Theorem lgsmulsqcoprm 15741

Description: The Legendre (Jacobi) symbol is preserved under multiplication with a square of an integer coprime to the second argument. Theorem 9.9(d) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.)

Assertion

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

Proof of Theorem lgsmulsqcoprm

Step	Hyp	Ref	Expression
1		zsqcl 10844	. . . . 5 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
2	1	adantr 276	. . . 4 |- ( ( A e. ZZ /\ A =/= 0 ) -> ( A ^ 2 ) e. ZZ )
3		simpl 109	. . . 4 |- ( ( B e. ZZ /\ B =/= 0 ) -> B e. ZZ )
4		simpl 109	. . . 4 |- ( ( N e. ZZ /\ ( A gcd N ) = 1 ) -> N e. ZZ )
5	2, 3, 4	3anim123i 1208	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( A ^ 2 ) e. ZZ /\ B e. ZZ /\ N e. ZZ ) )
6		zcn 9462	. . . . . . 7 |- ( A e. ZZ -> A e. CC )
7		sqne0 10839	. . . . . . 7 |- ( A e. CC -> ( ( A ^ 2 ) =/= 0 <-> A =/= 0 ) )
8	6, 7	syl 14	. . . . . 6 |- ( A e. ZZ -> ( ( A ^ 2 ) =/= 0 <-> A =/= 0 ) )
9	8	biimpar 297	. . . . 5 |- ( ( A e. ZZ /\ A =/= 0 ) -> ( A ^ 2 ) =/= 0 )
10		simpr 110	. . . . 5 |- ( ( B e. ZZ /\ B =/= 0 ) -> B =/= 0 )
11	9, 10	anim12i 338	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( ( A ^ 2 ) =/= 0 /\ B =/= 0 ) )
12	11	3adant3 1041	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( A ^ 2 ) =/= 0 /\ B =/= 0 ) )
13		lgsdir 15730	. . 3 |- ( ( ( ( A ^ 2 ) e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( ( A ^ 2 ) =/= 0 /\ B =/= 0 ) ) -> ( ( ( A ^ 2 ) x. B ) /L N ) = ( ( ( A ^ 2 ) /L N ) x. ( B /L N ) ) )
14	5, 12, 13	syl2anc 411	. 2 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( ( A ^ 2 ) x. B ) /L N ) = ( ( ( A ^ 2 ) /L N ) x. ( B /L N ) ) )
15		3anass 1006	. . . . . 6 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) <-> ( ( A e. ZZ /\ A =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) )
16	15	biimpri 133	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) )
17	16	3adant2 1040	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) )
18		lgssq 15735	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ 2 ) /L N ) = 1 )
19	17, 18	syl 14	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( A ^ 2 ) /L N ) = 1 )
20	19	oveq1d 6022	. 2 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( ( A ^ 2 ) /L N ) x. ( B /L N ) ) = ( 1 x. ( B /L N ) ) )
21	3, 4	anim12i 338	. . . . . 6 |- ( ( ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( B e. ZZ /\ N e. ZZ ) )
22	21	3adant1 1039	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( B e. ZZ /\ N e. ZZ ) )
23		lgscl 15709	. . . . 5 |- ( ( B e. ZZ /\ N e. ZZ ) -> ( B /L N ) e. ZZ )
24	22, 23	syl 14	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( B /L N ) e. ZZ )
25	24	zcnd 9581	. . 3 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( B /L N ) e. CC )
26	25	mulid2d 8176	. 2 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( 1 x. ( B /L N ) ) = ( B /L N ) )
27	14, 20, 26	3eqtrd 2266	1 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) /\ ( N e. ZZ /\ ( A gcd N ) = 1 ) ) -> ( ( ( A ^ 2 ) x. B ) /L N ) = ( B /L N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400  (class class class)co 6007   CCcc 8008   0cc0 8010   1c1 8011   x. cmul 8015   2c2 9172   ZZcz 9457   ^cexp 10772   gcd cgcd 12490   /Lclgs 15692

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 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

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 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-2o 6569  df-oadd 6572  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-ihash 11010  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-clim 11806  df-proddc 12078  df-dvds 12315  df-gcd 12491  df-prm 12646  df-phi 12749  df-pc 12824  df-lgs 15693

This theorem is referenced by: (None)