Theorem lgssq 15307

Description: The Legendre symbol at a square is equal to 1. Together with lgsmod 15293 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 1023	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A e. ZZ )
2		simp2 1000	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> N e. ZZ )
3		simp1r 1024	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A =/= 0 )
4		lgsdir 15302	. . 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 1257	. 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 9334	. . . . . 6 |- ( A e. ZZ -> A e. CC )
7	6	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ A =/= 0 ) -> A e. CC )
8	7	3ad2ant1 1020	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A e. CC )
9	8	sqvald 10765	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A ^ 2 ) = ( A x. A ) )
10	9	oveq1d 5938	. 2 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ 2 ) /L N ) = ( ( A x. A ) /L N ) )
11		lgscl 15281	. . . . . 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 9451	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. RR )
14		absresq 11246	. . . 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 15306	. . . . . . 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 1357	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( abs ` ( A /L N ) ) = 1 )
19	18	oveq1d 5938	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( 1 ^ 2 ) )
20		sq1 10728	. . . 4 |- ( 1 ^ 2 ) = 1
21	19, 20	eqtrdi 2245	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = 1 )
22	12	zcnd 9452	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. CC )
23	22	sqvald 10765	. . 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 2237	. 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 2239	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 980   = wceq 1364   e. wcel 2167   =/= wne 2367   ` cfv 5259  (class class class)co 5923   CCcc 7880   RRcr 7881   0cc0 7882   1c1 7883   x. cmul 7887   2c2 9044   ZZcz 9329   ^cexp 10633   abscabs 11165   gcd cgcd 12131   /Lclgs 15264

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7973  ax-resscn 7974  ax-1cn 7975  ax-1re 7976  ax-icn 7977  ax-addcl 7978  ax-addrcl 7979  ax-mulcl 7980  ax-mulrcl 7981  ax-addcom 7982  ax-mulcom 7983  ax-addass 7984  ax-mulass 7985  ax-distr 7986  ax-i2m1 7987  ax-0lt1 7988  ax-1rid 7989  ax-0id 7990  ax-rnegex 7991  ax-precex 7992  ax-cnre 7993  ax-pre-ltirr 7994  ax-pre-ltwlin 7995  ax-pre-lttrn 7996  ax-pre-apti 7997  ax-pre-ltadd 7998  ax-pre-mulgt0 7999  ax-pre-mulext 8000  ax-arch 8001  ax-caucvg 8002

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6200  df-2nd 6201  df-recs 6365  df-irdg 6430  df-frec 6451  df-1o 6476  df-2o 6477  df-oadd 6480  df-er 6594  df-en 6802  df-dom 6803  df-fin 6804  df-sup 7052  df-inf 7053  df-pnf 8066  df-mnf 8067  df-xr 8068  df-ltxr 8069  df-le 8070  df-sub 8202  df-neg 8203  df-reap 8605  df-ap 8612  df-div 8703  df-inn 8994  df-2 9052  df-3 9053  df-4 9054  df-5 9055  df-6 9056  df-7 9057  df-8 9058  df-9 9059  df-n0 9253  df-z 9330  df-uz 9605  df-q 9697  df-rp 9732  df-fz 10087  df-fzo 10221  df-fl 10363  df-mod 10418  df-seqfrec 10543  df-exp 10634  df-ihash 10871  df-cj 11010  df-re 11011  df-im 11012  df-rsqrt 11166  df-abs 11167  df-clim 11447  df-proddc 11719  df-dvds 11956  df-gcd 12132  df-prm 12287  df-phi 12390  df-pc 12465  df-lgs 15265

This theorem is referenced by:  1lgs  15310  lgsmulsqcoprm  15313  lgsquad2lem2  15349