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Mirrors > Home > MPE Home > Th. List > lgsle1 | Structured version Visualization version GIF version |
Description: The Legendre symbol has absolute value less than or equal to 1. Together with lgscl 25579 this implies that it takes values in {-1, 0, 1}. (Contributed by Mario Carneiro, 4-Feb-2015.) |
Ref | Expression |
---|---|
lgsle1 | ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴 /L 𝑁)) ≤ 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2772 | . . 3 ⊢ {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} | |
2 | 1 | lgscl2 25577 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1}) |
3 | fveq2 6493 | . . . . 5 ⊢ (𝑥 = (𝐴 /L 𝑁) → (abs‘𝑥) = (abs‘(𝐴 /L 𝑁))) | |
4 | 3 | breq1d 4933 | . . . 4 ⊢ (𝑥 = (𝐴 /L 𝑁) → ((abs‘𝑥) ≤ 1 ↔ (abs‘(𝐴 /L 𝑁)) ≤ 1)) |
5 | 4 | elrab 3589 | . . 3 ⊢ ((𝐴 /L 𝑁) ∈ {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} ↔ ((𝐴 /L 𝑁) ∈ ℤ ∧ (abs‘(𝐴 /L 𝑁)) ≤ 1)) |
6 | 5 | simprbi 489 | . 2 ⊢ ((𝐴 /L 𝑁) ∈ {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} → (abs‘(𝐴 /L 𝑁)) ≤ 1) |
7 | 2, 6 | syl 17 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴 /L 𝑁)) ≤ 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 {crab 3086 class class class wbr 4923 ‘cfv 6182 (class class class)co 6970 1c1 10328 ≤ cle 10467 ℤcz 11786 abscabs 14444 /L clgs 25562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-sup 8693 df-inf 8694 df-dju 9116 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-n0 11701 df-xnn0 11773 df-z 11787 df-uz 12052 df-q 12156 df-rp 12198 df-fz 12702 df-fzo 12843 df-fl 12970 df-mod 13046 df-seq 13178 df-exp 13238 df-hash 13499 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-dvds 15458 df-gcd 15694 df-prm 15862 df-phi 15949 df-pc 16020 df-lgs 25563 |
This theorem is referenced by: lgscl1 25588 lgsdirprm 25599 lgsabs1 25604 |
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