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Mirrors > Home > MPE Home > Th. List > lgsquad | Structured version Visualization version GIF version |
Description: The Law of Quadratic Reciprocity, see also theorem 9.8 in [ApostolNT] p. 185. If 𝑃 and 𝑄 are distinct odd primes, then the product of the Legendre symbols (𝑃 /L 𝑄) and (𝑄 /L 𝑃) is the parity of ((𝑃 − 1) / 2) · ((𝑄 − 1) / 2). This uses Eisenstein's proof, which also has a nice geometric interpretation - see https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity. This is Metamath 100 proof #7. (Contributed by Mario Carneiro, 19-Jun-2015.) |
Ref | Expression |
---|---|
lgsquad | ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑄 ∈ (ℙ ∖ {2}) ∧ 𝑃 ≠ 𝑄) → ((𝑃 /L 𝑄) · (𝑄 /L 𝑃)) = (-1↑(((𝑃 − 1) / 2) · ((𝑄 − 1) / 2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1130 | . 2 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑄 ∈ (ℙ ∖ {2}) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ (ℙ ∖ {2})) | |
2 | simp2 1131 | . 2 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑄 ∈ (ℙ ∖ {2}) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ (ℙ ∖ {2})) | |
3 | simp3 1132 | . 2 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑄 ∈ (ℙ ∖ {2}) ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄) | |
4 | eqid 2771 | . 2 ⊢ ((𝑃 − 1) / 2) = ((𝑃 − 1) / 2) | |
5 | eqid 2771 | . 2 ⊢ ((𝑄 − 1) / 2) = ((𝑄 − 1) / 2) | |
6 | eleq1w 2833 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (1...((𝑃 − 1) / 2)) ↔ 𝑧 ∈ (1...((𝑃 − 1) / 2)))) | |
7 | eleq1w 2833 | . . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (1...((𝑄 − 1) / 2)) ↔ 𝑤 ∈ (1...((𝑄 − 1) / 2)))) | |
8 | 6, 7 | bi2anan9 620 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑦 ∈ (1...((𝑄 − 1) / 2))) ↔ (𝑧 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑤 ∈ (1...((𝑄 − 1) / 2))))) |
9 | oveq1 6802 | . . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 · 𝑃) = (𝑤 · 𝑃)) | |
10 | oveq1 6802 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 · 𝑄) = (𝑧 · 𝑄)) | |
11 | 9, 10 | breqan12rd 4804 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑦 · 𝑃) < (𝑥 · 𝑄) ↔ (𝑤 · 𝑃) < (𝑧 · 𝑄))) |
12 | 8, 11 | anbi12d 616 | . . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑦 ∈ (1...((𝑄 − 1) / 2))) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)) ↔ ((𝑧 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑤 ∈ (1...((𝑄 − 1) / 2))) ∧ (𝑤 · 𝑃) < (𝑧 · 𝑄)))) |
13 | 12 | cbvopabv 4857 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑦 ∈ (1...((𝑄 − 1) / 2))) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} = {〈𝑧, 𝑤〉 ∣ ((𝑧 ∈ (1...((𝑃 − 1) / 2)) ∧ 𝑤 ∈ (1...((𝑄 − 1) / 2))) ∧ (𝑤 · 𝑃) < (𝑧 · 𝑄))} |
14 | 1, 2, 3, 4, 5, 13 | lgsquadlem3 25327 | 1 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑄 ∈ (ℙ ∖ {2}) ∧ 𝑃 ≠ 𝑄) → ((𝑃 /L 𝑄) · (𝑄 /L 𝑃)) = (-1↑(((𝑃 − 1) / 2) · ((𝑄 − 1) / 2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∖ cdif 3720 {csn 4317 class class class wbr 4787 {copab 4847 (class class class)co 6795 1c1 10142 · cmul 10146 < clt 10279 − cmin 10471 -cneg 10472 / cdiv 10889 2c2 11275 ...cfz 12532 ↑cexp 13066 ℙcprime 15591 /L clgs 25239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7099 ax-inf2 8705 ax-cnex 10197 ax-resscn 10198 ax-1cn 10199 ax-icn 10200 ax-addcl 10201 ax-addrcl 10202 ax-mulcl 10203 ax-mulrcl 10204 ax-mulcom 10205 ax-addass 10206 ax-mulass 10207 ax-distr 10208 ax-i2m1 10209 ax-1ne0 10210 ax-1rid 10211 ax-rnegex 10212 ax-rrecex 10213 ax-cnre 10214 ax-pre-lttri 10215 ax-pre-lttrn 10216 ax-pre-ltadd 10217 ax-pre-mulgt0 10218 ax-pre-sup 10219 ax-addf 10220 ax-mulf 10221 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-disj 4756 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6756 df-ov 6798 df-oprab 6799 df-mpt2 6800 df-of 7047 df-om 7216 df-1st 7318 df-2nd 7319 df-supp 7450 df-tpos 7507 df-wrecs 7562 df-recs 7624 df-rdg 7662 df-1o 7716 df-2o 7717 df-oadd 7720 df-er 7899 df-ec 7901 df-qs 7905 df-map 8014 df-en 8113 df-dom 8114 df-sdom 8115 df-fin 8116 df-fsupp 8435 df-sup 8507 df-inf 8508 df-oi 8574 df-card 8968 df-cda 9195 df-pnf 10281 df-mnf 10282 df-xr 10283 df-ltxr 10284 df-le 10285 df-sub 10473 df-neg 10474 df-div 10890 df-nn 11226 df-2 11284 df-3 11285 df-4 11286 df-5 11287 df-6 11288 df-7 11289 df-8 11290 df-9 11291 df-n0 11499 df-xnn0 11570 df-z 11584 df-dec 11700 df-uz 11893 df-q 11996 df-rp 12035 df-fz 12533 df-fzo 12673 df-fl 12800 df-mod 12876 df-seq 13008 df-exp 13067 df-hash 13321 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-clim 14426 df-sum 14624 df-dvds 15189 df-gcd 15424 df-prm 15592 df-phi 15677 df-pc 15748 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-starv 16163 df-sca 16164 df-vsca 16165 df-ip 16166 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-0g 16309 df-gsum 16310 df-imas 16375 df-qus 16376 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-mhm 17542 df-submnd 17543 df-grp 17632 df-minusg 17633 df-sbg 17634 df-mulg 17748 df-subg 17798 df-nsg 17799 df-eqg 17800 df-ghm 17865 df-cntz 17956 df-cmn 18401 df-abl 18402 df-mgp 18697 df-ur 18709 df-ring 18756 df-cring 18757 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-invr 18879 df-dvr 18890 df-rnghom 18924 df-drng 18958 df-field 18959 df-subrg 18987 df-lmod 19074 df-lss 19142 df-lsp 19184 df-sra 19386 df-rgmod 19387 df-lidl 19388 df-rsp 19389 df-2idl 19446 df-nzr 19472 df-rlreg 19497 df-domn 19498 df-idom 19499 df-cnfld 19961 df-zring 20033 df-zrh 20066 df-zn 20069 df-lgs 25240 |
This theorem is referenced by: lgsquad2 25331 |
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