Theorem lgsdinn0 25070

Description: Variation on lgsdi 25059 valid for all M , N but only for positive A. (The exact location of the failure of this law is for A = -u 1, M = 0, and some N in which case ( -u 1 /L 0 ) = 1 but ( -u 1 /L N ) = -u 1 when -u 1 is not a quadratic residue mod N.) (Contributed by Mario Carneiro, 28-Apr-2016.)

Assertion

Ref	Expression
lgsdinn0	|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Proof of Theorem lgsdinn0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1063	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
2		sq1 12958	. . . . . . . . . . . . . . . 16 |- ( 1 ^ 2 ) = 1
3	2	eqeq2i 2634	. . . . . . . . . . . . . . 15 |- ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> ( A ^ 2 ) = 1 )
4		nn0re 11301	. . . . . . . . . . . . . . . . 17 |- ( A e. NN0 -> A e. RR )
5		nn0ge0 11318	. . . . . . . . . . . . . . . . 17 |- ( A e. NN0 -> 0 <_ A )
6		1re 10039	. . . . . . . . . . . . . . . . . 18 |- 1 e. RR
7		0le1 10551	. . . . . . . . . . . . . . . . . 18 |- 0 <_ 1
8		sq11 12936	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( 1 e. RR /\ 0 <_ 1 ) ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
9	6, 7, 8	mpanr12 721	. . . . . . . . . . . . . . . . 17 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
10	4, 5, 9	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( A e. NN0 -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
11	10	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
12	3, 11	syl5bbr 274	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = 1 <-> A = 1 ) )
13	12	biimpa 501	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A = 1 )
14	13	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = ( 1 /L x ) )
15		1lgs 25065	. . . . . . . . . . . . 13 |- ( x e. ZZ -> ( 1 /L x ) = 1 )
16	15	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 /L x ) = 1 )
17	14, 16	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = 1 )
18	17	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( 1 x. ( A /L 0 ) ) )
19		nn0z 11400	. . . . . . . . . . . . . 14 |- ( A e. NN0 -> A e. ZZ )
20	19	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A e. ZZ )
21		0z 11388	. . . . . . . . . . . . 13 |- 0 e. ZZ
22		lgscl 25036	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A /L 0 ) e. ZZ )
23	20, 21, 22	sylancl 694	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. ZZ )
24	23	zcnd 11483	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. CC )
25	24	mulid2d 10058	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 x. ( A /L 0 ) ) = ( A /L 0 ) )
26	18, 25	eqtr2d 2657	. . . . . . . . 9 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
27		lgscl 25036	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ x e. ZZ ) -> ( A /L x ) e. ZZ )
28	19, 27	sylan 488	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. ZZ )
29	28	zcnd 11483	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. CC )
30	29	adantr 481	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L x ) e. CC )
31	30	mul01d 10235	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. 0 ) = 0 )
32	19	adantr 481	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ x e. ZZ ) -> A e. ZZ )
33		lgs0 25035	. . . . . . . . . . . . 13 |- ( A e. ZZ -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
34	32, 33	syl 17	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
35		ifnefalse 4098	. . . . . . . . . . . 12 |- ( ( A ^ 2 ) =/= 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 0 )
36	34, 35	sylan9eq 2676	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = 0 )
37	36	oveq2d 6666	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L x ) x. 0 ) )
38	31, 37, 36	3eqtr4rd 2667	. . . . . . . . 9 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
39	26, 38	pm2.61dane 2881	. . . . . . . 8 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
40	39	ralrimiva 2966	. . . . . . 7 |- ( A e. NN0 -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
41	40	3ad2ant1 1082	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
42		oveq2 6658	. . . . . . . . 9 |- ( x = N -> ( A /L x ) = ( A /L N ) )
43	42	oveq1d 6665	. . . . . . . 8 |- ( x = N -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L N ) x. ( A /L 0 ) ) )
44	43	eqeq2d 2632	. . . . . . 7 |- ( x = N -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) ) )
45	44	rspcv 3305	. . . . . 6 |- ( N e. ZZ -> ( A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) ) )
46	1, 41, 45	sylc 65	. . . . 5 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) )
47	46	adantr 481	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) )
48	19	3ad2ant1 1082	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A e. ZZ )
49	48, 21, 22	sylancl 694	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. ZZ )
50	49	zcnd 11483	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. CC )
51	50	adantr 481	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) e. CC )
52		lgscl 25036	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
53	48, 1, 52	syl2anc 693	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
54	53	zcnd 11483	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. CC )
55	54	adantr 481	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L N ) e. CC )
56	51, 55	mulcomd 10061	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L 0 ) x. ( A /L N ) ) = ( ( A /L N ) x. ( A /L 0 ) ) )
57	47, 56	eqtr4d 2659	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L 0 ) x. ( A /L N ) ) )
58		oveq1 6657	. . . . 5 |- ( M = 0 -> ( M x. N ) = ( 0 x. N ) )
59	1	zcnd 11483	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
60	59	mul02d 10234	. . . . 5 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
61	58, 60	sylan9eqr 2678	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M x. N ) = 0 )
62	61	oveq2d 6666	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) )
63		simpr 477	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> M = 0 )
64	63	oveq2d 6666	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L M ) = ( A /L 0 ) )
65	64	oveq1d 6665	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L 0 ) x. ( A /L N ) ) )
66	57, 62, 65	3eqtr4d 2666	. 2 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
67		simp2 1062	. . . . 5 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
68		oveq2 6658	. . . . . . . 8 |- ( x = M -> ( A /L x ) = ( A /L M ) )
69	68	oveq1d 6665	. . . . . . 7 |- ( x = M -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L M ) x. ( A /L 0 ) ) )
70	69	eqeq2d 2632	. . . . . 6 |- ( x = M -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) ) )
71	70	rspcv 3305	. . . . 5 |- ( M e. ZZ -> ( A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) ) )
72	67, 41, 71	sylc 65	. . . 4 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) )
73	72	adantr 481	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) )
74		oveq2 6658	. . . . 5 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
75	67	zcnd 11483	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
76	75	mul01d 10235	. . . . 5 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( M x. 0 ) = 0 )
77	74, 76	sylan9eqr 2678	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M x. N ) = 0 )
78	77	oveq2d 6666	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) )
79		simpr 477	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> N = 0 )
80	79	oveq2d 6666	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L N ) = ( A /L 0 ) )
81	80	oveq2d 6666	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L M ) x. ( A /L 0 ) ) )
82	73, 78, 81	3eqtr4d 2666	. 2 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
83		lgsdi 25059	. . 3 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
84	19, 83	syl3anl1 1374	. 2 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
85	66, 82, 84	pm2.61da2ne 2882	1 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   ifcif 4086   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   2c2 11070   NN0cn0 11292   ZZcz 11377   ^cexp 12860   /Lclgs 25019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-pc 15542  df-lgs 25020

This theorem is referenced by: (None)