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Mirrors > Home > MPE Home > Th. List > lgsqr | Structured version Visualization version Unicode version |
Description: The Legendre symbol for odd primes is iff the number is not a multiple of the prime (in which case it is , see lgsne0 25060) and the number is a quadratic residue (it is for nonresidues by the process of elimination from lgsabs1 25061). Given our definition of the Legendre symbol, this theorem is equivalent to Euler's criterion. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
lgsqr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 3732 | . . . . . . . . . . 11 | |
2 | 1 | adantl 482 | . . . . . . . . . 10 |
3 | prmz 15389 | . . . . . . . . . 10 | |
4 | 2, 3 | syl 17 | . . . . . . . . 9 |
5 | simpl 473 | . . . . . . . . 9 | |
6 | gcdcom 15235 | . . . . . . . . 9 | |
7 | 4, 5, 6 | syl2anc 693 | . . . . . . . 8 |
8 | 7 | eqeq1d 2624 | . . . . . . 7 |
9 | coprm 15423 | . . . . . . . 8 | |
10 | 2, 5, 9 | syl2anc 693 | . . . . . . 7 |
11 | lgsne0 25060 | . . . . . . . 8 | |
12 | 5, 4, 11 | syl2anc 693 | . . . . . . 7 |
13 | 8, 10, 12 | 3bitr4d 300 | . . . . . 6 |
14 | 13 | necon4bbid 2835 | . . . . 5 |
15 | 0ne1 11088 | . . . . . 6 | |
16 | neeq1 2856 | . . . . . 6 | |
17 | 15, 16 | mpbiri 248 | . . . . 5 |
18 | 14, 17 | syl6bi 243 | . . . 4 |
19 | 18 | necon2bd 2810 | . . 3 |
20 | lgsqrlem5 25075 | . . . 4 | |
21 | 20 | 3expia 1267 | . . 3 |
22 | 19, 21 | jcad 555 | . 2 |
23 | simprl 794 | . . . . . . . 8 | |
24 | 23 | zred 11482 | . . . . . . 7 |
25 | absresq 14042 | . . . . . . 7 | |
26 | 24, 25 | syl 17 | . . . . . 6 |
27 | 26 | oveq1d 6665 | . . . . 5 |
28 | simplr 792 | . . . . . . . . . . 11 | |
29 | 1 | ad3antlr 767 | . . . . . . . . . . . . 13 |
30 | 29, 3 | syl 17 | . . . . . . . . . . . 12 |
31 | zsqcl 12934 | . . . . . . . . . . . . 13 | |
32 | 23, 31 | syl 17 | . . . . . . . . . . . 12 |
33 | simplll 798 | . . . . . . . . . . . 12 | |
34 | simprr 796 | . . . . . . . . . . . 12 | |
35 | dvdssub2 15023 | . . . . . . . . . . . 12 | |
36 | 30, 32, 33, 34, 35 | syl31anc 1329 | . . . . . . . . . . 11 |
37 | 28, 36 | mtbird 315 | . . . . . . . . . 10 |
38 | 2nn 11185 | . . . . . . . . . . . 12 | |
39 | 38 | a1i 11 | . . . . . . . . . . 11 |
40 | prmdvdsexp 15427 | . . . . . . . . . . 11 | |
41 | 29, 23, 39, 40 | syl3anc 1326 | . . . . . . . . . 10 |
42 | 37, 41 | mtbid 314 | . . . . . . . . 9 |
43 | dvds0 14997 | . . . . . . . . . . . 12 | |
44 | 30, 43 | syl 17 | . . . . . . . . . . 11 |
45 | breq2 4657 | . . . . . . . . . . 11 | |
46 | 44, 45 | syl5ibrcom 237 | . . . . . . . . . 10 |
47 | 46 | necon3bd 2808 | . . . . . . . . 9 |
48 | 42, 47 | mpd 15 | . . . . . . . 8 |
49 | nnabscl 14065 | . . . . . . . 8 | |
50 | 23, 48, 49 | syl2anc 693 | . . . . . . 7 |
51 | 50 | nnzd 11481 | . . . . . 6 |
