Theorem lgsquad2 15943

Description: Extend lgsquad 15940 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)

Hypotheses

Ref	Expression
lgsquad2.1	|- ( ph -> M e. NN )
lgsquad2.2	|- ( ph -> -. 2 || M )
lgsquad2.3	|- ( ph -> N e. NN )
lgsquad2.4	|- ( ph -> -. 2 || N )
lgsquad2.5	|- ( ph -> ( M gcd N ) = 1 )

Assertion

Ref	Expression
lgsquad2	|- ( ph -> ( ( M /L N ) x. ( N /L M ) ) = ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) )

Proof of Theorem lgsquad2

Dummy variables m n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lgsquad2.1	. 2 |- ( ph -> M e. NN )
2		lgsquad2.2	. 2 |- ( ph -> -. 2 || M )
3		lgsquad2.3	. 2 |- ( ph -> N e. NN )
4		lgsquad2.4	. 2 |- ( ph -> -. 2 || N )
5		lgsquad2.5	. 2 |- ( ph -> ( M gcd N ) = 1 )
6	3	adantr 276	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> N e. NN )
7	4	adantr 276	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> -. 2 || N )
8		simprl 531	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. ( Prime \ { 2 } ) )
9		eldifi 3340	. . . . . 6 |- ( m e. ( Prime \ { 2 } ) -> m e. Prime )
10	8, 9	syl 14	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. Prime )
11		prmnn 12800	. . . . 5 |- ( m e. Prime -> m e. NN )
12	10, 11	syl 14	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. NN )
13		eldifsni 3821	. . . . . . . 8 |- ( m e. ( Prime \ { 2 } ) -> m =/= 2 )
14	8, 13	syl 14	. . . . . . 7 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m =/= 2 )
15	14	necomd 2498	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> 2 =/= m )
16	15	neneqd 2433	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> -. 2 = m )
17		2z 9601	. . . . . . 7 |- 2 e. ZZ
18		uzid 9864	. . . . . . 7 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
19	17, 18	ax-mp 5	. . . . . 6 |- 2 e. ( ZZ>= ` 2 )
20		dvdsprm 12827	. . . . . 6 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ m e. Prime ) -> ( 2 || m <-> 2 = m ) )
21	19, 10, 20	sylancr 414	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( 2 || m <-> 2 = m ) )
22	16, 21	mtbird 680	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> -. 2 || m )
23	6	nnzd 9695	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> N e. ZZ )
24	12	nnzd 9695	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. ZZ )
25	23, 24	gcdcomd 12663	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( N gcd m ) = ( m gcd N ) )
26		simprr 533	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( m gcd N ) = 1 )
27	25, 26	eqtrd 2265	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( N gcd m ) = 1 )
28		simprl 531	. . . . 5 |- ( ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) /\ ( n e. ( Prime \ { 2 } ) /\ ( n gcd m ) = 1 ) ) -> n e. ( Prime \ { 2 } ) )
29	8	adantr 276	. . . . 5 |- ( ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) /\ ( n e. ( Prime \ { 2 } ) /\ ( n gcd m ) = 1 ) ) -> m e. ( Prime \ { 2 } ) )
30		eldifi 3340	. . . . . . . 8 |- ( n e. ( Prime \ { 2 } ) -> n e. Prime )
31		prmrp 12835	. . . . . . . 8 |- ( ( n e. Prime /\ m e. Prime ) -> ( ( n gcd m ) = 1 <-> n =/= m ) )
32	30, 10, 31	syl2anr 290	. . . . . . 7 |- ( ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) /\ n e. ( Prime \ { 2 } ) ) -> ( ( n gcd m ) = 1 <-> n =/= m ) )
33	32	biimpd 144	. . . . . 6 |- ( ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) /\ n e. ( Prime \ { 2 } ) ) -> ( ( n gcd m ) = 1 -> n =/= m ) )
34	33	impr 379	. . . . 5 |- ( ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) /\ ( n e. ( Prime \ { 2 } ) /\ ( n gcd m ) = 1 ) ) -> n =/= m )
35		lgsquad 15940	. . . . 5 |- ( ( n e. ( Prime \ { 2 } ) /\ m e. ( Prime \ { 2 } ) /\ n =/= m ) -> ( ( n /L m ) x. ( m /L n ) ) = ( -u 1 ^ ( ( ( n - 1 ) / 2 ) x. ( ( m - 1 ) / 2 ) ) ) )
