑁))) → ((𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1))‘𝑥) ∈ ℤ)
77		zmulcl 12647	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 · 𝑦) ∈ ℤ)
78	77	adantl 480	. . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 · 𝑦) ∈ ℤ)
79	71, 76, 78	seqcl 14025	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)) ∈ ℤ)
80	79	zcnd 12703	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)) ∈ ℂ)
81	65, 67, 80	mulassd 11273	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ((if(𝐴 < 0, -1, 1) · if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1)) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁))) = (if(𝐴 < 0, -1, 1) · (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)))))
82	62, 81	eqtrd 2767	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (if((-𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt -𝑁)), 1)))‘(abs‘-𝑁))) = (if(𝐴 < 0, -1, 1) · (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)))))
83		simp1 1133	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℤ)
84		znegcl 12633	. . . 4 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
85	84	3ad2ant2 1131	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → -𝑁 ∈ ℤ)
86		simp3 1135	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝑁 ≠ 0)
87	59, 86	negne0d 11605	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → -𝑁 ≠ 0)
88		eqid 2727	. . . 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))
89	88	lgsval4 27268	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ -𝑁 ∈ ℤ ∧ -𝑁 ≠ 0) → (𝐴 /_L -𝑁) = (if((-𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt -𝑁)), 1)))‘(abs‘-𝑁))))
90	83, 85, 87, 89	syl3anc 1368	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /_L -𝑁) = (if((-𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt -𝑁)), 1)))‘(abs‘-𝑁))))
91	72	lgsval4 27268	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /_L 𝑁) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁))))
92	91	oveq2d 7440	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (if(𝐴 < 0, -1, 1) · (𝐴 /_L 𝑁)) = (if(𝐴 < 0, -1, 1) · (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /_L 𝑛)↑(𝑛 pCnt 𝑁)), 1)))‘(abs‘𝑁)))))
93	82, 90, 92	3eqtr4d 2777	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /_L -𝑁) = (if(𝐴 < 0, -1, 1) · (𝐴 /_L 𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2936 ifcif 4530 class class class wbr 5150 ↦ cmpt 5233 ⟶wf 6547 ‘cfv 6551 (class class class)co 7424 ℂcc 11142 ℝcr 11143 0cc0 11144 1c1 11145 · cmul 11149 < clt 11284 ≤ cle 11285 -cneg 11481 ℕcn 12248 ℤcz 12594 ℤ_≥cuz 12858 ℚcq 12968 ...cfz 13522 seqcseq 14004 ↑cexp 14064 abscabs 15219 ℙcprime 16647 pCnt cpc 16810 /_L clgs 27245

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-oadd 8495 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-inf 9472 df-dju 9930 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-xnn0 12581 df-z 12595 df-uz 12859 df-q 12969 df-rp 13013 df-fz 13523 df-fzo 13666 df-fl 13795 df-mod 13873 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-dvds 16237 df-gcd 16475 df-prm 16648 df-phi 16740 df-pc 16811 df-lgs 27246

This theorem is referenced by: lgsneg1 27273

W3C validator