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Mirrors > Home > ILE Home > Th. List > lgscllem | GIF version |
Description: The Legendre symbol is an element of 𝑍. (Contributed by Mario Carneiro, 4-Feb-2015.) |
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)) |
lgsfcl2.z | ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} |
Ref | Expression |
---|---|
lgscllem | ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lgsval.1 | . . 3 ⊢ 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)) | |
2 | 1 | lgsval 14072 | . 2 ⊢ ((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))))) |
3 | lgsfcl2.z | . . . . . . . 8 ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} | |
4 | 3 | lgslem2 14069 | . . . . . . 7 ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍) |
5 | 4 | simp3i 1008 | . . . . . 6 ⊢ 1 ∈ 𝑍 |
6 | 5 | a1i 9 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 1 ∈ 𝑍) |
7 | 4 | simp2i 1007 | . . . . . 6 ⊢ 0 ∈ 𝑍 |
8 | 7 | a1i 9 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 0 ∈ 𝑍) |
9 | zsqcl 10576 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) | |
10 | 1zzd 9269 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℤ) | |
11 | zdceq 9317 | . . . . . 6 ⊢ (((𝐴↑2) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝐴↑2) = 1) | |
12 | 9, 10, 11 | syl2an2r 595 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝐴↑2) = 1) |
13 | 6, 8, 12 | ifcldcd 3569 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → if((𝐴↑2) = 1, 1, 0) ∈ 𝑍) |
14 | 13 | adantr 276 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → if((𝐴↑2) = 1, 1, 0) ∈ 𝑍) |
15 | 4 | simp1i 1006 | . . . . . 6 ⊢ -1 ∈ 𝑍 |
16 | 15 | a1i 9 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → -1 ∈ 𝑍) |
17 | simpr 110 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ) | |
18 | 0zd 9254 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 0 ∈ ℤ) | |
19 | zdclt 9319 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 < 0) | |
20 | 17, 18, 19 | syl2anc 411 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 < 0) |
21 | zdclt 9319 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝐴 < 0) | |
22 | 18, 21 | syldan 282 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝐴 < 0) |
23 | dcan2 934 | . . . . . 6 ⊢ (DECID 𝑁 < 0 → (DECID 𝐴 < 0 → DECID (𝑁 < 0 ∧ 𝐴 < 0))) | |
24 | 20, 22, 23 | sylc 62 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑁 < 0 ∧ 𝐴 < 0)) |
25 | 16, 6, 24 | ifcldcd 3569 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) ∈ 𝑍) |
26 | nnuz 9552 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
27 | 1zzd 9269 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → 1 ∈ ℤ) | |
28 | df-ne 2348 | . . . . . . . 8 ⊢ (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0) | |
29 | 1, 3 | lgsfcl2 14074 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶𝑍) |
30 | 29 | 3expa 1203 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶𝑍) |
31 | 28, 30 | sylan2br 288 | . . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → 𝐹:ℕ⟶𝑍) |
32 | 31 | ffvelcdmda 5647 | . . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) ∧ 𝑦 ∈ ℕ) → (𝐹‘𝑦) ∈ 𝑍) |
33 | 3 | lgslem3 14070 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝑍 ∧ 𝑧 ∈ 𝑍) → (𝑦 · 𝑧) ∈ 𝑍) |
34 | 33 | adantl 277 | . . . . . 6 ⊢ ((((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) ∧ (𝑦 ∈ 𝑍 ∧ 𝑧 ∈ 𝑍)) → (𝑦 · 𝑧) ∈ 𝑍) |
35 | 26, 27, 32, 34 | seqf 10447 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → seq1( · , 𝐹):ℕ⟶𝑍) |
36 | simplr 528 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → 𝑁 ∈ ℤ) | |
37 | simpr 110 | . . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → ¬ 𝑁 = 0) | |
38 | 37 | neqned 2354 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → 𝑁 ≠ 0) |
39 | nnabscl 11093 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (abs‘𝑁) ∈ ℕ) | |
40 | 36, 38, 39 | syl2anc 411 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → (abs‘𝑁) ∈ ℕ) |
41 | 35, 40 | ffvelcdmd 5648 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → (seq1( · , 𝐹)‘(abs‘𝑁)) ∈ 𝑍) |
42 | 3 | lgslem3 14070 | . . . 4 ⊢ ((if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) ∈ 𝑍 ∧ (seq1( · , 𝐹)‘(abs‘𝑁)) ∈ 𝑍) → (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) ∈ 𝑍) |
43 | 25, 41, 42 | syl2an2r 595 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ 𝑁 = 0) → (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))) ∈ 𝑍) |
44 | zdceq 9317 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0) | |
45 | 17, 18, 44 | syl2anc 411 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑁 = 0) |
46 | 14, 43, 45 | ifcldadc 3563 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))) ∈ 𝑍) |
47 | 2, 46 | eqeltrd 2254 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ 𝑍) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 {crab 2459 ifcif 3534 {cpr 3592 class class class wbr 4000 ↦ cmpt 4061 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 0cc0 7802 1c1 7803 + caddc 7805 · cmul 7807 < clt 7982 ≤ cle 7983 − cmin 8118 -cneg 8119 / cdiv 8618 ℕcn 8908 2c2 8959 7c7 8964 8c8 8965 ℤcz 9242 mod cmo 10308 seqcseq 10431 ↑cexp 10505 abscabs 10990 ∥ cdvds 11778 ℙcprime 12090 pCnt cpc 12267 /L clgs 14065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-xor 1376 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-frec 6386 df-1o 6411 df-2o 6412 df-oadd 6415 df-er 6529 df-en 6735 df-dom 6736 df-fin 6737 df-sup 6977 df-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-5 8970 df-6 8971 df-7 8972 df-8 8973 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-fz 9996 df-fzo 10129 df-fl 10256 df-mod 10309 df-seqfrec 10432 df-exp 10506 df-ihash 10740 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-clim 11271 df-proddc 11543 df-dvds 11779 df-gcd 11927 df-prm 12091 df-phi 12194 df-pc 12268 df-lgs 14066 |
This theorem is referenced by: lgscl2 14080 |
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