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Theorem lgsfvalg 16038
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 2301 . . 3 |- ( n = M -> ( n e. Prime <-> M e. Prime ) )
3 eqeq1 2245 . . . . 5 |- ( n = M -> ( n = 2 <-> M = 2 ) )
4 oveq1 6082 . . . . . . . . . 10 |- ( n = M -> ( n - 1 ) = ( M - 1 ) )
54oveq1d 6090 . . . . . . . . 9 |- ( n = M -> ( ( n - 1 ) / 2 ) = ( ( M - 1 ) / 2 ) )
65oveq2d 6091 . . . . . . . 8 |- ( n = M -> ( A ^ ( ( n - 1 ) / 2 ) ) = ( A ^ ( ( M - 1 ) / 2 ) ) )
76oveq1d 6090 . . . . . . 7 |- ( n = M -> ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) = ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) )
8 id 19 . . . . . . 7 |- ( n = M -> n = M )
97, 8oveq12d 6093 . . . . . 6 |- ( n = M -> ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) = ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) )
109oveq1d 6090 . . . . 5 |- ( n = M -> ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) = ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) )
113, 10ifbieq2d 3662 . . . 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 6082 . . . 4 |- ( n = M -> ( n pCnt N ) = ( M pCnt N ) )
1311, 12oveq12d 6093 . . 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 3660 . 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 1030 . 2 |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> M e. NN )
16 0zd 9635 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> 0 e. ZZ )
17 1zzd 9650 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> 1 e. ZZ )
18 neg1z 9655 . . . . . . . 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 9451 . . . . . . . . . . . . . . 15 |- 8 e. NN
2221a1i 9 . . . . . . . . . . . . . 14 |- ( A e. ZZ -> 8 e. NN )
2320, 22zmodcld 10760 . . . . . . . . . . . . 13 |- ( A e. ZZ -> ( A mod 8 ) e. NN0 )
2423nn0zd 9745 . . . . . . . . . . . 12 |- ( A e. ZZ -> ( A mod 8 ) e. ZZ )
25 1zzd 9650 . . . . . . . . . . . 12 |- ( A e. ZZ -> 1 e. ZZ )
26 zdceq 9699 . . . . . . . . . . . 12 |- ( ( ( A mod 8 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A mod 8 ) = 1 )
2724, 25, 26syl2anc 415 . . . . . . . . . . 11 |- ( A e. ZZ -> DECID ( A mod 8 ) = 1 )
28 7nn 9450 . . . . . . . . . . . . 13 |- 7 e. NN
2928nnzi 9644 . . . . . . . . . . . 12 |- 7 e. ZZ
30 zdceq 9699 . . . . . . . . . . . 12 |- ( ( ( A mod 8 ) e. ZZ /\ 7 e. ZZ ) -> DECID ( A mod 8 ) = 7 )
3124, 29, 30sylancl 417 . . . . . . . . . . 11 |- ( A e. ZZ -> DECID ( A mod 8 ) = 7 )
32 dcor 948 . . . . . . . . . . 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 3725 . . . . . . . . . . . 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 850 . . . . . . . . . 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 1049 . . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> DECID ( A mod 8 ) e. { 1 , 7 } )
3938ad2antrr 492 . . . . . . 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 3675 . . . . . 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 9445 . . . . . . . 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 1067 . . . . . . 7 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> A e. ZZ )
44 dvdsdc 12543 . . . . . . 7 |- ( ( 2 e. NN /\ A e. ZZ ) -> DECID 2 || A )
4542, 43, 44syl2anc 415 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ M = 2 ) -> DECID 2 || A )
4616, 40, 45ifcldcd 3675 . . . . 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 1067 . . . . . . . . . 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 12882 . . . . . . . . . . . . . 14 |- ( M e. Prime -> ( M = 2 \/ -. 2 || M ) )
5049orcomd 741 . . . . . . . . . . . . 13 |- ( M e. Prime -> ( -. 2 || M \/ M = 2 ) )
5150ad2antlr 493 . . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( -. 2 || M \/ M = 2 ) )
5248, 51ecased 1390 . . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> -. 2 || M )
5315ad2antrr 492 . . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> M e. NN )
5453nnnn0d 9599 . . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> M e. NN0 )
55 nn0oddm1d2 12654 . . . . . . . . . . . 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 10969 . . . . . . . . . 10 |- ( ( A e. ZZ /\ ( ( M - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( M - 1 ) / 2 ) ) e. ZZ )
5947, 57, 58syl2anc 415 . . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( A ^ ( ( M - 1 ) / 2 ) ) e. ZZ )
6059peano2zd 9750 . . . . . . . 8 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) e. ZZ )
6160, 53zmodcld 10760 . . . . . . 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 9745 . . . . . 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 9650 . . . . . 6 |- ( ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) /\ -. M = 2 ) -> 1 e. ZZ )
6462, 63zsubcld 9752 . . . . 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 1033 . . . . . . 7 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. NN )
6665nnzd 9746 . . . . . 6 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> M e. ZZ )
67 2z 9651 . . . . . 6 |- 2 e. ZZ
68 zdceq 9699 . . . . . 6 |- ( ( M e. ZZ /\ 2 e. ZZ ) -> DECID M = 2 )
6966, 67, 68sylancl 417 . . . . 5 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> DECID M = 2 )
7046, 64, 69ifcldadc 3667 . . . 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 1032 . . . . 5 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> N e. NN )
7371, 72pccld 13057 . . . 4 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ M e. Prime ) -> ( M pCnt N ) e. NN0 )
74 zexpcl 10969 . . . 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 415 . . 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 9650 . . 3 |- ( ( ( A e. ZZ /\ N e. NN /\ M e. NN ) /\ -. M e. Prime ) -> 1 e. ZZ )
77 prmdc 12886 . . . 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 3667 . 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 5793 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 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   ifcif 3635   {cpr 3706   class class class wbr 4125   |-> cmpt 4187   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   - cmin 8487   -ucneg 8488   / cdiv 8992   NNcn 9283   2c2 9334   7c7 9339   8c8 9340   NN0cn0 9542   ZZcz 9623   mod cmo 10737   ^cexp 10953   || cdvds 12532   Primecprime 12863   pCnt cpc 13041
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289
This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-2o 6678  df-er 6797  df-en 7013  df-fin 7015  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533  df-gcd 12709  df-prm 12864  df-pc 13042
This theorem is referenced by:  lgsval2lem  16043
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