Theorem lgsquad2 15973

Description: Extend lgsquad 15970 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 3343	. . . . . 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 12811	. . . . 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 3824	. . . . . . . 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 2500	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> 2 =/= m )
16	15	neneqd 2435	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> -. 2 = m )
17		2z 9607	. . . . . . 7 |- 2 e. ZZ
18		uzid 9871	. . . . . . 7 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
19	17, 18	ax-mp 5	. . . . . 6 |- 2 e. ( ZZ>= ` 2 )
20		dvdsprm 12838	. . . . . 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 9702	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> N e. ZZ )
24	12	nnzd 9702	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. ZZ )
25	23, 24	gcdcomd 12674	. . . . 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 2267	. . . 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 3343	. . . . . . . 8 |- ( n e. ( Prime \ { 2 } ) -> n e. Prime )
31		prmrp 12846	. . . . . . . 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 15970	. . . . 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 15972	. . 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 15904	. . . . 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 15904	. . . . 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 9584	. . . . 5 |- ( ( m /L N ) e. ZZ -> ( m /L N ) e. CC )
44		zcn 9584	. . . . 5 |- ( ( N /L m ) e. ZZ -> ( N /L m ) e. CC )
45		mulcom 8258	. . . . 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 9253	. . . . . . 7 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. CC )
49		ax-1cn 8222	. . . . . . 7 |- 1 e. CC
50		subcl 8474	. . . . . . 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 9485	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( m - 1 ) / 2 ) e. CC )
53	6	nncnd 9253	. . . . . . 7 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> N e. CC )
54		subcl 8474	. . . . . . 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 9485	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( N - 1 ) / 2 ) e. CC )
57	52, 56	mulcomd 8297	. . . 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 6068	. . 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 2277	. 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 15972	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 2205   =/= wne 2414   A.wral 2522   \ cdif 3210   {csn 3691   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   CCcc 8127   1c1 8130   x. cmul 8134   - cmin 8446   -ucneg 8447   / cdiv 8948   NNcn 9239   2c2 9290   ZZcz 9579   ZZ>=cuz 9856   ...cfz 10345   ^cexp 10904   || cdvds 12477   gcd cgcd 12653   Primecprime 12808   /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  ax-addf 8251  ax-mulf 8252

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-tp 3699  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-disj 4088  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-of 6268  df-1st 6336  df-2nd 6337  df-tpos 6478  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-2o 6650  df-oadd 6653  df-er 6769  df-ec 6771  df-qs 6775  df-map 6886  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-dec 9713  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-sumdc 12043  df-proddc 12241  df-dvds 12478  df-gcd 12654  df-prm 12809  df-phi 12912  df-pc 12987  df-struct 13231  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-plusg 13320  df-mulr 13321  df-starv 13322  df-sca 13323  df-vsca 13324  df-ip 13325  df-tset 13326  df-ple 13327  df-ds 13329  df-unif 13330  df-0g 13488  df-igsum 13489  df-topgen 13490  df-iimas 13532  df-qus 13533  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-mhm 13689  df-submnd 13690  df-grp 13733  df-minusg 13734  df-sbg 13735  df-mulg 13854  df-subg 13904  df-nsg 13905  df-eqg 13906  df-ghm 13975  df-cmn 14020  df-abl 14021  df-mgp 14082  df-rng 14094  df-ur 14121  df-srg 14125  df-ring 14159  df-cring 14160  df-oppr 14229  df-dvdsr 14250  df-unit 14251  df-invr 14283  df-dvr 14294  df-rhm 14314  df-nzr 14342  df-subrg 14381  df-domn 14421  df-idom 14422  df-lmod 14454  df-lssm 14518  df-lsp 14552  df-sra 14600  df-rgmod 14601  df-lidl 14634  df-rsp 14635  df-2idl 14665  df-bl 14711  df-mopn 14712  df-fg 14714  df-metu 14715  df-cnfld 14722  df-zring 14756  df-zrh 14779  df-zn 14781  df-lgs 15888

This theorem is referenced by:  lgsquad3  15974