MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lgsval Structured version   Visualization version   GIF version

Theorem lgsval 24926
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 477 . . . 4 ((𝑎 = 𝐴𝑚 = 𝑁) → 𝑚 = 𝑁)
21eqeq1d 2623 . . 3 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑚 = 0 ↔ 𝑁 = 0))
3 simpl 473 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑁) → 𝑎 = 𝐴)
43oveq1d 6619 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑎↑2) = (𝐴↑2))
54eqeq1d 2623 . . . 4 ((𝑎 = 𝐴𝑚 = 𝑁) → ((𝑎↑2) = 1 ↔ (𝐴↑2) = 1))
65ifbid 4080 . . 3 ((𝑎 = 𝐴𝑚 = 𝑁) → if((𝑎↑2) = 1, 1, 0) = if((𝐴↑2) = 1, 1, 0))
71breq1d 4623 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑚 < 0 ↔ 𝑁 < 0))
83breq1d 4623 . . . . . 6 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑎 < 0 ↔ 𝐴 < 0))
97, 8anbi12d 746 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑁) → ((𝑚 < 0 ∧ 𝑎 < 0) ↔ (𝑁 < 0 ∧ 𝐴 < 0)))
109ifbid 4080 . . . 4 ((𝑎 = 𝐴𝑚 = 𝑁) → if((𝑚 < 0 ∧ 𝑎 < 0), -1, 1) = if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1))
113breq2d 4625 . . . . . . . . . . . 12 ((𝑎 = 𝐴𝑚 = 𝑁) → (2 ∥ 𝑎 ↔ 2 ∥ 𝐴))
123oveq1d 6619 . . . . . . . . . . . . . 14 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑎 mod 8) = (𝐴 mod 8))
1312eleq1d 2683 . . . . . . . . . . . . 13 ((𝑎 = 𝐴𝑚 = 𝑁) → ((𝑎 mod 8) ∈ {1, 7} ↔ (𝐴 mod 8) ∈ {1, 7}))
1413ifbid 4080 . . . . . . . . . . . 12 ((𝑎 = 𝐴𝑚 = 𝑁) → if((𝑎 mod 8) ∈ {1, 7}, 1, -1) = if((𝐴 mod 8) ∈ {1, 7}, 1, -1))
1511, 14ifbieq2d 4083 . . . . . . . . . . 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 6619 . . . . . . . . . . . . . 14 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑎↑((𝑛 − 1) / 2)) = (𝐴↑((𝑛 − 1) / 2)))
1716oveq1d 6619 . . . . . . . . . . . . 13 ((𝑎 = 𝐴𝑚 = 𝑁) → ((𝑎↑((𝑛 − 1) / 2)) + 1) = ((𝐴↑((𝑛 − 1) / 2)) + 1))
1817oveq1d 6619 . . . . . . . . . . . 12 ((𝑎 = 𝐴𝑚 = 𝑁) → (((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) = (((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛))
1918oveq1d 6619 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑚 = 𝑁) → ((((𝑎↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1) = ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))
2015, 19ifeq12d 4078 . . . . . . . . . 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 6620 . . . . . . . . . 10 ((𝑎 = 𝐴𝑚 = 𝑁) → (𝑛 pCnt 𝑚) = (𝑛 pCnt 𝑁))
2220, 21oveq12d 6622 . . . . . . . . 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 4076 . . . . . . . 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 4705 . . . . . . 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, 25syl6eqr 2673 . . . . . 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 12749 . . . . 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 6152 . . . . 5 ((𝑎 = 𝐴𝑚 = 𝑁) → (abs‘𝑚) = (abs‘𝑁))
2927, 28fveq12d 6154 . . . 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 6622 . . 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 4085 . 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 24920 . 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 11252 . . . . 5 1 ∈ ℕ0
34 0nn0 11251 . . . . 5 0 ∈ ℕ0
3533, 34keepel 4127 . . . 4 if((𝐴↑2) = 1, 1, 0) ∈ ℕ0
3635elexi 3199 . . 3 if((𝐴↑2) = 1, 1, 0) ∈ V
37 ovex 6632 . . 3 (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) ∈ V
3836, 37ifex 4128 . 2 if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))) ∈ V
3931, 32, 38ovmpt2a 6744 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 384   = wceq 1480  wcel 1987  ifcif 4058  {cpr 4150   class class class wbr 4613  cmpt 4673  cfv 5847  (class class class)co 6604  0cc0 9880  1c1 9881   + caddc 9883   · cmul 9885   < clt 10018  cmin 10210  -cneg 10211   / cdiv 10628  cn 10964  2c2 11014  7c7 11019  8c8 11020  0cn0 11236  cz 11321   mod cmo 12608  seqcseq 12741  cexp 12800  abscabs 13908  cdvds 14907  cprime 15309   pCnt cpc 15465   /L clgs 24919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-mulcl 9942  ax-i2m1 9948
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-nn 10965  df-n0 11237  df-seq 12742  df-lgs 24920
This theorem is referenced by:  lgscllem  24929  lgsval2lem  24932  lgs0  24935  lgsval4  24942
  Copyright terms: Public domain W3C validator