Theorem lgssq 13541

Description: The Legendre symbol at a square is equal to 1. Together with lgsmod 13527 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 1011	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A e. ZZ )
2		simp2 988	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> N e. ZZ )
3		simp1r 1012	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A =/= 0 )
4		lgsdir 13536	. . 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 1236	. 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 9192	. . . . . 6 |- ( A e. ZZ -> A e. CC )
7	6	adantr 274	. . . . 5 |- ( ( A e. ZZ /\ A =/= 0 ) -> A e. CC )
8	7	3ad2ant1 1008	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A e. CC )
9	8	sqvald 10581	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A ^ 2 ) = ( A x. A ) )
10	9	oveq1d 5856	. 2 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ 2 ) /L N ) = ( ( A x. A ) /L N ) )
11		lgscl 13515	. . . . . 6 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
12	1, 2, 11	syl2anc 409	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. ZZ )
13	12	zred 9309	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. RR )
14		absresq 11016	. . . 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 13540	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( ( abs ` ( A /L N ) ) = 1 <-> ( A gcd N ) = 1 ) )
17	16	adantlr 469	. . . . . 6 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ ) -> ( ( abs ` ( A /L N ) ) = 1 <-> ( A gcd N ) = 1 ) )
18	17	biimp3ar 1336	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( abs ` ( A /L N ) ) = 1 )
19	18	oveq1d 5856	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( 1 ^ 2 ) )
20		sq1 10544	. . . 4 |- ( 1 ^ 2 ) = 1
21	19, 20	eqtrdi 2214	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = 1 )
22	12	zcnd 9310	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. CC )
23	22	sqvald 10581	. . 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 2206	. 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 2208	1 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ 2 ) /L N ) = 1 )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2335   ` cfv 5187  (class class class)co 5841   CCcc 7747   RRcr 7748   0cc0 7749   1c1 7750   x. cmul 7754   2c2 8904   ZZcz 9187   ^cexp 10450   abscabs 10935   gcd cgcd 11871   /Lclgs 13498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-xor 1366  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-irdg 6334  df-frec 6355  df-1o 6380  df-2o 6381  df-oadd 6384  df-er 6497  df-en 6703  df-dom 6704  df-fin 6705  df-sup 6945  df-inf 6946  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-5 8915  df-6 8916  df-7 8917  df-8 8918  df-9 8919  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-ihash 10685  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-clim 11216  df-proddc 11488  df-dvds 11724  df-gcd 11872  df-prm 12036  df-phi 12139  df-pc 12213  df-lgs 13499

This theorem is referenced by:  1lgs  13544  lgsmulsqcoprm  13547