Theorem lgsprme0 15721

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 12633	. . . 4 |- ( P e. Prime -> P e. ZZ )
2		lgscl 15693	. . . 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 9457	. . 3 |- 0 e. ZZ
5		zdceq 9522	. . 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 12632	. . . . . 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 10567	. . . 4 |- ( ( A e. ZZ /\ P e. Prime ) -> ( A mod P ) e. NN0 )
11	10	nn0zd 9567	. . 3 |- ( ( A e. ZZ /\ P e. Prime ) -> ( A mod P ) e. ZZ )
12		zdceq 9522	. . 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 15717	. . . . . 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 12666	. . . . . . 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 12494	. . . . . . . 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 2238	. . . . . 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 12302	. . . . . . . 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 671	. . . . 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 2473	. 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 839   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4083  (class class class)co 6001   0cc0 7999   1c1 8000   NNcn 9110   ZZcz 9446   mod cmo 10544   || cdvds 12298   gcd cgcd 12474   Primecprime 12629   /Lclgs 15676

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-2o 6563  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-sup 7151  df-inf 7152  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-proddc 12062  df-dvds 12299  df-gcd 12475  df-prm 12630  df-phi 12733  df-pc 12808  df-lgs 15677

This theorem is referenced by: (None)