Theorem lgsval 27300

Description: Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.)

Hypothesis

Ref	Expression
lgsval.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))

Assertion

Ref	Expression
lgsval	⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) = if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))))

Distinct variable groups:   𝐴,𝑛   𝑛,𝑁

Allowed substitution hint:   𝐹(𝑛)

Proof of Theorem lgsval

Dummy variables 𝑎 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → 𝑚 = 𝑁)
2	1	eqeq1d 2736	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑚 = 0 ↔ 𝑁 = 0))
3		simpl 482	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → 𝑎 = 𝐴)
4	3	oveq1d 7429	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑎↑2) = (𝐴↑2))
5	4	eqeq1d 2736	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → ((𝑎↑2) = 1 ↔ (𝐴↑2) = 1))
6	5	ifbid 4531	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → if((𝑎↑2) = 1, 1, 0) = if((𝐴↑2) = 1, 1, 0))
7	1	breq1d 5135	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑚 < 0 ↔ 𝑁 < 0))
8	3	breq1d 5135	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑎 < 0 ↔ 𝐴 < 0))
9	7, 8	anbi12d 632	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → ((𝑚 < 0 ∧ 𝑎 < 0) ↔ (𝑁 < 0 ∧ 𝐴 < 0)))
10	9	ifbid 4531	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → if((𝑚 < 0 ∧ 𝑎 < 0), -1, 1) = if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1))
11	3	breq2d 5137	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (2 ∥ 𝑎 ↔ 2 ∥ 𝐴))
12	3	oveq1d 7429	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑎 mod 8) = (𝐴 mod 8))
13	12	eleq1d 2818	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → ((𝑎 mod 8) ∈ {1, 7} ↔ (𝐴 mod 8) ∈ {1, 7}))
14	13	ifbid 4531	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → if((𝑎 mod 8) ∈ {1, 7}, 1, -1) = if((𝐴 mod 8) ∈ {1, 7}, 1, -1))
15	11, 14	ifbieq2d 4534	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)) = if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)))
16	3	oveq1d 7429	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑎↑((𝑛 − 1) / 2)) = (𝐴↑((𝑛 − 1) / 2)))
17	16	oveq1d 7429	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → ((𝑎↑((𝑛 − 1) / 2)) + 1) = ((𝐴↑((𝑛 − 1) / 2)) + 1))
18	17	oveq1d 7429	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) = (((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛))
19	18	oveq1d 7429	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1) = ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))
20	15, 19	ifeq12d 4529	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → if(𝑛 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1)) = if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1)))
21	1	oveq2d 7430	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑛 pCnt 𝑚) = (𝑛 pCnt 𝑁))
22	20, 21	oveq12d 7432	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (if(𝑛 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑚)) = (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)))
23	22	ifeq1d 4527	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑚)), 1) = if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))
24	23	mpteq2dv 5226	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑚)), 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1)))
25		lgsval.1	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1))
26	24, 25	eqtr4di 2787	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑚)), 1)) = 𝐹)
27	26	seqeq3d 14033	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑚)), 1))) = seq1( · , 𝐹))
28	1	fveq2d 6891	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (abs‘𝑚) = (abs‘𝑁))
29	27, 28	fveq12d 6894	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑚)), 1)))‘(abs‘𝑚)) = (seq1( · , 𝐹)‘(abs‘𝑁)))
30	10, 29	oveq12d 7432	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (if((𝑚 < 0 ∧ 𝑎 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑚)), 1)))‘(abs‘𝑚))) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))))
31	2, 6, 30	ifbieq12d 4536	. 2 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → if(𝑚 = 0, if((𝑎↑2) = 1, 1, 0), (if((𝑚 < 0 ∧ 𝑎 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑚)), 1)))‘(abs‘𝑚)))) = if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))))
32		df-lgs 27294	. 2 ⊢ /L = (𝑎 ∈ ℤ, 𝑚 ∈ ℤ ↦ if(𝑚 = 0, if((𝑎↑2) = 1, 1, 0), (if((𝑚 < 0 ∧ 𝑎 < 0), -1, 1) · (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑚)), 1)))‘(abs‘𝑚)))))
33		1nn0 12526	. . . . 5 ⊢ 1 ∈ ℕ0
34		0nn0 12525	. . . . 5 ⊢ 0 ∈ ℕ0
35	33, 34	ifcli 4555	. . . 4 ⊢ if((𝐴↑2) = 1, 1, 0) ∈ ℕ0
36	35	elexi 3487	. . 3 ⊢ if((𝐴↑2) = 1, 1, 0) ∈ V
37		ovex 7447	. . 3 ⊢ (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) ∈ V
38	36, 37	ifex 4558	. 2 ⊢ if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))) ∈ V
39	31, 32, 38	ovmpoa 7571	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) = if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ifcif 4507  {cpr 4610   class class class wbr 5125   ↦ cmpt 5207  ‘cfv 6542  (class class class)co 7414  0cc0 11138  1c1 11139   + caddc 11141   · cmul 11143   < clt 11278   − cmin 11475  -cneg 11476   / cdiv 11903  ℕcn 12249  2c2 12304  7c7 12309  8c8 12310  ℕ₀cn0 12510  ℤcz 12597   mod cmo 13892  seqcseq 14025  ↑cexp 14085  abscabs 15256   ∥ cdvds 16273  ℙcprime 16691   pCnt cpc 16857   /_L clgs 27293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414  ax-un 7738  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-mulcl 11200  ax-i2m1 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7871  df-2nd 7998  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433  df-nn 12250  df-n0 12511  df-seq 14026  df-lgs 27294

This theorem is referenced by:  lgscllem  27303  lgsval2lem  27306  lgs0  27309  lgsval4  27316