Theorem lgsquad2 15841

Description: Extend lgsquad 15838 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 3328	. . . . . 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 12705	. . . . 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 3803	. . . . . . . 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 2487	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> 2 =/= m )
16	15	neneqd 2422	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> -. 2 = m )
17		2z 9512	. . . . . . 7 |- 2 e. ZZ
18		uzid 9775	. . . . . . 7 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
19	17, 18	ax-mp 5	. . . . . 6 |- 2 e. ( ZZ>= ` 2 )
20		dvdsprm 12732	. . . . . 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 679	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> -. 2 || m )
23	6	nnzd 9606	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> N e. ZZ )
24	12	nnzd 9606	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. ZZ )
25	23, 24	gcdcomd 12568	. . . . 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 2263	. . . 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 3328	. . . . . . . 8 |- ( n e. ( Prime \ { 2 } ) -> n e. Prime )
31		prmrp 12740	. . . . . . . 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 15838	. . . . 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 1273	. . . 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 15840	. . 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 15772	. . . . 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 15772	. . . . 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 9489	. . . . 5 |- ( ( m /L N ) e. ZZ -> ( m /L N ) e. CC )
44		zcn 9489	. . . . 5 |- ( ( N /L m ) e. ZZ -> ( N /L m ) e. CC )
45		mulcom 8166	. . . . 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 9162	. . . . . . 7 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. CC )
49		ax-1cn 8130	. . . . . . 7 |- 1 e. CC
50		subcl 8383	. . . . . . 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 9394	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( m - 1 ) / 2 ) e. CC )
53	6	nncnd 9162	. . . . . . 7 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> N e. CC )
54		subcl 8383	. . . . . . 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 9394	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( N - 1 ) / 2 ) e. CC )
57	52, 56	mulcomd 8206	. . . 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 6039	. . 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 2273	. 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 15840	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 1397   e. wcel 2201   =/= wne 2401   A.wral 2509   \ cdif 3196   {csn 3670   class class class wbr 4089   ` cfv 5328  (class class class)co 6023   CCcc 8035   1c1 8038   x. cmul 8042   - cmin 8355   -ucneg 8356   / cdiv 8857   NNcn 9148   2c2 9199   ZZcz 9484   ZZ>=cuz 9760   ...cfz 10248   ^cexp 10806   || cdvds 12371   gcd cgcd 12547   Primecprime 12702   /Lclgs 15755

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155  ax-arch 8156  ax-caucvg 8157  ax-addf 8159  ax-mulf 8160

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-tp 3678  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-disj 4066  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-isom 5337  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-of 6240  df-1st 6308  df-2nd 6309  df-tpos 6416  df-recs 6476  df-irdg 6541  df-frec 6562  df-1o 6587  df-2o 6588  df-oadd 6591  df-er 6707  df-ec 6709  df-qs 6713  df-map 6824  df-en 6915  df-dom 6916  df-fin 6917  df-sup 7188  df-inf 7189  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-z 9485  df-dec 9617  df-uz 9761  df-q 9859  df-rp 9894  df-fz 10249  df-fzo 10383  df-fl 10536  df-mod 10591  df-seqfrec 10716  df-exp 10807  df-ihash 11044  df-cj 11425  df-re 11426  df-im 11427  df-rsqrt 11581  df-abs 11582  df-clim 11862  df-sumdc 11937  df-proddc 12135  df-dvds 12372  df-gcd 12548  df-prm 12703  df-phi 12806  df-pc 12881  df-struct 13107  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112  df-iress 13113  df-plusg 13196  df-mulr 13197  df-starv 13198  df-sca 13199  df-vsca 13200  df-ip 13201  df-tset 13202  df-ple 13203  df-ds 13205  df-unif 13206  df-0g 13364  df-igsum 13365  df-topgen 13366  df-iimas 13408  df-qus 13409  df-mgm 13462  df-sgrp 13508  df-mnd 13523  df-mhm 13565  df-submnd 13566  df-grp 13609  df-minusg 13610  df-sbg 13611  df-mulg 13730  df-subg 13780  df-nsg 13781  df-eqg 13782  df-ghm 13851  df-cmn 13896  df-abl 13897  df-mgp 13958  df-rng 13970  df-ur 13997  df-srg 14001  df-ring 14035  df-cring 14036  df-oppr 14105  df-dvdsr 14126  df-unit 14127  df-invr 14159  df-dvr 14170  df-rhm 14190  df-nzr 14218  df-subrg 14257  df-domn 14297  df-idom 14298  df-lmod 14327  df-lssm 14391  df-lsp 14425  df-sra 14473  df-rgmod 14474  df-lidl 14507  df-rsp 14508  df-2idl 14538  df-bl 14584  df-mopn 14585  df-fg 14587  df-metu 14588  df-cnfld 14595  df-zring 14629  df-zrh 14652  df-zn 14654  df-lgs 15756

This theorem is referenced by:  lgsquad3  15842