Theorem lgssq 15842

Description: The Legendre symbol at a square is equal to 1. Together with lgsmod 15828 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 1048	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A e. ZZ )
2		simp2 1025	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> N e. ZZ )
3		simp1r 1049	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A =/= 0 )
4		lgsdir 15837	. . 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 1282	. 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 9528	. . . . . 6 |- ( A e. ZZ -> A e. CC )
7	6	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ A =/= 0 ) -> A e. CC )
8	7	3ad2ant1 1045	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A e. CC )
9	8	sqvald 10978	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A ^ 2 ) = ( A x. A ) )
10	9	oveq1d 6043	. 2 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ 2 ) /L N ) = ( ( A x. A ) /L N ) )
11		lgscl 15816	. . . . . 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 9646	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. RR )
14		absresq 11701	. . . 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 15841	. . . . . . 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 1383	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( abs ` ( A /L N ) ) = 1 )
19	18	oveq1d 6043	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( 1 ^ 2 ) )
20		sq1 10941	. . . 4 |- ( 1 ^ 2 ) = 1
21	19, 20	eqtrdi 2280	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = 1 )
22	12	zcnd 9647	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. CC )
23	22	sqvald 10978	. . 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 2272	. 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 2274	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 1005   = wceq 1398   e. wcel 2202   =/= wne 2403   ` cfv 5333  (class class class)co 6028   CCcc 8073   RRcr 8074   0cc0 8075   1c1 8076   x. cmul 8080   2c2 9236   ZZcz 9523   ^cexp 10846   abscabs 11620   gcd cgcd 12587   /Lclgs 15799

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-2o 6626  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-sup 7226  df-inf 7227  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-fl 10576  df-mod 10631  df-seqfrec 10756  df-exp 10847  df-ihash 11084  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902  df-proddc 12175  df-dvds 12412  df-gcd 12588  df-prm 12743  df-phi 12846  df-pc 12921  df-lgs 15800

This theorem is referenced by:  1lgs  15845  lgsmulsqcoprm  15848  lgsquad2lem2  15884