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Mirrors > Home > ILE Home > Th. List > lgsabs1 | GIF version |
Description: The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.) |
Ref | Expression |
---|---|
lgsabs1 | ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ (𝐴 gcd 𝑁) = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lgscl 14082 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℤ) | |
2 | 1 | zcnd 9365 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℂ) |
3 | 2 | abscld 11174 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴 /L 𝑁)) ∈ ℝ) |
4 | 1re 7947 | . . 3 ⊢ 1 ∈ ℝ | |
5 | letri3 8028 | . . 3 ⊢ (((abs‘(𝐴 /L 𝑁)) ∈ ℝ ∧ 1 ∈ ℝ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ ((abs‘(𝐴 /L 𝑁)) ≤ 1 ∧ 1 ≤ (abs‘(𝐴 /L 𝑁))))) | |
6 | 3, 4, 5 | sylancl 413 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ ((abs‘(𝐴 /L 𝑁)) ≤ 1 ∧ 1 ≤ (abs‘(𝐴 /L 𝑁))))) |
7 | lgsle1 14083 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴 /L 𝑁)) ≤ 1) | |
8 | 7 | biantrurd 305 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1 ≤ (abs‘(𝐴 /L 𝑁)) ↔ ((abs‘(𝐴 /L 𝑁)) ≤ 1 ∧ 1 ≤ (abs‘(𝐴 /L 𝑁))))) |
9 | nnne0 8936 | . . . 4 ⊢ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ → (abs‘(𝐴 /L 𝑁)) ≠ 0) | |
10 | neneq 2369 | . . . . . . 7 ⊢ ((abs‘(𝐴 /L 𝑁)) ≠ 0 → ¬ (abs‘(𝐴 /L 𝑁)) = 0) | |
11 | 10 | adantl 277 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (abs‘(𝐴 /L 𝑁)) ≠ 0) → ¬ (abs‘(𝐴 /L 𝑁)) = 0) |
12 | nn0abscl 11078 | . . . . . . . . 9 ⊢ ((𝐴 /L 𝑁) ∈ ℤ → (abs‘(𝐴 /L 𝑁)) ∈ ℕ0) | |
13 | 1, 12 | syl 14 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴 /L 𝑁)) ∈ ℕ0) |
14 | elnn0 9167 | . . . . . . . 8 ⊢ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ0 ↔ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ∨ (abs‘(𝐴 /L 𝑁)) = 0)) | |
15 | 13, 14 | sylib 122 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ∨ (abs‘(𝐴 /L 𝑁)) = 0)) |
16 | 15 | adantr 276 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (abs‘(𝐴 /L 𝑁)) ≠ 0) → ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ∨ (abs‘(𝐴 /L 𝑁)) = 0)) |
17 | 11, 16 | ecased 1349 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (abs‘(𝐴 /L 𝑁)) ≠ 0) → (abs‘(𝐴 /L 𝑁)) ∈ ℕ) |
18 | 17 | ex 115 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ≠ 0 → (abs‘(𝐴 /L 𝑁)) ∈ ℕ)) |
19 | 9, 18 | impbid2 143 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ↔ (abs‘(𝐴 /L 𝑁)) ≠ 0)) |
20 | elnnnn0c 9210 | . . . . 5 ⊢ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ↔ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ0 ∧ 1 ≤ (abs‘(𝐴 /L 𝑁)))) | |
21 | 20 | baib 919 | . . . 4 ⊢ ((abs‘(𝐴 /L 𝑁)) ∈ ℕ0 → ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ↔ 1 ≤ (abs‘(𝐴 /L 𝑁)))) |
22 | 13, 21 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ∈ ℕ ↔ 1 ≤ (abs‘(𝐴 /L 𝑁)))) |
23 | abs00 11057 | . . . . . 6 ⊢ ((𝐴 /L 𝑁) ∈ ℂ → ((abs‘(𝐴 /L 𝑁)) = 0 ↔ (𝐴 /L 𝑁) = 0)) | |
24 | 23 | necon3bid 2388 | . . . . 5 ⊢ ((𝐴 /L 𝑁) ∈ ℂ → ((abs‘(𝐴 /L 𝑁)) ≠ 0 ↔ (𝐴 /L 𝑁) ≠ 0)) |
25 | 2, 24 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ≠ 0 ↔ (𝐴 /L 𝑁) ≠ 0)) |
26 | lgsne0 14106 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 /L 𝑁) ≠ 0 ↔ (𝐴 gcd 𝑁) = 1)) | |
27 | 25, 26 | bitrd 188 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) ≠ 0 ↔ (𝐴 gcd 𝑁) = 1)) |
28 | 19, 22, 27 | 3bitr3d 218 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1 ≤ (abs‘(𝐴 /L 𝑁)) ↔ (𝐴 gcd 𝑁) = 1)) |
29 | 6, 8, 28 | 3bitr2d 216 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ (𝐴 gcd 𝑁) = 1)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 ℂcc 7800 ℝcr 7801 0cc0 7802 1c1 7803 ≤ cle 7983 ℕcn 8908 ℕ0cn0 9165 ℤcz 9242 abscabs 10990 gcd cgcd 11926 /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: lgssq 14108 lgssq2 14109 |
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