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Mirrors > Home > MPE Home > Th. List > lgssq | Structured version Visualization version GIF version |
Description: The Legendre symbol at a square is equal to 1. Together with lgsmod 25504 this implies that the Legendre symbol takes value 1 at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.) (Revised by AV, 20-Jul-2021.) |
Ref | Expression |
---|---|
lgssq | ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑2) /L 𝑁) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1211 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝐴 ∈ ℤ) | |
2 | simp2 1128 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝑁 ∈ ℤ) | |
3 | simp1r 1212 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝐴 ≠ 0) | |
4 | lgsdir 25513 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐴 ≠ 0 ∧ 𝐴 ≠ 0)) → ((𝐴 · 𝐴) /L 𝑁) = ((𝐴 /L 𝑁) · (𝐴 /L 𝑁))) | |
5 | 1, 1, 2, 3, 3, 4 | syl32anc 1446 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴 · 𝐴) /L 𝑁) = ((𝐴 /L 𝑁) · (𝐴 /L 𝑁))) |
6 | zcn 11737 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
7 | 6 | adantr 474 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ) |
8 | 7 | 3ad2ant1 1124 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝐴 ∈ ℂ) |
9 | 8 | sqvald 13328 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (𝐴↑2) = (𝐴 · 𝐴)) |
10 | 9 | oveq1d 6939 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑2) /L 𝑁) = ((𝐴 · 𝐴) /L 𝑁)) |
11 | lgscl 25492 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℤ) | |
12 | 1, 2, 11 | syl2anc 579 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (𝐴 /L 𝑁) ∈ ℤ) |
13 | 12 | zred 11838 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (𝐴 /L 𝑁) ∈ ℝ) |
14 | absresq 14453 | . . . 4 ⊢ ((𝐴 /L 𝑁) ∈ ℝ → ((abs‘(𝐴 /L 𝑁))↑2) = ((𝐴 /L 𝑁)↑2)) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((abs‘(𝐴 /L 𝑁))↑2) = ((𝐴 /L 𝑁)↑2)) |
16 | lgsabs1 25517 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ (𝐴 gcd 𝑁) = 1)) | |
17 | 16 | adantlr 705 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ (𝐴 gcd 𝑁) = 1)) |
18 | 17 | biimp3ar 1543 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (abs‘(𝐴 /L 𝑁)) = 1) |
19 | 18 | oveq1d 6939 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((abs‘(𝐴 /L 𝑁))↑2) = (1↑2)) |
20 | sq1 13281 | . . . 4 ⊢ (1↑2) = 1 | |
21 | 19, 20 | syl6eq 2830 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((abs‘(𝐴 /L 𝑁))↑2) = 1) |
22 | 12 | zcnd 11839 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (𝐴 /L 𝑁) ∈ ℂ) |
23 | 22 | sqvald 13328 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴 /L 𝑁)↑2) = ((𝐴 /L 𝑁) · (𝐴 /L 𝑁))) |
24 | 15, 21, 23 | 3eqtr3d 2822 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 1 = ((𝐴 /L 𝑁) · (𝐴 /L 𝑁))) |
25 | 5, 10, 24 | 3eqtr4d 2824 | 1 ⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑2) /L 𝑁) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 · cmul 10279 2c2 11434 ℤcz 11732 ↑cexp 13182 abscabs 14385 gcd cgcd 15626 /L clgs 25475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-inf 8639 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-xnn0 11719 df-z 11733 df-uz 11997 df-q 12100 df-rp 12142 df-fz 12648 df-fzo 12789 df-fl 12916 df-mod 12992 df-seq 13124 df-exp 13183 df-hash 13440 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-dvds 15392 df-gcd 15627 df-prm 15795 df-phi 15879 df-pc 15950 df-lgs 25476 |
This theorem is referenced by: 1lgs 25521 lgsmulsqcoprm 25524 lgsqr 25532 lgsquad2lem2 25566 |
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