36	28, 29, 34, 35	syl3anc 1274	. . . 4 |- ( ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) /\ ( n e. ( Prime \ { 2 } ) /\ ( n gcd m ) = 1 ) ) -> ( ( n /L m ) x. ( m /L n ) ) = ( -u 1 ^ ( ( ( n - 1 ) / 2 ) x. ( ( m - 1 ) / 2 ) ) ) )
37		biid 171	. . . 4 |- ( A. x e. ( 1 ... y ) ( ( x gcd ( 2 x. m ) ) = 1 -> ( ( x /L m ) x. ( m /L x ) ) = ( -u 1 ^ ( ( ( x - 1 ) / 2 ) x. ( ( m - 1 ) / 2 ) ) ) ) <-> A. x e. ( 1 ... y ) ( ( x gcd ( 2 x. m ) ) = 1 -> ( ( x /L m ) x. ( m /L x ) ) = ( -u 1 ^ ( ( ( x - 1 ) / 2 ) x. ( ( m - 1 ) / 2 ) ) ) ) )
38	6, 7, 12, 22, 27, 36, 37	lgsquad2lem2 15942	. . 3 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( N /L m ) x. ( m /L N ) ) = ( -u 1 ^ ( ( ( N - 1 ) / 2 ) x. ( ( m - 1 ) / 2 ) ) ) )
39		lgscl 15874	. . . . 5 |- ( ( m e. ZZ /\ N e. ZZ ) -> ( m /L N ) e. ZZ )
40	24, 23, 39	syl2anc 411	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( m /L N ) e. ZZ )
41		lgscl 15874	. . . . 5 |- ( ( N e. ZZ /\ m e. ZZ ) -> ( N /L m ) e. ZZ )
42	23, 24, 41	syl2anc 411	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( N /L m ) e. ZZ )
43		zcn 9578	. . . . 5 |- ( ( m /L N ) e. ZZ -> ( m /L N ) e. CC )
44		zcn 9578	. . . . 5 |- ( ( N /L m ) e. ZZ -> ( N /L m ) e. CC )
45		mulcom 8252	. . . . 5 |- ( ( ( m /L N ) e. CC /\ ( N /L m ) e. CC ) -> ( ( m /L N ) x. ( N /L m ) ) = ( ( N /L m ) x. ( m /L N ) ) )
46	43, 44, 45	syl2an 289	. . . 4 |- ( ( ( m /L N ) e. ZZ /\ ( N /L m ) e. ZZ ) -> ( ( m /L N ) x. ( N /L m ) ) = ( ( N /L m ) x. ( m /L N ) ) )
47	40, 42, 46	syl2anc 411	. . 3 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( m /L N ) x. ( N /L m ) ) = ( ( N /L m ) x. ( m /L N ) ) )
48	12	nncnd 9247	. . . . . . 7 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. CC )
49		ax-1cn 8216	. . . . . . 7 |- 1 e. CC
50		subcl 8468	. . . . . . 7 |- ( ( m e. CC /\ 1 e. CC ) -> ( m - 1 ) e. CC )
51	48, 49, 50	sylancl 413	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( m - 1 ) e. CC )
52	51	halfcld 9479	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( m - 1 ) / 2 ) e. CC )
53	6	nncnd 9247	. . . . . . 7 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> N e. CC )
54		subcl 8468	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - 1 ) e. CC )
55	53, 49, 54	sylancl 413	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( N - 1 ) e. CC )
56	55	halfcld 9479	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( N - 1 ) / 2 ) e. CC )
57	52, 56	mulcomd 8291	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( ( m - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) = ( ( ( N - 1 ) / 2 ) x. ( ( m - 1 ) / 2 ) ) )
58	57	oveq2d 6065	. . 3 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( -u 1 ^ ( ( ( m - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) = ( -u 1 ^ ( ( ( N - 1 ) / 2 ) x. ( ( m - 1 ) / 2 ) ) ) )
59	38, 47, 58	3eqtr4d 2275	. 2 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( m /L N ) x. ( N /L m ) ) = ( -u 1 ^ ( ( ( m - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) )
60		biid 171	. 2 |- ( A. x e. ( 1 ... y ) ( ( x gcd ( 2 x. N ) ) = 1 -> ( ( x /L N ) x. ( N /L x ) ) = ( -u 1 ^ ( ( ( x - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) ) <-> A. x e. ( 1 ... y ) ( ( x gcd ( 2 x. N ) ) = 1 -> ( ( x /L N ) x. ( N /L x ) ) = ( -u 1 ^ ( ( ( x - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) ) )
61	1, 2, 3, 4, 5, 59, 60	lgsquad2lem2 15942	1 |- ( ph -> ( ( M /L N ) x. ( N /L M ) ) = ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   =/= wne 2412   A.wral 2520   \ cdif 3207   {csn 3688   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   CCcc 8121   1c1 8124   x. cmul 8128   - cmin 8440   -ucneg 8441   / cdiv 8942   NNcn 9233   2c2 9284   ZZcz 9573   ZZ>=cuz 9849   ...cfz 10338   ^cexp 10896   || cdvds 12466   gcd cgcd 12642   Primecprime 12797   /Lclgs 15857

