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Theorem lgsfvalg 15684
Description: Value of the function F which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by Jim Kingdon, 4-Nov-2024.)
Hypothesis
Ref Expression
lgsval.1 |- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )
Assertion
Ref Expression
lgsfvalg |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> ( F ` M ) = if ( M e. Prime , ( if ( M = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) ) ^ ( M pCnt N ) ) , 1 ) )
Distinct variable groups:   A, n   n, M   n, N
Allowed substitution hint:   F( n)
Proof of Theorem lgsfvalg
StepHypRef Expression
1 lgsval.1 . 2 |- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )
2 eleq1 2292 . . 3 |- ( n = M -> ( n e. Prime <-> M e. Prime ) )
3 eqeq1 2236 . . . . 5 |- ( n = M -> ( n = 2 <-> M = 2 ) )
4 oveq1 6008 . . . . . . . . . 10 |- ( n = M -> ( n - 1 ) = ( M - 1 ) )
54oveq1d 6016 . . . . . . . . 9 |- ( n = M -> ( ( n - 1 ) / 2 ) = ( ( M - 1 ) / 2 ) )
65oveq2d 6017 . . . . . . . 8 |- ( n = M -> ( A ^ ( ( n - 1 ) / 2 ) ) = ( A ^ ( ( M - 1 ) / 2 ) ) )
76oveq1d 6016 . . . . . . 7 |- ( n = M -> ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) = ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) )
8 id 19 . . . . . . 7 |- ( n = M -> n = M )
97, 8oveq12d 6019 . . . . . 6 |- ( n = M -> ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) = ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) )
109oveq1d 6016 . . . . 5 |- ( n = M -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) = ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) )
113, 10ifbieq2d 3627 . . . 4 |- ( n = M -> if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) = if ( M = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) ) )
12 oveq1 6008 . . . 4 |- ( n = M -> ( n pCnt N ) = ( M pCnt N ) )
1311, 12oveq12d 6019 . . 3 |- ( n = M -> ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) = ( if ( M = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) ) ^ ( M pCnt N ) ) )
142, 13ifbieq1d 3625 . 2 |- ( n = M -> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) = if ( M e. Prime , ( if ( M = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) ) ^ ( M pCnt N ) ) , 1 ) )
15 simp3 1023 . 2 |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> M e. NN )
16 0zd 9458 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> 0 e. ZZ )
17 1zzd 9473 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> 1 e. ZZ )
18 neg1z 9478 . . . . . . . 8 |- -u 1 e. ZZ
1918a1i 9 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> -u 1 e. ZZ )
20 id 19 . . . . . . . . . . . . . 14 |- ( A e. ZZ -> A e. ZZ )
21 8nn 9278 . . . . . . . . . . . . . . 15 |- 8 e. NN
2221a1i 9 . . . . . . . . . . . . . 14 |- ( A e. ZZ -> 8 e. NN )
2320, 22zmodcld 10567 . . . . . . . . . . . . 13 |- ( A e. ZZ -> ( A mod 8 ) e. NN0 )
2423nn0zd 9567 . . . . . . . . . . . 12 |- ( A e. ZZ -> ( A mod 8 ) e. ZZ )
25 1zzd 9473 . . . . . . . . . . . 12 |- ( A e. ZZ -> 1 e. ZZ )
26 zdceq 9522 . . . . . . . . . . . 12 |- ( ( ( A mod 8 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A mod 8 ) = 1 )
2724, 25, 26syl2anc 411 . . . . . . . . . . 11 |- ( A e. ZZ -> DECID ( A mod 8 ) = 1 )
28 7nn 9277 . . . . . . . . . . . . 13 |- 7 e. NN
2928nnzi 9467 . . . . . . . . . . . 12 |- 7 e. ZZ
30 zdceq 9522 . . . . . . . . . . . 12 |- ( ( ( A mod 8 ) e. ZZ /\ 7 e. ZZ ) -> DECID ( A mod 8 ) = 7 )
3124, 29, 30sylancl 413 . . . . . . . . . . 11 |- ( A e. ZZ -> DECID ( A mod 8 ) = 7 )
32 dcor 941 . . . . . . . . . . 11 |- (DECID ( A mod 8 ) = 1 -> (DECID ( A mod 8 ) = 7 -> DECID ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) ) )
3327, 31, 32sylc 62 . . . . . . . . . 10 |- ( A e. ZZ -> DECID ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) )
34 elprg 3686 . . . . . . . . . . . 12 |- ( ( A mod 8 ) e. NN0 -> ( ( A mod 8 ) e. { 1 , 7 } <-> ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) ) )
3523, 34syl 14 . . . . . . . . . . 11 |- ( A e. ZZ -> ( ( A mod 8 ) e. { 1 , 7 } <-> ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) ) )
3635dcbid 843 . . . . . . . . . 10 |- ( A e. ZZ -> (DECID ( A mod 8 ) e. { 1 , 7 } <-> DECID ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) ) )
3733, 36mpbird 167 . . . . . . . . 9 |- ( A e. ZZ -> DECID ( A mod 8 ) e. { 1 , 7 } )
38373ad2ant1 1042 . . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> DECID ( A mod 8 ) e. { 1 , 7 } )
3938ad2antrr 488 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> DECID ( A mod 8 ) e. { 1 , 7 } )
4017, 19, 39ifcldcd 3640 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) e. ZZ )
41 2nn 9272 . . . . . . . 8 |- 2 e. NN
4241a1i 9 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> 2 e. NN )
