Theorem lgsprme0 16041

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 12833	. . . 4 |- ( P e. Prime -> P e. ZZ )
2		lgscl 16013	. . . 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 9605	. . 3 |- 0 e. ZZ
5		zdceq 9670	. . 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 12832	. . . . . 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 10731	. . . 4 |- ( ( A e. ZZ /\ P e. Prime ) -> ( A mod P ) e. NN0 )
11	10	nn0zd 9716	. . 3 |- ( ( A e. ZZ /\ P e. Prime ) -> ( A mod P ) e. ZZ )
12		zdceq 9670	. . 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 16037	. . . . . 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 12866	. . . . . . 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 12694	. . . . . . . 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 12502	. . . . . . . 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 4114  (class class class)co 6058   0cc0 8143   1c1 8144   NNcn 9254   ZZcz 9594   mod cmo 10708   || cdvds 12498   gcd cgcd 12674   Primecprime 12829   /Lclgs 15996

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-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

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 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-proddc 12262  df-dvds 12499  df-gcd 12675  df-prm 12830  df-phi 12933  df-pc 13008  df-lgs 15997

This theorem is referenced by: (None)