Theorem lgsdirnn0 25069

Description: Variation on lgsdir 25057 valid for all A , B but only for positive N. (The exact location of the failure of this law is for A = 0, B < 0, N = -u 1 in which case ( 0 /L -u 1 ) = 1 but ( B /L -u 1 ) = -u 1.) (Contributed by Mario Carneiro, 28-Apr-2016.)

Assertion

Ref	Expression
lgsdirnn0	|- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )

Proof of Theorem lgsdirnn0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. ZZ )
2		id 22	. . . . . . . . . . . . . 14 |- ( x e. ZZ -> x e. ZZ )
3		nn0z 11400	. . . . . . . . . . . . . 14 |- ( N e. NN0 -> N e. ZZ )
4		lgscl 25036	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ N e. ZZ ) -> ( x /L N ) e. ZZ )
5	2, 3, 4	syl2anr 495	. . . . . . . . . . . . 13 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( x /L N ) e. ZZ )
6	5	zcnd 11483	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( x /L N ) e. CC )
7	6	adantr 481	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( x /L N ) e. CC )
8	7	mul01d 10235	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( ( x /L N ) x. 0 ) = 0 )
9		simpr 477	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( 0 /L N ) = 0 )
10	9	oveq2d 6666	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( x /L N ) x. 0 ) )
11	8, 10, 9	3eqtr4rd 2667	. . . . . . . . 9 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
12		0z 11388	. . . . . . . . . . . . . 14 |- 0 e. ZZ
13	3	adantr 481	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ x e. ZZ ) -> N e. ZZ )
14		lgsne0 25060	. . . . . . . . . . . . . 14 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( ( 0 /L N ) =/= 0 <-> ( 0 gcd N ) = 1 ) )
15	12, 13, 14	sylancr 695	. . . . . . . . . . . . 13 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 /L N ) =/= 0 <-> ( 0 gcd N ) = 1 ) )
16		gcdcom 15235	. . . . . . . . . . . . . . . . 17 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 gcd N ) = ( N gcd 0 ) )
17	12, 13, 16	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 gcd N ) = ( N gcd 0 ) )
18		nn0gcdid0 15242	. . . . . . . . . . . . . . . . 17 |- ( N e. NN0 -> ( N gcd 0 ) = N )
19	18	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( N gcd 0 ) = N )
20	17, 19	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 gcd N ) = N )
21	20	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 gcd N ) = 1 <-> N = 1 ) )
22		lgs1 25066	. . . . . . . . . . . . . . . 16 |- ( x e. ZZ -> ( x /L 1 ) = 1 )
23	22	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( x /L 1 ) = 1 )
24		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( N = 1 -> ( x /L N ) = ( x /L 1 ) )
25	24	eqeq1d 2624	. . . . . . . . . . . . . . 15 |- ( N = 1 -> ( ( x /L N ) = 1 <-> ( x /L 1 ) = 1 ) )
26	23, 25	syl5ibrcom 237	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( N = 1 -> ( x /L N ) = 1 ) )
27	21, 26	sylbid 230	. . . . . . . . . . . . 13 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 gcd N ) = 1 -> ( x /L N ) = 1 ) )
28	15, 27	sylbid 230	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 /L N ) =/= 0 -> ( x /L N ) = 1 ) )
29	28	imp 445	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( x /L N ) = 1 )
30	29	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( ( x /L N ) x. ( 0 /L N ) ) = ( 1 x. ( 0 /L N ) ) )
31	3	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> N e. ZZ )
32		lgscl 25036	. . . . . . . . . . . . 13 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 /L N ) e. ZZ )
33	12, 31, 32	sylancr 695	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 0 /L N ) e. ZZ )
34	33	zcnd 11483	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 0 /L N ) e. CC )
35	34	mulid2d 10058	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 1 x. ( 0 /L N ) ) = ( 0 /L N ) )
36	30, 35	eqtr2d 2657	. . . . . . . . 9 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
37	11, 36	pm2.61dane 2881	. . . . . . . 8 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
38	37	ralrimiva 2966	. . . . . . 7 |- ( N e. NN0 -> A. x e. ZZ ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
39	38	3ad2ant3 1084	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A. x e. ZZ ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
40		oveq1 6657	. . . . . . . . 9 |- ( x = B -> ( x /L N ) = ( B /L N ) )
