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Theorem lgsfvalg 13506
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 2228 . . 3 |- ( n = M -> ( n e. Prime <-> M e. Prime ) )
3 eqeq1 2172 . . . . 5 |- ( n = M -> ( n = 2 <-> M = 2 ) )
4 oveq1 5848 . . . . . . . . . 10 |- ( n = M -> ( n - 1 ) = ( M - 1 ) )
54oveq1d 5856 . . . . . . . . 9 |- ( n = M -> ( ( n - 1 ) / 2 ) = ( ( M - 1 ) / 2 ) )
65oveq2d 5857 . . . . . . . 8 |- ( n = M -> ( A ^ ( ( n - 1 ) / 2 ) ) = ( A ^ ( ( M - 1 ) / 2 ) ) )
76oveq1d 5856 . . . . . . 7 |- ( n = M -> ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) = ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) )
8 id 19 . . . . . . 7 |- ( n = M -> n = M )
97, 8oveq12d 5859 . . . . . 6 |- ( n = M -> ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) = ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) )
109oveq1d 5856 . . . . 5 |- ( n = M -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) = ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) )
113, 10ifbieq2d 3543 . . . 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 5848 . . . 4 |- ( n = M -> ( n pCnt N ) = ( M pCnt N ) )
1311, 12oveq12d 5859 . . 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 3541 . 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 989 . 2 |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> M e. NN )
16 0zd 9199 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> 0 e. ZZ )
17 1zzd 9214 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> 1 e. ZZ )
18 neg1z 9219 . . . . . . . 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 9020 . . . . . . . . . . . . . . 15 |- 8 e. NN
2221a1i 9 . . . . . . . . . . . . . 14 |- ( A e. ZZ -> 8 e. NN )
2320, 22zmodcld 10276 . . . . . . . . . . . . 13 |- ( A e. ZZ -> ( A mod 8 ) e. NN0 )
2423nn0zd 9307 . . . . . . . . . . . 12 |- ( A e. ZZ -> ( A mod 8 ) e. ZZ )
25 1zzd 9214 . . . . . . . . . . . 12 |- ( A e. ZZ -> 1 e. ZZ )
26 zdceq 9262 . . . . . . . . . . . 12 |- ( ( ( A mod 8 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A mod 8 ) = 1 )
2724, 25, 26syl2anc 409 . . . . . . . . . . 11 |- ( A e. ZZ -> DECID ( A mod 8 ) = 1 )
28 7nn 9019 . . . . . . . . . . . . 13 |- 7 e. NN
2928nnzi 9208 . . . . . . . . . . . 12 |- 7 e. ZZ
30 zdceq 9262 . . . . . . . . . . . 12 |- ( ( ( A mod 8 ) e. ZZ /\ 7 e. ZZ ) -> DECID ( A mod 8 ) = 7 )
3124, 29, 30sylancl 410 . . . . . . . . . . 11 |- ( A e. ZZ -> DECID ( A mod 8 ) = 7 )
32 dcor 925 . . . . . . . . . . 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 3595 . . . . . . . . . . . 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 828 . . . . . . . . . 10 |- ( A e. ZZ -> (DECID ( A mod 8 ) e. { 1 , 7 } <-> DECID ( ( A mod 8 ) = 1 \/ ( A mod 8 ) = 7 ) ) )
3733, 36mpbird 166 . . . . . . . . 9 |- ( A e. ZZ -> DECID ( A mod 8 ) e. { 1 , 7 } )
38373ad2ant1 1008 . . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> DECID ( A mod 8 ) e. { 1 , 7 } )
3938ad2antrr 480 . . . . . . 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 3554 . . . . . 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 9014 . . . . . . . 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 1026 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> A e. ZZ )
44 dvdsdc 11734 . . . . . . 7 |- ( ( 2 e. NN /\ A e. ZZ ) -> DECID 2 || A )
4542, 43, 44syl2anc 409 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> DECID 2 || A )
4616, 40, 45ifcldcd 3554 . . . . 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 1026 . . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> A e. ZZ )
48 simpr 109 . . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> -. M = 2 )
49 prm2orodd 12054 . . . . . . . . . . . . . 14 |- ( M e. Prime -> ( M = 2 \/ -. 2 || M ) )
5049orcomd 719 . . . . . . . . . . . . 13 |- ( M e. Prime -> ( -. 2 || M \/ M = 2 ) )
5150ad2antlr 481 . . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( -. 2 || M \/ M = 2 ) )
5248, 51ecased 1339 . . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> -. 2 || M )
5315ad2antrr 480 . . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> M e. NN )
5453nnnn0d 9163 . . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> M e. NN0 )
55 nn0oddm1d2 11842 . . . . . . . . . . . 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 146 . . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( ( M - 1 ) / 2 ) e. NN0 )
58 zexpcl 10466 . . . . . . . . . 10 |- ( ( A e. ZZ /\ ( ( M - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( M - 1 ) / 2 ) ) e. ZZ )
5947, 57, 58syl2anc 409 . . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( A ^ ( ( M - 1 ) / 2 ) ) e. ZZ )
6059peano2zd 9312 . . . . . . . 8 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) e. ZZ )
6160, 53zmodcld 10276 . . . . . . 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 9307 . . . . . 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 9214 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> 1 e. ZZ )
6462, 63zsubcld 9314 . . . . 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 992 . . . . . . 7 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. NN )
6665nnzd 9308 . . . . . 6 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. ZZ )
67 2z 9215 . . . . . 6 |- 2 e. ZZ
68 zdceq 9262 . . . . . 6 |- ( ( M e. ZZ /\ 2 e. ZZ ) -> DECID M = 2 )
6966, 67, 68sylancl 410 . . . . 5 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> DECID M = 2 )
7046, 64, 69ifcldadc 3548 . . . 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 109 . . . . 5 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. Prime )
72 simpl2 991 . . . . 5 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> N e. NN )
7371, 72pccld 12228 . . . 4 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> ( M pCnt N ) e. NN0 )
74 zexpcl 10466 . . . 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 409 . . 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 9214 . . 3 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ -. M e. Prime ) -> 1 e. ZZ )
77 prmdc 12058 . . . 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 3548 . 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 5578 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 103   <-> wb 104   \/ wo 698  DECID wdc 824   /\ w3a 968   = wceq 1343   e. wcel 2136   ifcif 3519   {cpr 3576   class class class wbr 3981   |-> cmpt 4042   ` cfv 5187  (class class class)co 5841   0cc0 7749   1c1 7750   + caddc 7752   - cmin 8065   -ucneg 8066   / cdiv 8564   NNcn 8853   2c2 8904   7c7 8909   8c8 8910   NN0cn0 9110   ZZcz 9187   mod cmo 10253   ^cexp 10450   || cdvds 11723   Primecprime 12035   pCnt cpc 12212
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869
This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-xor 1366  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-1o 6380  df-2o 6381  df-er 6497  df-en 6703  df-fin 6705  df-sup 6945  df-inf 6946  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-5 8915  df-6 8916  df-7 8917  df-8 8918  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-dvds 11724  df-gcd 11872  df-prm 12036  df-pc 12213
This theorem is referenced by:  lgsval2lem  13511
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