Theorem lgsmulsqcoprm 14114

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 10576	. . . . 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 1184	. . 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 9247	. . . . . . 7 |- ( A e. ZZ -> A e. CC )
7		sqne0 10571	. . . . . . 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 1017	. . 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 14103	. . 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 982	. . . . . 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 1016	. . . 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 14108	. . . 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 5884	. 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 1015	. . . . 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 14082	. . . . 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 9365	. . 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 7966	. 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 2214	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 978   = wceq 1353   e. wcel 2148   =/= wne 2347  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803   x. cmul 7807   2c2 8959   ZZcz 9242   ^cexp 10505   gcd cgcd 11926   /Lclgs 14065

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-2o 6412  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-9 8974  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543  df-dvds 11779  df-gcd 11927  df-prm 12091  df-phi 12194  df-pc 12268  df-lgs 14066

This theorem is referenced by: (None)