Theorem lgsmulsqcoprm 15936

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 10976	. . . . 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 1211	. . 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 9584	. . . . . . 7 |- ( A e. ZZ -> A e. CC )
7		sqne0 10971	. . . . . . 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 1044	. . 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 15925	. . 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 1009	. . . . . 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 1043	. . . 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 15930	. . . 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 6067	. 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 1042	. . . . 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 15904	. . . . 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 9704	. . 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	mullidd 8294	. 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 2271	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 1005   = wceq 1398   e. wcel 2205   =/= wne 2414  (class class class)co 6052   CCcc 8127   0cc0 8129   1c1 8130   x. cmul 8134   2c2 9290   ZZcz 9579   ^cexp 10904   gcd cgcd 12653   /Lclgs 15887

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-2o 6650  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-sup 7277  df-inf 7278  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-fzo 10481  df-fl 10634  df-mod 10689  df-seqfrec 10814  df-exp 10905  df-ihash 11143  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-clim 11968  df-proddc 12241  df-dvds 12478  df-gcd 12654  df-prm 12809  df-phi 12912  df-pc 12987  df-lgs 15888

This theorem is referenced by: (None)