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Mirrors > Home > MPE Home > Th. List > lgs0 | Structured version Visualization version GIF version |
Description: The Legendre symbol when the second argument is zero. (Contributed by Mario Carneiro, 4-Feb-2015.) |
Ref | Expression |
---|---|
lgs0 | ⊢ (𝐴 ∈ ℤ → (𝐴 /L 0) = if((𝐴↑2) = 1, 1, 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11993 | . . 3 ⊢ 0 ∈ ℤ | |
2 | eqid 2821 | . . . 4 ⊢ (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 0)), 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 0)), 1)) | |
3 | 2 | lgsval 25877 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 0 ∈ ℤ) → (𝐴 /L 0) = if(0 = 0, if((𝐴↑2) = 1, 1, 0), (if((0 < 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 0)), 1)))‘(abs‘0))))) |
4 | 1, 3 | mpan2 689 | . 2 ⊢ (𝐴 ∈ ℤ → (𝐴 /L 0) = if(0 = 0, if((𝐴↑2) = 1, 1, 0), (if((0 < 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 0)), 1)))‘(abs‘0))))) |
5 | eqid 2821 | . . 3 ⊢ 0 = 0 | |
6 | 5 | iftruei 4474 | . 2 ⊢ if(0 = 0, if((𝐴↑2) = 1, 1, 0), (if((0 < 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 0)), 1)))‘(abs‘0)))) = if((𝐴↑2) = 1, 1, 0) |
7 | 4, 6 | syl6eq 2872 | 1 ⊢ (𝐴 ∈ ℤ → (𝐴 /L 0) = if((𝐴↑2) = 1, 1, 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ifcif 4467 {cpr 4569 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 − cmin 10870 -cneg 10871 / cdiv 11297 ℕcn 11638 2c2 11693 7c7 11698 8c8 11699 ℤcz 11982 mod cmo 13238 seqcseq 13370 ↑cexp 13430 abscabs 14593 ∥ cdvds 15607 ℙcprime 16015 pCnt cpc 16173 /L clgs 25870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-i2m1 10605 ax-rnegex 10608 ax-cnre 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-seq 13371 df-lgs 25871 |
This theorem is referenced by: lgsdir 25908 lgsne0 25911 lgsdinn0 25921 |
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