Theorem lgsprme0 16047

Description: The Legendre symbol at any prime (even at 2) is 0 iff the prime does not divide the first argument. See definition in [ApostolNT] p. 179. (Contributed by AV, 20-Jul-2021.)

Assertion

Ref	Expression
lgsprme0	|- ( ( A e. ZZ /\ P e. Prime ) -> ( ( A /L P ) = 0 <-> ( A mod P ) = 0 ) )

Proof of Theorem lgsprme0

Step	Hyp	Ref	Expression
1		prmz 12839	. . . 4 |- ( P e. Prime -> P e. ZZ )
2		lgscl 16019	. . . 4 |- ( ( A e. ZZ /\ P e. ZZ ) -> ( A /L P ) e. ZZ )
3	1, 2	sylan2 286	. . 3 |- ( ( A e. ZZ /\ P e. Prime ) -> ( A /L P ) e. ZZ )
4		0z 9610	. . 3 |- 0 e. ZZ
5		zdceq 9675	. . 3 |- ( ( ( A /L P ) e. ZZ /\ 0 e. ZZ ) -> DECID ( A /L P ) = 0 )
6	3, 4, 5	sylancl 413	. 2 |- ( ( A e. ZZ /\ P e. Prime ) -> DECID ( A /L P ) = 0 )
7		simpl 109	. . . . 5 |- ( ( A e. ZZ /\ P e. Prime ) -> A e. ZZ )
8		prmnn 12838	. . . . . 6 |- ( P e. Prime -> P e. NN )
9	8	adantl 277	. . . . 5 |- ( ( A e. ZZ /\ P e. Prime ) -> P e. NN )
10	7, 9	zmodcld 10736	. . . 4 |- ( ( A e. ZZ /\ P e. Prime ) -> ( A mod P ) e. NN0 )
11	10	nn0zd 9721	. . 3 |- ( ( A e. ZZ /\ P e. Prime ) -> ( A mod P ) e. ZZ )
12		zdceq 9675	. . 3 |- ( ( ( A mod P ) e. ZZ /\ 0 e. ZZ ) -> DECID ( A mod P ) = 0 )
13	11, 4, 12	sylancl 413	. 2 |- ( ( A e. ZZ /\ P e. Prime ) -> DECID ( A mod P ) = 0 )
14		lgsne0 16043	. . . . . 6 |- ( ( A e. ZZ /\ P e. ZZ ) -> ( ( A /L P ) =/= 0 <-> ( A gcd P ) = 1 ) )
15	1, 14	sylan2 286	. . . . 5 |- ( ( A e. ZZ /\ P e. Prime ) -> ( ( A /L P ) =/= 0 <-> ( A gcd P ) = 1 ) )
16		coprm 12872	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
17	16	ancoms 268	. . . . . 6 |- ( ( A e. ZZ /\ P e. Prime ) -> ( -. P || A <-> ( P gcd A ) = 1 ) )
18	1	anim1i 340	. . . . . . . . 9 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P e. ZZ /\ A e. ZZ ) )
19	18	ancoms 268	. . . . . . . 8 |- ( ( A e. ZZ /\ P e. Prime ) -> ( P e. ZZ /\ A e. ZZ ) )
20		gcdcom 12700	. . . . . . . 8 |- ( ( P e. ZZ /\ A e. ZZ ) -> ( P gcd A ) = ( A gcd P ) )
21	19, 20	syl 14	. . . . . . 7 |- ( ( A e. ZZ /\ P e. Prime ) -> ( P gcd A ) = ( A gcd P ) )
22	21	eqeq1d 2243	. . . . . 6 |- ( ( A e. ZZ /\ P e. Prime ) -> ( ( P gcd A ) = 1 <-> ( A gcd P ) = 1 ) )
23	17, 22	bitr2d 189	. . . . 5 |- ( ( A e. ZZ /\ P e. Prime ) -> ( ( A gcd P ) = 1 <-> -. P || A ) )
24		dvdsval3 12508	. . . . . . . 8 |- ( ( P e. NN /\ A e. ZZ ) -> ( P || A <-> ( A mod P ) = 0 ) )
25	8, 24	sylan 283	. . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P || A <-> ( A mod P ) = 0 ) )
26	25	ancoms 268	. . . . . 6 |- ( ( A e. ZZ /\ P e. Prime ) -> ( P || A <-> ( A mod P ) = 0 ) )
27	26	notbid 673	. . . . 5 |- ( ( A e. ZZ /\ P e. Prime ) -> ( -. P || A <-> -. ( A mod P ) = 0 ) )
28	15, 23, 27	3bitrd 214	. . . 4 |- ( ( A e. ZZ /\ P e. Prime ) -> ( ( A /L P ) =/= 0 <-> -. ( A mod P ) = 0 ) )
29	28	2a1d 23	. . 3 |- ( ( A e. ZZ /\ P e. Prime ) -> (DECID ( A /L P ) = 0 -> (DECID ( A mod P ) = 0 -> ( ( A /L P ) =/= 0 <-> -. ( A mod P ) = 0 ) ) ) )
30	29	necon4abiddc 2487	. 2 |- ( ( A e. ZZ /\ P e. Prime ) -> (DECID ( A /L P ) = 0 -> (DECID ( A mod P ) = 0 -> ( ( A /L P ) = 0 <-> ( A mod P ) = 0 ) ) ) )
31	6, 13, 30	mp2d 47	1 |- ( ( A e. ZZ /\ P e. Prime ) -> ( ( A /L P ) = 0 <-> ( A mod P ) = 0 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   class class class wbr 4115  (class class class)co 6060   0cc0 8145   1c1 8146   NNcn 9259   ZZcz 9599   mod cmo 10713   || cdvds 12504   gcd cgcd 12680   Primecprime 12835   /Lclgs 16002

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-iinf 4717  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-mulrcl 8244  ax-addcom 8245  ax-mulcom 8246  ax-addass 8247  ax-mulass 8248  ax-distr 8249  ax-i2m1 8250  ax-0lt1 8251  ax-1rid 8252  ax-0id 8253  ax-rnegex 8254  ax-precex 8255  ax-cnre 8256  ax-pre-ltirr 8257  ax-pre-ltwlin 8258  ax-pre-lttrn 8259  ax-pre-apti 8260  ax-pre-ltadd 8261  ax-pre-mulgt0 8262  ax-pre-mulext 8263  ax-arch 8264  ax-caucvg 8265

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4720  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-isom 5368  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-1st 6349  df-2nd 6350  df-recs 6551  df-irdg 6616  df-frec 6637  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6782  df-en 6991  df-dom 6992  df-fin 6993  df-sup 7290  df-inf 7291  df-pnf 8328  df-mnf 8329  df-xr 8330  df-ltxr 8331  df-le 8332  df-sub 8465  df-neg 8466  df-reap 8869  df-ap 8876  df-div 8969  df-inn 9260  df-2 9318  df-3 9319  df-4 9320  df-5 9321  df-6 9322  df-7 9323  df-8 9324  df-n0 9519  df-z 9600  df-uz 9877  df-q 9975  df-rp 10010  df-fz 10367  df-fzo 10504  df-fl 10659  df-mod 10714  df-seqfrec 10839  df-exp 10930  df-ihash 11169  df-cj 11557  df-re 11558  df-im 11559  df-rsqrt 11714  df-abs 11715  df-clim 11995  df-proddc 12268  df-dvds 12505  df-gcd 12681  df-prm 12836  df-phi 12939  df-pc 13014  df-lgs 16003

This theorem is referenced by: (None)