Theorem lgsval 27138

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 2726	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑚 = 0 ↔ 𝑁 = 0))
3		simpl 482	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → 𝑎 = 𝐴)
4	3	oveq1d 7416	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑎↑2) = (𝐴↑2))
5	4	eqeq1d 2726	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → ((𝑎↑2) = 1 ↔ (𝐴↑2) = 1))
6	5	ifbid 4543	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → if((𝑎↑2) = 1, 1, 0) = if((𝐴↑2) = 1, 1, 0))
7	1	breq1d 5148	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑚 < 0 ↔ 𝑁 < 0))
8	3	breq1d 5148	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑎 < 0 ↔ 𝐴 < 0))
9	7, 8	anbi12d 630	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → ((𝑚 < 0 ∧ 𝑎 < 0) ↔ (𝑁 < 0 ∧ 𝐴 < 0)))
10	9	ifbid 4543	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → if((𝑚 < 0 ∧ 𝑎 < 0), -1, 1) = if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1))
11	3	breq2d 5150	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (2 ∥ 𝑎 ↔ 2 ∥ 𝐴))
12	3	oveq1d 7416	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑎 mod 8) = (𝐴 mod 8))
13	12	eleq1d 2810	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → ((𝑎 mod 8) ∈ {1, 7} ↔ (𝐴 mod 8) ∈ {1, 7}))
14	13	ifbid 4543	. . . . . . . . . . . 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 4546	. . . . . . . . . . 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 7416	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑎↑((𝑛 − 1) / 2)) = (𝐴↑((𝑛 − 1) / 2)))
17	16	oveq1d 7416	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → ((𝑎↑((𝑛 − 1) / 2)) + 1) = ((𝐴↑((𝑛 − 1) / 2)) + 1))
18	17	oveq1d 7416	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) = (((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛))
19	18	oveq1d 7416	. . . . . . . . . . 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 4541	. . . . . . . . . 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 7417	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (𝑛 pCnt 𝑚) = (𝑛 pCnt 𝑁))
22	20, 21	oveq12d 7419	. . . . . . . . 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 4539	. . . . . . . 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 5240	. . . . . . 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 2782	. . . . . 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 13970	. . . . 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 6885	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑚 = 𝑁) → (abs‘𝑚) = (abs‘𝑁))
29	27, 28	fveq12d 6888	. . . 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 7419	. . 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 4548	. 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 27132	. 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 12484	. . . . 5 ⊢ 1 ∈ ℕ0
34		0nn0 12483	. . . . 5 ⊢ 0 ∈ ℕ0
35	33, 34	ifcli 4567	. . . 4 ⊢ if((𝐴↑2) = 1, 1, 0) ∈ ℕ0
36	35	elexi 3486	. . 3 ⊢ if((𝐴↑2) = 1, 1, 0) ∈ V
37		ovex 7434	. . 3 ⊢ (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) ∈ V
38	36, 37	ifex 4570	. 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 7555	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 1533   ∈ wcel 2098  ifcif 4520  {cpr 4622   class class class wbr 5138   ↦ cmpt 5221  ‘cfv 6533  (class class class)co 7401  0cc0 11105  1c1 11106   + caddc 11108   · cmul 11110   < clt 11244   − cmin 11440  -cneg 11441   / cdiv 11867  ℕcn 12208  2c2 12263  7c7 12268  8c8 12269  ℕ₀cn0 12468  ℤcz 12554   mod cmo 13830  seqcseq 13962  ↑cexp 14023  abscabs 15177   ∥ cdvds 16193  ℙcprime 16604   pCnt cpc 16765   /_L clgs 27131

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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-mulcl 11167  ax-i2m1 11173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-nn 12209  df-n0 12469  df-seq 13963  df-lgs 27132

This theorem is referenced by:  lgscllem  27141  lgsval2lem  27144  lgs0  27147  lgsval4  27154