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242  ax-caucvg 8243  ax-addf 8245  ax-mulf 8246

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-tp 3696  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-disj 4085  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-of 6265  df-1st 6333  df-2nd 6334  df-tpos 6475  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-ec 6768  df-qs 6772  df-map 6883  df-en 6975  df-dom 6976  df-fin 6977  df-sup 7274  df-inf 7275  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-z 9574  df-dec 9706  df-uz 9850  df-q 9948  df-rp 9983  df-fz 10339  df-fzo 10473  df-fl 10626  df-mod 10681  df-seqfrec 10806  df-exp 10897  df-ihash 11134  df-cj 11520  df-re 11521  df-im 11522  df-rsqrt 11676  df-abs 11677  df-clim 11957  df-sumdc 12032  df-proddc 12230  df-dvds 12467  df-gcd 12643  df-prm 12798  df-phi 12901  df-pc 12976  df-struct 13203  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-iress 13209  df-plusg 13292  df-mulr 13293  df-starv 13294  df-sca 13295  df-vsca 13296  df-ip 13297  df-tset 13298  df-ple 13299  df-ds 13301  df-unif 13302  df-0g 13460  df-igsum 13461  df-topgen 13462  df-iimas 13504  df-qus 13505  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-mhm 13661  df-submnd 13662  df-grp 13705  df-minusg 13706  df-sbg 13707  df-mulg 13826  df-subg 13876  df-nsg 13877  df-eqg 13878  df-ghm 13947  df-cmn 13992  df-abl 13993  df-mgp 14054  df-rng 14066  df-ur 14093  df-srg 14097  df-ring 14131  df-cring 14132  df-oppr 14201  df-dvdsr 14222  df-unit 14223  df-invr 14255  df-dvr 14266  df-rhm 14286  df-nzr 14314  df-subrg 14353  df-domn 14393  df-idom 14394  df-lmod 14424  df-lssm 14488  df-lsp 14522  df-sra 14570  df-rgmod 14571  df-lidl 14604  df-rsp 14605  df-2idl 14635  df-bl 14681  df-mopn 14682  df-fg 14684  df-metu 14685  df-cnfld 14692  df-zring 14726  df-zrh 14749  df-zn 14751  df-lgs 15858

This theorem is referenced by:  lgsquad3  15944