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Theorem lgsfvalg 15246
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 2259 . . 3 |- ( n = M -> ( n e. Prime <-> M e. Prime ) )
3 eqeq1 2203 . . . . 5 |- ( n = M -> ( n = 2 <-> M = 2 ) )
4 oveq1 5929 . . . . . . . . . 10 |- ( n = M -> ( n - 1 ) = ( M - 1 ) )
54oveq1d 5937 . . . . . . . . 9 |- ( n = M -> ( ( n - 1 ) / 2 ) = ( ( M - 1 ) / 2 ) )
65oveq2d 5938 . . . . . . . 8 |- ( n = M -> ( A ^ ( ( n - 1 ) / 2 ) ) = ( A ^ ( ( M - 1 ) / 2 ) ) )
76oveq1d 5937 . . . . . . 7 |- ( n = M -> ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) = ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) )
8 id 19 . . . . . . 7 |- ( n = M -> n = M )
97, 8oveq12d 5940 . . . . . 6 |- ( n = M -> ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) = ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) )
109oveq1d 5937 . . . . 5 |- ( n = M -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) = ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) )
113, 10ifbieq2d 3585 . . . 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 5929 . . . 4 |- ( n = M -> ( n pCnt N ) = ( M pCnt N ) )
1311, 12oveq12d 5940 . . 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 3583 . 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 1001 . 2 |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> M e. NN )
16 0zd 9338 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> 0 e. ZZ )
17 1zzd 9353 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> 1 e. ZZ )
18 neg1z 9358 . . . . . . . 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 9158 . . . . . . . . . . . . . . 15 |- 8 e. NN
2221a1i 9 . . . . . . . . . . . . . 14 |- ( A e. ZZ -> 8 e. NN )
2320, 22zmodcld 10437 . . . . . . . . . . . . 13 |- ( A e. ZZ -> ( A mod 8 ) e. NN0 )
2423nn0zd 9446 . . . . . . . . . . . 12 |- ( A e. ZZ -> ( A mod 8 ) e. ZZ )
25 1zzd 9353 . . . . . . . . . . . 12 |- ( A e. ZZ -> 1 e. ZZ )
26 zdceq 9401 . . . . . . . . . . . 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 9157 . . . . . . . . . . . . 13 |- 7 e. NN
2928nnzi 9347 . . . . . . . . . . . 12 |- 7 e. ZZ
30 zdceq 9401 . . . . . . . . . . . 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 937 . . . . . . . . . . 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 3642 . . . . . . . . . . . 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 839 . . . . . . . . . 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 1020 . . . . . . . 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 3597 . . . . . 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 9152 . . . . . . . 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 1038 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> A e. ZZ )
44 dvdsdc 11963 . . . . . . 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 3597 . . . . 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 1038 . . . . . . . . . 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 12294 . . . . . . . . . . . . . 14 |- ( M e. Prime -> ( M = 2 \/ -. 2 || M ) )
5049orcomd 730 . . . . . . . . . . . . 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 1360 . . . . . . . . . . 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 9302 . . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> M e. NN0 )
55 nn0oddm1d2 12074 . . . . . . . . . . . 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 10646 . . . . . . . . . 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 9451 . . . . . . . 8 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) e. ZZ )
6160, 53zmodcld 10437 . . . . . . 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 9446 . . . . . 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 9353 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> 1 e. ZZ )
6462, 63zsubcld 9453 . . . . 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 1004 . . . . . . 7 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. NN )
6665nnzd 9447 . . . . . 6 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. ZZ )
67 2z 9354 . . . . . 6 |- 2 e. ZZ
68 zdceq 9401 . . . . . 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 3590 . . . 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 1003 . . . . 5 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> N e. NN )
7371, 72pccld 12469 . . . 4 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> ( M pCnt N ) e. NN0 )
74 zexpcl 10646 . . . 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 9353 . . 3 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ -. M e. Prime ) -> 1 e. ZZ )
77 prmdc 12298 . . . 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 3590 . 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 5655 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 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2167   ifcif 3561   {cpr 3623   class class class wbr 4033   |-> cmpt 4094   ` cfv 5258  (class class class)co 5922   0cc0 7879   1c1 7880   + caddc 7882   - cmin 8197   -ucneg 8198   / cdiv 8699   NNcn 8990   2c2 9041   7c7 9046   8c8 9047   NN0cn0 9249   ZZcz 9326   mod cmo 10414   ^cexp 10630   || cdvds 11952   Primecprime 12275   pCnt cpc 12453
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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-2o 6475  df-er 6592  df-en 6800  df-fin 6802  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-gcd 12121  df-prm 12276  df-pc 12454
This theorem is referenced by:  lgsval2lem  15251
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