43 simpll1 1060 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> A e. ZZ )
44 dvdsdc 12309 . . . . . . 7 |- ( ( 2 e. NN /\ A e. ZZ ) -> DECID 2 || A )
4542, 43, 44syl2anc 411 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> DECID 2 || A )
4616, 40, 45ifcldcd 3640 . . . . 5 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) e. ZZ )
47 simpll1 1060 . . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> A e. ZZ )
48 simpr 110 . . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> -. M = 2 )
49 prm2orodd 12648 . . . . . . . . . . . . . 14 |- ( M e. Prime -> ( M = 2 \/ -. 2 || M ) )
5049orcomd 734 . . . . . . . . . . . . 13 |- ( M e. Prime -> ( -. 2 || M \/ M = 2 ) )
5150ad2antlr 489 . . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( -. 2 || M \/ M = 2 ) )
5248, 51ecased 1383 . . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> -. 2 || M )
5315ad2antrr 488 . . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> M e. NN )
5453nnnn0d 9422 . . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> M e. NN0 )
55 nn0oddm1d2 12420 . . . . . . . . . . . 12 |- ( M e. NN0 -> ( -. 2 || M <-> ( ( M - 1 ) / 2 ) e. NN0 ) )
5654, 55syl 14 . . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( -. 2 || M <-> ( ( M - 1 ) / 2 ) e. NN0 ) )
5752, 56mpbid 147 . . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( ( M - 1 ) / 2 ) e. NN0 )
58 zexpcl 10776 . . . . . . . . . 10 |- ( ( A e. ZZ /\ ( ( M - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( M - 1 ) / 2 ) ) e. ZZ )
5947, 57, 58syl2anc 411 . . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( A ^ ( ( M - 1 ) / 2 ) ) e. ZZ )
6059peano2zd 9572 . . . . . . . 8 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) e. ZZ )
6160, 53zmodcld 10567 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) e. NN0 )
6261nn0zd 9567 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) e. ZZ )
63 1zzd 9473 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> 1 e. ZZ )
6462, 63zsubcld 9574 . . . . 5 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) e. ZZ )
65 simpl3 1026 . . . . . . 7 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. NN )
6665nnzd 9568 . . . . . 6 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. ZZ )
67 2z 9474 . . . . . 6 |- 2 e. ZZ
68 zdceq 9522 . . . . . 6 |- ( ( M e. ZZ /\ 2 e. ZZ ) -> DECID M = 2 )
6966, 67, 68sylancl 413 . . . . 5 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> DECID M = 2 )
7046, 64, 69ifcldadc 3632 . . . 4 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> if ( M = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) ) e. ZZ )
71 simpr 110 . . . . 5 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. Prime )
72 simpl2 1025 . . . . 5 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> N e. NN )
7371, 72pccld 12823 . . . 4 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> ( M pCnt N ) e. NN0 )
74 zexpcl 10776 . . . 4 |- ( ( if ( M = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) ) e. ZZ /\ ( M pCnt N ) e. NN0 ) -> ( if ( M = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) ) ^ ( M pCnt N ) ) e. ZZ )
7570, 73, 74syl2anc 411 . . 3 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> ( if ( M = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) ) ^ ( M pCnt N ) ) e. ZZ )
76 1zzd 9473 . . 3 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ -. M e. Prime ) -> 1 e. ZZ )
77 prmdc 12652 . . . 4 |- ( M e. NN -> DECID M e. Prime )
7815, 77syl 14 . . 3 |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> DECID M e. Prime )
7975, 76, 78ifcldadc 3632 . 2 |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> if ( M e. Prime , ( if ( M = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) ) ^ ( M pCnt N ) ) , 1 ) e. ZZ )
801, 14, 15, 79fvmptd3 5728 1 |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> ( F ` M ) = if ( M e. Prime , ( if ( M = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) ) ^ ( M pCnt N ) ) , 1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   ifcif 3602   {cpr 3667   class class class wbr 4083   |-> cmpt 4145   ` cfv 5318  (class class class)co 6001   0cc0 7999   1c1 8000   + caddc 8002   - cmin 8317   -ucneg 8318   / cdiv 8819   NNcn 9110   2c2 9161   7c7 9166   8c8 9167   NN0cn0 9369   ZZcz 9446   mod cmo 10544   ^cexp 10760   || cdvds 12298   Primecprime 12629   pCnt cpc 12807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119
This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-2o 6563  df-er 6680  df-en 6888  df-fin 6890  df-sup 7151  df-inf 7152  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-gcd 12475  df-prm 12630  df-pc 12808
This theorem is referenced by:  lgsval2lem  15689
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