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Theorem lgsval 26449
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.
StepHypRef Expression
1 simpr 485 . . . 4 ((𝑎 = 𝐴𝑚 = 𝑁) → 𝑚 = 𝑁)
21eqeq1d 2740 . . 3 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑚 = 0 ↔ 𝑁 = 0))
3 simpl 483 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑁) → 𝑎 = 𝐴)
43oveq1d 7290 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑎↑2) = (𝐴↑2))
54eqeq1d 2740 . . . 4 ((𝑎 = 𝐴𝑚 = 𝑁) → ((𝑎↑2) = 1 ↔ (𝐴↑2) = 1))
65ifbid 4482 . . 3 ((𝑎 = 𝐴𝑚 = 𝑁) → if((𝑎↑2) = 1, 1, 0) = if((𝐴↑2) = 1, 1, 0))
71breq1d 5084 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑚 < 0 ↔ 𝑁 < 0))
83breq1d 5084 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑎 < 0 ↔ 𝐴 < 0))
97, 8anbi12d 631 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑁) → ((𝑚 < 0 ∧ 𝑎 < 0) ↔ (𝑁 < 0 ∧ 𝐴 < 0)))
109ifbid 4482 . . . 4 ((𝑎 = 𝐴𝑚 = 𝑁) → if((𝑚 < 0 ∧ 𝑎 < 0), -1, 1) = if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1))
113breq2d 5086 . . . . . . . . . . . 12 ((𝑎 = 𝐴𝑚 = 𝑁) → (2 ∥ 𝑎 ↔ 2 ∥ 𝐴))
123oveq1d 7290 . . . . . . . . . . . . . 14 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑎 mod 8) = (𝐴 mod 8))
1312eleq1d 2823 . . . . . . . . . . . . 13 ((𝑎 = 𝐴𝑚 = 𝑁) → ((𝑎 mod 8) ∈ {1, 7} ↔ (𝐴 mod 8) ∈ {1, 7}))
1413ifbid 4482 . . . . . . . . . . . 12 ((𝑎 = 𝐴𝑚 = 𝑁) → if((𝑎 mod 8) ∈ {1, 7}, 1, -1) = if((𝐴 mod 8) ∈ {1, 7}, 1, -1))
1511, 14ifbieq2d 4485 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑚 = 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)))
163oveq1d 7290 . . . . . . . . . . . . . 14 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑎↑((𝑛 − 1) / 2)) = (𝐴↑((𝑛 − 1) / 2)))
1716oveq1d 7290 . . . . . . . . . . . . 13 ((𝑎 = 𝐴𝑚 = 𝑁) → ((𝑎↑((𝑛 − 1) / 2)) + 1) = ((𝐴↑((𝑛 − 1) / 2)) + 1))
1817oveq1d 7290 . . . . . . . . . . . 12 ((𝑎 = 𝐴𝑚 = 𝑁) → (((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) = (((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛))
1918oveq1d 7290 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑚 = 𝑁) → ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1) = ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))
2015, 19ifeq12d 4480 . . . . . . . . . 10 ((𝑎 = 𝐴𝑚 = 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)))
211oveq2d 7291 . . . . . . . . . 10 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑛 pCnt 𝑚) = (𝑛 pCnt 𝑁))
2220, 21oveq12d 7293 . . . . . . . . 9 ((𝑎 = 𝐴𝑚 = 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)))
2322ifeq1d 4478 . . . . . . . 8 ((𝑎 = 𝐴𝑚 = 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))
2423mpteq2dv 5176 . . . . . . 7 ((𝑎 = 𝐴𝑚 = 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))
2624, 25eqtr4di 2796 . . . . . 6 ((𝑎 = 𝐴𝑚 = 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)
2726seqeq3d 13729 . . . . 5 ((𝑎 = 𝐴𝑚 = 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))
281fveq2d 6778 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑁) → (abs‘𝑚) = (abs‘𝑁))
2927, 28fveq12d 6781 . . . 4 ((𝑎 = 𝐴𝑚 = 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)))
3010, 29oveq12d 7293 . . 3 ((𝑎 = 𝐴𝑚 = 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))))
312, 6, 30ifbieq12d 4487 . 2 ((𝑎 = 𝐴𝑚 = 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 26443 . 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 12249 . . . . 5 1 ∈ ℕ0
34 0nn0 12248 . . . . 5 0 ∈ ℕ0
3533, 34ifcli 4506 . . . 4 if((𝐴↑2) = 1, 1, 0) ∈ ℕ0
3635elexi 3451 . . 3 if((𝐴↑2) = 1, 1, 0) ∈ V
37 ovex 7308 . . 3 (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) ∈ V
3836, 37ifex 4509 . 2 if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))) ∈ V
3931, 32, 38ovmpoa 7428 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)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  ifcif 4459  {cpr 4563   class class class wbr 5074  cmpt 5157  cfv 6433  (class class class)co 7275  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876   < clt 11009  cmin 11205  -cneg 11206   / cdiv 11632  cn 11973  2c2 12028  7c7 12033  8c8 12034  0cn0 12233  cz 12319   mod cmo 13589  seqcseq 13721  cexp 13782  abscabs 14945  cdvds 15963  cprime 16376   pCnt cpc 16537   /L clgs 26442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-mulcl 10933  ax-i2m1 10939
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-nn 11974  df-n0 12234  df-seq 13722  df-lgs 26443
This theorem is referenced by:  lgscllem  26452  lgsval2lem  26455  lgs0  26458  lgsval4  26465
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