Theorem lgsquad2 15805

Description: Extend lgsquad 15802 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 529	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. ( Prime \ { 2 } ) )
9		eldifi 3327	. . . . . 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 12675	. . . . 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 3800	. . . . . . . 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 2486	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> 2 =/= m )
16	15	neneqd 2421	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> -. 2 = m )
17		2z 9500	. . . . . . 7 |- 2 e. ZZ
18		uzid 9763	. . . . . . 7 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
19	17, 18	ax-mp 5	. . . . . 6 |- 2 e. ( ZZ>= ` 2 )
20		dvdsprm 12702	. . . . . 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 677	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> -. 2 || m )
23	6	nnzd 9594	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> N e. ZZ )
24	12	nnzd 9594	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. ZZ )
25	23, 24	gcdcomd 12538	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( N gcd m ) = ( m gcd N ) )
26		simprr 531	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( m gcd N ) = 1 )
27	25, 26	eqtrd 2262	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( N gcd m ) = 1 )
28		simprl 529	. . . . 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 3327	. . . . . . . 8 |- ( n e. ( Prime \ { 2 } ) -> n e. Prime )
31		prmrp 12710	. . . . . . . 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 15802	. . . . 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 1271	. . . 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 15804	. . 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 15736	. . . . 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 15736	. . . . 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 9477	. . . . 5 |- ( ( m /L N ) e. ZZ -> ( m /L N ) e. CC )
44		zcn 9477	. . . . 5 |- ( ( N /L m ) e. ZZ -> ( N /L m ) e. CC )
45		mulcom 8154	. . . . 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 9150	. . . . . . 7 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. CC )
49		ax-1cn 8118	. . . . . . 7 |- 1 e. CC
50		subcl 8371	. . . . . . 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 9382	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( m - 1 ) / 2 ) e. CC )
53	6	nncnd 9150	. . . . . . 7 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> N e. CC )
54		subcl 8371	. . . . . . 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 9382	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( N - 1 ) / 2 ) e. CC )
57	52, 56	mulcomd 8194	. . . 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 6029	. . 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 2272	. 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 15804	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 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   \ cdif 3195   {csn 3667   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8023   1c1 8026   x. cmul 8030   - cmin 8343   -ucneg 8344   / cdiv 8845   NNcn 9136   2c2 9187   ZZcz 9472   ZZ>=cuz 9748   ...cfz 10236   ^cexp 10793   || cdvds 12341   gcd cgcd 12517   Primecprime 12672   /Lclgs 15719

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145  ax-addf 8147  ax-mulf 8148

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-tp 3675  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-disj 4063  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230  df-1st 6298  df-2nd 6299  df-tpos 6406  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-2o 6578  df-oadd 6581  df-er 6697  df-ec 6699  df-qs 6703  df-map 6814  df-en 6905  df-dom 6906  df-fin 6907  df-sup 7177  df-inf 7178  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-z 9473  df-dec 9605  df-uz 9749  df-q 9847  df-rp 9882  df-fz 10237  df-fzo 10371  df-fl 10523  df-mod 10578  df-seqfrec 10703  df-exp 10794  df-ihash 11031  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-clim 11833  df-sumdc 11908  df-proddc 12105  df-dvds 12342  df-gcd 12518  df-prm 12673  df-phi 12776  df-pc 12851  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13081  df-sets 13082  df-iress 13083  df-plusg 13166  df-mulr 13167  df-starv 13168  df-sca 13169  df-vsca 13170  df-ip 13171  df-tset 13172  df-ple 13173  df-ds 13175  df-unif 13176  df-0g 13334  df-igsum 13335  df-topgen 13336  df-iimas 13378  df-qus 13379  df-mgm 13432  df-sgrp 13478  df-mnd 13493  df-mhm 13535  df-submnd 13536  df-grp 13579  df-minusg 13580  df-sbg 13581  df-mulg 13700  df-subg 13750  df-nsg 13751  df-eqg 13752  df-ghm 13821  df-cmn 13866  df-abl 13867  df-mgp 13927  df-rng 13939  df-ur 13966  df-srg 13970  df-ring 14004  df-cring 14005  df-oppr 14074  df-dvdsr 14095  df-unit 14096  df-invr 14128  df-dvr 14139  df-rhm 14159  df-nzr 14187  df-subrg 14226  df-domn 14266  df-idom 14267  df-lmod 14296  df-lssm 14360  df-lsp 14394  df-sra 14442  df-rgmod 14443  df-lidl 14476  df-rsp 14477  df-2idl 14507  df-bl 14553  df-mopn 14554  df-fg 14556  df-metu 14557  df-cnfld 14564  df-zring 14598  df-zrh 14621  df-zn 14623  df-lgs 15720

This theorem is referenced by:  lgsquad3  15806