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Mirrors > Home > ILE Home > Th. List > lgsfvalg | Unicode version |
Description: Value of the function which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by Jim Kingdon, 4-Nov-2024.) |
Ref | Expression |
---|---|
lgsval.1 |
Ref | Expression |
---|---|
lgsfvalg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lgsval.1 | . 2 | |
2 | eleq1 2233 | . . 3 | |
3 | eqeq1 2177 | . . . . 5 | |
4 | oveq1 5858 | . . . . . . . . . 10 | |
5 | 4 | oveq1d 5866 | . . . . . . . . 9 |
6 | 5 | oveq2d 5867 | . . . . . . . 8 |
7 | 6 | oveq1d 5866 | . . . . . . 7 |
8 | id 19 | . . . . . . 7 | |
9 | 7, 8 | oveq12d 5869 | . . . . . 6 |
10 | 9 | oveq1d 5866 | . . . . 5 |
11 | 3, 10 | ifbieq2d 3549 | . . . 4 |
12 | oveq1 5858 | . . . 4 | |
13 | 11, 12 | oveq12d 5869 | . . 3 |
14 | 2, 13 | ifbieq1d 3547 | . 2 |
15 | simp3 994 | . 2 | |
16 | 0zd 9217 | . . . . . 6 | |
17 | 1zzd 9232 | . . . . . . 7 | |
18 | neg1z 9237 | . . . . . . . 8 | |
19 | 18 | a1i 9 | . . . . . . 7 |
20 | id 19 | . . . . . . . . . . . . . 14 | |
21 | 8nn 9038 | . . . . . . . . . . . . . . 15 | |
22 | 21 | a1i 9 | . . . . . . . . . . . . . 14 |
23 | 20, 22 | zmodcld 10294 | . . . . . . . . . . . . 13 |
24 | 23 | nn0zd 9325 | . . . . . . . . . . . 12 |
25 | 1zzd 9232 | . . . . . . . . . . . 12 | |
26 | zdceq 9280 | . . . . . . . . . . . 12 DECID | |
27 | 24, 25, 26 | syl2anc 409 | . . . . . . . . . . 11 DECID |
28 | 7nn 9037 | . . . . . . . . . . . . 13 | |
29 | 28 | nnzi 9226 | . . . . . . . . . . . 12 |
30 | zdceq 9280 | . . . . . . . . . . . 12 DECID | |
31 | 24, 29, 30 | sylancl 411 | . . . . . . . . . . 11 DECID |
32 | dcor 930 | . . . . . . . . . . 11 DECID DECID DECID | |
33 | 27, 31, 32 | sylc 62 | . . . . . . . . . 10 DECID |
34 | elprg 3601 | . . . . . . . . . . . 12 | |
35 | 23, 34 | syl 14 | . . . . . . . . . . 11 |
36 | 35 | dcbid 833 | . . . . . . . . . 10 DECID DECID |
37 | 33, 36 | mpbird 166 | . . . . . . . . 9 DECID |
38 | 37 | 3ad2ant1 1013 | . . . . . . . 8 DECID |
39 | 38 | ad2antrr 485 | . . . . . . 7 DECID |
40 | 17, 19, 39 | ifcldcd 3560 | . . . . . 6 |
41 | 2nn 9032 | . . . . . . . 8 | |
42 | 41 | a1i 9 | . . . . . . 7 |
43 | simpll1 1031 | . . . . . . 7 | |
44 | dvdsdc 11753 | . . . . . . 7 DECID | |
45 | 42, 43, 44 | syl2anc 409 | . . . . . 6 DECID |
46 | 16, 40, 45 | ifcldcd 3560 | . . . . 5 |
47 | simpll1 1031 | . . . . . . . . . 10 | |
48 | simpr 109 | . . . . . . . . . . . 12 | |
49 | prm2orodd 12073 | . . . . . . . . . . . . . 14 | |
50 | 49 | orcomd 724 | . . . . . . . . . . . . 13 |
51 | 50 | ad2antlr 486 | . . . . . . . . . . . 12 |
52 | 48, 51 | ecased 1344 | . . . . . . . . . . 11 |
53 | 15 | ad2antrr 485 | . . . . . . . . . . . . 13 |
54 | 53 | nnnn0d 9181 | . . . . . . . . . . . 12 |
55 | nn0oddm1d2 11861 | . . . . . . . . . . . 12 | |
56 | 54, 55 | syl 14 | . . . . . . . . . . 11 |
57 | 52, 56 | mpbid 146 | . . . . . . . . . 10 |
58 | zexpcl 10484 | . . . . . . . . . 10 | |
59 | 47, 57, 58 | syl2anc 409 | . . . . . . . . 9 |
60 | 59 | peano2zd 9330 | . . . . . . . 8 |
61 | 60, 53 | zmodcld 10294 | . . . . . . 7 |
62 | 61 | nn0zd 9325 | . . . . . 6 |
63 | 1zzd 9232 | . . . . . 6 | |
64 | 62, 63 | zsubcld 9332 | . . . . 5 |
65 | simpl3 997 | . . . . . . 7 | |
66 | 65 | nnzd 9326 | . . . . . 6 |
67 | 2z 9233 | . . . . . 6 | |
68 | zdceq 9280 | . . . . . 6 DECID | |
69 | 66, 67, 68 | sylancl 411 | . . . . 5 DECID |
70 | 46, 64, 69 | ifcldadc 3554 | . . . 4 |
71 | simpr 109 | . . . . 5 | |
72 | simpl2 996 | . . . . 5 | |
73 | 71, 72 | pccld 12247 | . . . 4 |
74 | zexpcl 10484 | . . . 4 | |
75 | 70, 73, 74 | syl2anc 409 | . . 3 |
76 | 1zzd 9232 | . . 3 | |
77 | prmdc 12077 | . . . 4 DECID | |
78 | 15, 77 | syl 14 | . . 3 DECID |
79 | 75, 76, 78 | ifcldadc 3554 | . 2 |
80 | 1, 14, 15, 79 | fvmptd3 5587 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 DECID wdc 829 w3a 973 wceq 1348 wcel 2141 cif 3525 cpr 3582 class class class wbr 3987 cmpt 4048 cfv 5196 (class class class)co 5851 cc0 7767 c1 7768 caddc 7770 cmin 8083 cneg 8084 cdiv 8582 cn 8871 c2 8922 c7 8927 c8 8928 cn0 9128 cz 9205 cmo 10271 cexp 10468 cdvds 11742 cprime 12054 cpc 12231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 ax-arch 7886 ax-caucvg 7887 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-xor 1371 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-1o 6393 df-2o 6394 df-er 6511 df-en 6717 df-fin 6719 df-sup 6959 df-inf 6960 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-5 8933 df-6 8934 df-7 8935 df-8 8936 df-n0 9129 df-z 9206 df-uz 9481 df-q 9572 df-rp 9604 df-fz 9959 df-fzo 10092 df-fl 10219 df-mod 10272 df-seqfrec 10395 df-exp 10469 df-cj 10799 df-re 10800 df-im 10801 df-rsqrt 10955 df-abs 10956 df-dvds 11743 df-gcd 11891 df-prm 12055 df-pc 12232 |
This theorem is referenced by: lgsval2lem 13670 |
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