52 | 50 | nnne0d 11065 | . . . . . 6 |
53 | gcdcom 15235 | . . . . . . . 8 | |
54 | 51, 30, 53 | syl2anc 693 | . . . . . . 7 |
55 | dvdsabsb 15001 | . . . . . . . . . 10 | |
56 | 30, 23, 55 | syl2anc 693 | . . . . . . . . 9 |
57 | 42, 56 | mtbid 314 | . . . . . . . 8 |
58 | coprm 15423 | . . . . . . . . 9 | |
59 | 29, 51, 58 | syl2anc 693 | . . . . . . . 8 |
60 | 57, 59 | mpbid 222 | . . . . . . 7 |
61 | 54, 60 | eqtrd 2656 | . . . . . 6 |
62 | lgssq 25062 | . . . . . 6 | |
63 | 51, 52, 30, 61, 62 | syl211anc 1332 | . . . . 5 |
64 | prmnn 15388 | . . . . . . . . . 10 | |
65 | 29, 64 | syl 17 | . . . . . . . . 9 |
66 | moddvds 14991 | . . . . . . . . 9 | |
67 | 65, 32, 33, 66 | syl3anc 1326 | . . . . . . . 8 |
68 | 34, 67 | mpbird 247 | . . . . . . 7 |
69 | 68 | oveq1d 6665 | . . . . . 6 |
70 | eldifsni 4320 | . . . . . . . . . 10 | |
71 | 70 | ad3antlr 767 | . . . . . . . . 9 |
72 | 71 | necomd 2849 | . . . . . . . 8 |
73 | 2z 11409 | . . . . . . . . . 10 | |
74 | uzid 11702 | . . . . . . . . . 10 | |
75 | 73, 74 | ax-mp 5 | . . . . . . . . 9 |
76 | dvdsprm 15415 | . . . . . . . . . 10 | |
77 | 76 | necon3bbid 2831 | . . . . . . . . 9 |
78 | 75, 29, 77 | sylancr 695 | . . . . . . . 8 |
79 | 72, 78 | mpbird 247 | . . . . . . 7 |
80 | lgsmod 25048 | . . . . . . 7 | |
81 | 32, 65, 79, 80 | syl3anc 1326 | . . . . . 6 |
82 | lgsmod 25048 | . . . . . . 7 | |
83 | 33, 65, 79, 82 | syl3anc 1326 | . . . . . 6 |
84 | 69, 81, 83 | 3eqtr3d 2664 | . . . . 5 |
85 | 27, 63, 84 | 3eqtr3rd 2665 | . . . 4 |
86 | 85 | rexlimdvaa 3032 | . . 3 |
87 | 86 | expimpd 629 | . 2 |
88 | 22, 87 | impbid 202 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 wrex 2913 cdif 3571 csn 4177 class class class wbr 4653 cfv 5888 (class class class)co 6650 cr 9935 cc0 9936 c1 9937 cmin 10266 cn 11020 c2 11070 cz 11377 cuz 11687 cmo 12668 cexp 12860 cabs 13974 cdvds 14983 cgcd 15216 cprime 15385 clgs 25019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-phi 15471 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-imas 16168 df-qus 16169 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-nsg 17592 df-eqg 17593 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-field 18750 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-rsp 19175 df-2idl 19232 df-nzr 19258 df-rlreg 19283 df-domn 19284 df-idom 19285 df-assa 19312 df-asp 19313 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-evls 19506 df-evl 19507 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-coe1 19553 df-evl1 19681 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-zn 19855 df-mdeg 23815 df-deg1 23816 df-mon1 23890 df-uc1p 23891 df-q1p 23892 df-r1p 23893 df-lgs 25020 |
This theorem is referenced by: lgsqrmod 25077 2sqlem11 25154 2sqblem 25156 |
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