Theorem lgsquad2 16214

Description: Extend lgsquad 16211 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 535	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. ( Prime \ { 2 } ) )
9		eldifi 3351	. . . . . 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 12890	. . . . 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 3843	. . . . . . . 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 2506	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> 2 =/= m )
16	15	neneqd 2441	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> -. 2 = m )
17		2z 9674	. . . . . . 7 |- 2 e. ZZ
18		uzid 9938	. . . . . . 7 |- ( 2 e. ZZ -> 2 e. ( ZZ>= ` 2 ) )
19	17, 18	ax-mp 5	. . . . . 6 |- 2 e. ( ZZ>= ` 2 )
20		dvdsprm 12917	. . . . . 6 |- ( ( 2 e. ( ZZ>= ` 2 ) /\ m e. Prime ) -> ( 2 || m <-> 2 = m ) )
21	19, 10, 20	sylancr 418	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( 2 || m <-> 2 = m ) )
22	16, 21	mtbird 684	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> -. 2 || m )
23	6	nnzd 9769	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> N e. ZZ )
24	12	nnzd 9769	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. ZZ )
25	23, 24	gcdcomd 12753	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( N gcd m ) = ( m gcd N ) )
26		simprr 537	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( m gcd N ) = 1 )
27	25, 26	eqtrd 2271	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( N gcd m ) = 1 )
28		simprl 535	. . . . 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 3351	. . . . . . . 8 |- ( n e. ( Prime \ { 2 } ) -> n e. Prime )
31		prmrp 12925	. . . . . . . 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 16211	. . . . 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 1278	. . . 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 16213	. . 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 16145	. . . . 5 |- ( ( m e. ZZ /\ N e. ZZ ) -> ( m /L N ) e. ZZ )
40	24, 23, 39	syl2anc 415	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( m /L N ) e. ZZ )
41		lgscl 16145	. . . . 5 |- ( ( N e. ZZ /\ m e. ZZ ) -> ( N /L m ) e. ZZ )
42	23, 24, 41	syl2anc 415	. . . 4 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( N /L m ) e. ZZ )
43		zcn 9651	. . . . 5 |- ( ( m /L N ) e. ZZ -> ( m /L N ) e. CC )
44		zcn 9651	. . . . 5 |- ( ( N /L m ) e. ZZ -> ( N /L m ) e. CC )
45		mulcom 8308	. . . . 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 415	. . 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 9319	. . . . . . 7 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> m e. CC )
49		ax-1cn 8272	. . . . . . 7 |- 1 e. CC
50		subcl 8525	. . . . . . 7 |- ( ( m e. CC /\ 1 e. CC ) -> ( m - 1 ) e. CC )
51	48, 49, 50	sylancl 417	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( m - 1 ) e. CC )
52	51	halfcld 9552	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( m - 1 ) / 2 ) e. CC )
53	6	nncnd 9319	. . . . . . 7 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> N e. CC )
54		subcl 8525	. . . . . . 7 |- ( ( N e. CC /\ 1 e. CC ) -> ( N - 1 ) e. CC )
55	53, 49, 54	sylancl 417	. . . . . 6 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( N - 1 ) e. CC )
56	55	halfcld 9552	. . . . 5 |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( N - 1 ) / 2 ) e. CC )
57	52, 56	mulcomd 8347	. . . 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 6101	. . 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 2281	. 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 16213	1 |- ( ph -> ( ( M /L N ) x. ( N /L M ) ) = ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   \ cdif 3217   {csn 3709   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   1c1 8180   x. cmul 8184   - cmin 8497   -ucneg 8498   / cdiv 9003   NNcn 9305   2c2 9356   ZZcz 9646   ZZ>=cuz 9923   ...cfz 10413   ^cexp 10977   || cdvds 12556   gcd cgcd 12732   Primecprime 12887   /Lclgs 16128

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-addf 8301  ax-mulf 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-tpos 6516  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-ec 6809  df-qs 6813  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-z 9647  df-dec 9780  df-uz 9924  df-q 10022  df-rp 10057  df-fz 10414  df-fzo 10552  df-fl 10707  df-mod 10762  df-seqfrec 10887  df-exp 10978  df-ihash 11217  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-clim 12047  df-sumdc 12122  df-proddc 12320  df-dvds 12557  df-gcd 12733  df-prm 12888  df-phi 12991  df-pc 13066  df-struct 13356  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-mulr 13447  df-starv 13448  df-sca 13449  df-vsca 13450  df-ip 13451  df-tset 13452  df-ple 13453  df-ds 13455  df-unif 13456  df-0g 13614  df-gzsum 13615  df-topgen 13616  df-iimas 13626  df-qus 13627  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-mhm 13768  df-submnd 13769  df-grp 13810  df-minusg 13811  df-sbg 13812  df-mulg 13925  df-subg 13975  df-nsg 13976  df-eqg 13977  df-ghm 14046  df-cmn 14091  df-abl 14092  df-gsumfi 14153  df-mgp 14220  df-rng 14234  df-ur 14265  df-srg 14270  df-ring 14304  df-cring 14305  df-oppr 14375  df-dvdsr 14397  df-unit 14398  df-invr 14430  df-dvr 14441  df-rhm 14461  df-nzr 14489  df-subrg 14529  df-domn 14569  df-idom 14570  df-lmod 14627  df-lssm 14692  df-lsp 14726  df-sra 14774  df-rgmod 14775  df-lidl 14808  df-rsp 14809  df-2idl 14839  df-bl 14885  df-mopn 14886  df-fg 14888  df-metu 14889  df-cnfld 14896  df-zring 14928  df-zrh 14951  df-zn 14953  df-lgs 16129

This theorem is used by:  lgsquad3  16215