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Mirrors > Home > MPE Home > Th. List > lgsqrmod | Structured version Visualization version GIF version |
Description: If the Legendre symbol of an integer for an odd prime is 1, then the number is a quadratic residue mod 𝑃. (Contributed by AV, 20-Aug-2021.) |
Ref | Expression |
---|---|
lgsqrmod | ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) = 1 → ∃𝑥 ∈ ℤ ((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lgsqr 26087 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) = 1 ↔ (¬ 𝑃 ∥ 𝐴 ∧ ∃𝑥 ∈ ℤ 𝑃 ∥ ((𝑥↑2) − 𝐴)))) | |
2 | eldifi 4017 | . . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ) | |
3 | prmnn 16115 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
4 | 2, 3 | syl 17 | . . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ) |
5 | 4 | ad2antlr 727 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑥 ∈ ℤ) → 𝑃 ∈ ℕ) |
6 | zsqcl 13586 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → (𝑥↑2) ∈ ℤ) | |
7 | 6 | adantl 485 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑥 ∈ ℤ) → (𝑥↑2) ∈ ℤ) |
8 | simpll 767 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑥 ∈ ℤ) → 𝐴 ∈ ℤ) | |
9 | moddvds 15710 | . . . . . 6 ⊢ ((𝑃 ∈ ℕ ∧ (𝑥↑2) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃) ↔ 𝑃 ∥ ((𝑥↑2) − 𝐴))) | |
10 | 5, 7, 8, 9 | syl3anc 1372 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑥 ∈ ℤ) → (((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃) ↔ 𝑃 ∥ ((𝑥↑2) − 𝐴))) |
11 | 10 | biimprd 251 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑥 ∈ ℤ) → (𝑃 ∥ ((𝑥↑2) − 𝐴) → ((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃))) |
12 | 11 | reximdva 3184 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (∃𝑥 ∈ ℤ 𝑃 ∥ ((𝑥↑2) − 𝐴) → ∃𝑥 ∈ ℤ ((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃))) |
13 | 12 | adantld 494 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((¬ 𝑃 ∥ 𝐴 ∧ ∃𝑥 ∈ ℤ 𝑃 ∥ ((𝑥↑2) − 𝐴)) → ∃𝑥 ∈ ℤ ((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃))) |
14 | 1, 13 | sylbid 243 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) = 1 → ∃𝑥 ∈ ℤ ((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∃wrex 3054 ∖ cdif 3840 {csn 4516 class class class wbr 5030 (class class class)co 7170 1c1 10616 − cmin 10948 ℕcn 11716 2c2 11771 ℤcz 12062 mod cmo 13328 ↑cexp 13521 ∥ cdvds 15699 ℙcprime 16112 /L clgs 26030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-ofr 7426 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-tpos 7921 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-oadd 8135 df-er 8320 df-ec 8322 df-qs 8326 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-sup 8979 df-inf 8980 df-oi 9047 df-dju 9403 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-xnn0 12049 df-z 12063 df-dec 12180 df-uz 12325 df-q 12431 df-rp 12473 df-fz 12982 df-fzo 13125 df-fl 13253 df-mod 13329 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-dvds 15700 df-gcd 15938 df-prm 16113 df-phi 16203 df-pc 16274 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-hom 16692 df-cco 16693 df-0g 16818 df-gsum 16819 df-prds 16824 df-pws 16826 df-imas 16884 df-qus 16885 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-mulg 18343 df-subg 18394 df-nsg 18395 df-eqg 18396 df-ghm 18474 df-cntz 18565 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-srg 19375 df-ring 19418 df-cring 19419 df-oppr 19495 df-dvdsr 19513 df-unit 19514 df-invr 19544 df-dvr 19555 df-rnghom 19589 df-drng 19623 df-field 19624 df-subrg 19652 df-lmod 19755 df-lss 19823 df-lsp 19863 df-sra 20063 df-rgmod 20064 df-lidl 20065 df-rsp 20066 df-2idl 20124 df-nzr 20150 df-rlreg 20175 df-domn 20176 df-idom 20177 df-cnfld 20218 df-zring 20290 df-zrh 20324 df-zn 20327 df-assa 20669 df-asp 20670 df-ascl 20671 df-psr 20722 df-mvr 20723 df-mpl 20724 df-opsr 20726 df-evls 20886 df-evl 20887 df-psr1 20955 df-vr1 20956 df-ply1 20957 df-coe1 20958 df-evl1 21086 df-mdeg 24805 df-deg1 24806 df-mon1 24883 df-uc1p 24884 df-q1p 24885 df-r1p 24886 df-lgs 26031 |
This theorem is referenced by: lgsqrmodndvds 26089 |
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