Theorem lgssq 14480

Description: The Legendre symbol at a square is equal to 1. Together with lgsmod 14466 this implies that the Legendre symbol takes value 1 at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.) (Revised by AV, 20-Jul-2021.)

Assertion

Ref	Expression
lgssq	|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ 2 ) /L N ) = 1 )

Proof of Theorem lgssq

Step	Hyp	Ref	Expression
1		simp1l 1021	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A e. ZZ )
2		simp2 998	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> N e. ZZ )
3		simp1r 1022	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A =/= 0 )
4		lgsdir 14475	. . 3 |- ( ( ( A e. ZZ /\ A e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ A =/= 0 ) ) -> ( ( A x. A ) /L N ) = ( ( A /L N ) x. ( A /L N ) ) )
5	1, 1, 2, 3, 3, 4	syl32anc 1246	. 2 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A x. A ) /L N ) = ( ( A /L N ) x. ( A /L N ) ) )
6		zcn 9260	. . . . . 6 |- ( A e. ZZ -> A e. CC )
7	6	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ A =/= 0 ) -> A e. CC )
8	7	3ad2ant1 1018	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A e. CC )
9	8	sqvald 10653	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A ^ 2 ) = ( A x. A ) )
10	9	oveq1d 5892	. 2 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ 2 ) /L N ) = ( ( A x. A ) /L N ) )
11		lgscl 14454	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
12	1, 2, 11	syl2anc 411	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. ZZ )
13	12	zred 9377	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. RR )
14		absresq 11089	. . . 4 |- ( ( A /L N ) e. RR -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( ( A /L N ) ^ 2 ) )
15	13, 14	syl 14	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( ( A /L N ) ^ 2 ) )
16		lgsabs1 14479	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( ( abs ` ( A /L N ) ) = 1 <-> ( A gcd N ) = 1 ) )
17	16	adantlr 477	. . . . . 6 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ ) -> ( ( abs ` ( A /L N ) ) = 1 <-> ( A gcd N ) = 1 ) )
18	17	biimp3ar 1346	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( abs ` ( A /L N ) ) = 1 )
19	18	oveq1d 5892	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( 1 ^ 2 ) )
20		sq1 10616	. . . 4 |- ( 1 ^ 2 ) = 1
21	19, 20	eqtrdi 2226	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = 1 )
22	12	zcnd 9378	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. CC )
23	22	sqvald 10653	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A /L N ) ^ 2 ) = ( ( A /L N ) x. ( A /L N ) ) )
24	15, 21, 23	3eqtr3d 2218	. 2 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> 1 = ( ( A /L N ) x. ( A /L N ) ) )
25	5, 10, 24	3eqtr4d 2220	1 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ 2 ) /L N ) = 1 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   ` cfv 5218  (class class class)co 5877   CCcc 7811   RRcr 7812   0cc0 7813   1c1 7814   x. cmul 7818   2c2 8972   ZZcz 9255   ^cexp 10521   abscabs 11008   gcd cgcd 11945   /Lclgs 14437

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-2o 6420  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-7 8985  df-8 8986  df-9 8987  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-ihash 10758  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-proddc 11561  df-dvds 11797  df-gcd 11946  df-prm 12110  df-phi 12213  df-pc 12287  df-lgs 14438

This theorem is referenced by:  1lgs  14483  lgsmulsqcoprm  14486