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Theorem lgsfvalg 15753
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 2294 . . 3 |- ( n = M -> ( n e. Prime <-> M e. Prime ) )
3 eqeq1 2238 . . . . 5 |- ( n = M -> ( n = 2 <-> M = 2 ) )
4 oveq1 6025 . . . . . . . . . 10 |- ( n = M -> ( n - 1 ) = ( M - 1 ) )
54oveq1d 6033 . . . . . . . . 9 |- ( n = M -> ( ( n - 1 ) / 2 ) = ( ( M - 1 ) / 2 ) )
65oveq2d 6034 . . . . . . . 8 |- ( n = M -> ( A ^ ( ( n - 1 ) / 2 ) ) = ( A ^ ( ( M - 1 ) / 2 ) ) )
76oveq1d 6033 . . . . . . 7 |- ( n = M -> ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) = ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) )
8 id 19 . . . . . . 7 |- ( n = M -> n = M )
97, 8oveq12d 6036 . . . . . 6 |- ( n = M -> ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) = ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) )
109oveq1d 6033 . . . . 5 |- ( n = M -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) = ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) )
113, 10ifbieq2d 3630 . . . 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 6025 . . . 4 |- ( n = M -> ( n pCnt N ) = ( M pCnt N ) )
1311, 12oveq12d 6036 . . 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 3628 . 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 1025 . 2 |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> M e. NN )
16 0zd 9491 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> 0 e. ZZ )
17 1zzd 9506 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> 1 e. ZZ )
18 neg1z 9511 . . . . . . . 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 9311 . . . . . . . . . . . . . . 15 |- 8 e. NN
2221a1i 9 . . . . . . . . . . . . . 14 |- ( A e. ZZ -> 8 e. NN )
2320, 22zmodcld 10608 . . . . . . . . . . . . 13 |- ( A e. ZZ -> ( A mod 8 ) e. NN0 )
2423nn0zd 9600 . . . . . . . . . . . 12 |- ( A e. ZZ -> ( A mod 8 ) e. ZZ )
25 1zzd 9506 . . . . . . . . . . . 12 |- ( A e. ZZ -> 1 e. ZZ )
26 zdceq 9555 . . . . . . . . . . . 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 9310 . . . . . . . . . . . . 13 |- 7 e. NN
2928nnzi 9500 . . . . . . . . . . . 12 |- 7 e. ZZ
30 zdceq 9555 . . . . . . . . . . . 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 943 . . . . . . . . . . 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 3689 . . . . . . . . . . . 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 845 . . . . . . . . . 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 1044 . . . . . . . 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 3643 . . . . . 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 9305 . . . . . . . 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 1062 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> A e. ZZ )
44 dvdsdc 12377 . . . . . . 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 3643 . . . . 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 1062 . . . . . . . . . 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 12716 . . . . . . . . . . . . . 14 |- ( M e. Prime -> ( M = 2 \/ -. 2 || M ) )
5049orcomd 736 . . . . . . . . . . . . 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 1385 . . . . . . . . . . 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 9455 . . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> M e. NN0 )
55 nn0oddm1d2 12488 . . . . . . . . . . . 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 10817 . . . . . . . . . 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 9605 . . . . . . . 8 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) e. ZZ )
6160, 53zmodcld 10608 . . . . . . 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 9600 . . . . . 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 9506 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> 1 e. ZZ )
6462, 63zsubcld 9607 . . . . 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 1028 . . . . . . 7 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. NN )
6665nnzd 9601 . . . . . 6 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. ZZ )
67 2z 9507 . . . . . 6 |- 2 e. ZZ
68 zdceq 9555 . . . . . 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 3635 . . . 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 1027 . . . . 5 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> N e. NN )
7371, 72pccld 12891 . . . 4 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> ( M pCnt N ) e. NN0 )
74 zexpcl 10817 . . . 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 9506 . . 3 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ -. M e. Prime ) -> 1 e. ZZ )
77 prmdc 12720 . . . 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 3635 . 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 5740 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 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   ifcif 3605   {cpr 3670   class class class wbr 4088   |-> cmpt 4150   ` cfv 5326  (class class class)co 6018   0cc0 8032   1c1 8033   + caddc 8035   - cmin 8350   -ucneg 8351   / cdiv 8852   NNcn 9143   2c2 9194   7c7 9199   8c8 9200   NN0cn0 9402   ZZcz 9479   mod cmo 10585   ^cexp 10801   || cdvds 12366   Primecprime 12697   pCnt cpc 12875
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152
This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-2o 6583  df-er 6702  df-en 6910  df-fin 6912  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-dvds 12367  df-gcd 12543  df-prm 12698  df-pc 12876
This theorem is referenced by:  lgsval2lem  15758
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