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| 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 12624 | . . 3 ⊢ 0 ∈ ℤ | |
| 2 | eqid 2737 | . . . 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 27345 | . . 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 691 | . 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 2737 | . . 3 ⊢ 0 = 0 | |
| 6 | 5 | iftruei 4532 | . 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 | eqtrdi 2793 | 1 ⊢ (𝐴 ∈ ℤ → (𝐴 /L 0) = if((𝐴↑2) = 1, 1, 0)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ifcif 4525 {cpr 4628 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 − cmin 11492 -cneg 11493 / cdiv 11920 ℕcn 12266 2c2 12321 7c7 12326 8c8 12327 ℤcz 12613 mod cmo 13909 seqcseq 14042 ↑cexp 14102 abscabs 15273 ∥ cdvds 16290 ℙcprime 16708 pCnt cpc 16874 /L clgs 27338 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-i2m1 11223 ax-rnegex 11226 ax-cnre 11228 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-seq 14043 df-lgs 27339 | 
| This theorem is referenced by: lgsdir 27376 lgsne0 27379 lgsdinn0 27389 | 
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