41	40	oveq1d 6665	. . . . . . . 8 |- ( x = B -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( B /L N ) x. ( 0 /L N ) ) )
42	41	eqeq2d 2632	. . . . . . 7 |- ( x = B -> ( ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) <-> ( 0 /L N ) = ( ( B /L N ) x. ( 0 /L N ) ) ) )
43	42	rspcv 3305	. . . . . 6 |- ( B e. ZZ -> ( A. x e. ZZ ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) -> ( 0 /L N ) = ( ( B /L N ) x. ( 0 /L N ) ) ) )
44	1, 39, 43	sylc 65	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 /L N ) = ( ( B /L N ) x. ( 0 /L N ) ) )
45	44	adantr 481	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( 0 /L N ) = ( ( B /L N ) x. ( 0 /L N ) ) )
46	3	3ad2ant3 1084	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> N e. ZZ )
47	12, 46, 32	sylancr 695	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 /L N ) e. ZZ )
48	47	zcnd 11483	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 /L N ) e. CC )
49	48	adantr 481	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( 0 /L N ) e. CC )
50		lgscl 25036	. . . . . . . 8 |- ( ( B e. ZZ /\ N e. ZZ ) -> ( B /L N ) e. ZZ )
51	1, 46, 50	syl2anc 693	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( B /L N ) e. ZZ )
52	51	zcnd 11483	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( B /L N ) e. CC )
53	52	adantr 481	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( B /L N ) e. CC )
54	49, 53	mulcomd 10061	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( ( 0 /L N ) x. ( B /L N ) ) = ( ( B /L N ) x. ( 0 /L N ) ) )
55	45, 54	eqtr4d 2659	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( 0 /L N ) = ( ( 0 /L N ) x. ( B /L N ) ) )
56		oveq1 6657	. . . . 5 |- ( A = 0 -> ( A x. B ) = ( 0 x. B ) )
57		zcn 11382	. . . . . . 7 |- ( B e. ZZ -> B e. CC )
58	57	3ad2ant2 1083	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. CC )
59	58	mul02d 10234	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 x. B ) = 0 )
60	56, 59	sylan9eqr 2678	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A x. B ) = 0 )
61	60	oveq1d 6665	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( ( A x. B ) /L N ) = ( 0 /L N ) )
62		simpr 477	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> A = 0 )
63	62	oveq1d 6665	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A /L N ) = ( 0 /L N ) )
64	63	oveq1d 6665	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( ( A /L N ) x. ( B /L N ) ) = ( ( 0 /L N ) x. ( B /L N ) ) )
65	55, 61, 64	3eqtr4d 2666	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )
66		simp1 1061	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. ZZ )
67		oveq1 6657	. . . . . . . 8 |- ( x = A -> ( x /L N ) = ( A /L N ) )
68	67	oveq1d 6665	. . . . . . 7 |- ( x = A -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( A /L N ) x. ( 0 /L N ) ) )
69	68	eqeq2d 2632	. . . . . 6 |- ( x = A -> ( ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) <-> ( 0 /L N ) = ( ( A /L N ) x. ( 0 /L N ) ) ) )
70	69	rspcv 3305	. . . . 5 |- ( A e. ZZ -> ( A. x e. ZZ ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) -> ( 0 /L N ) = ( ( A /L N ) x. ( 0 /L N ) ) ) )
71	66, 39, 70	sylc 65	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 /L N ) = ( ( A /L N ) x. ( 0 /L N ) ) )
72	71	adantr 481	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( 0 /L N ) = ( ( A /L N ) x. ( 0 /L N ) ) )
73		oveq2 6658	. . . . 5 |- ( B = 0 -> ( A x. B ) = ( A x. 0 ) )
74	66	zcnd 11483	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. CC )
75	74	mul01d 10235	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( A x. 0 ) = 0 )
76	73, 75	sylan9eqr 2678	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( A x. B ) = 0 )
77	76	oveq1d 6665	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( ( A x. B ) /L N ) = ( 0 /L N ) )
78		simpr 477	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> B = 0 )
79	78	oveq1d 6665	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( B /L N ) = ( 0 /L N ) )
80	79	oveq2d 6666	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( ( A /L N ) x. ( B /L N ) ) = ( ( A /L N ) x. ( 0 /L N ) ) )
81	72, 77, 80	3eqtr4d 2666	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )
82		lgsdir 25057	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )
83	3, 82	syl3anl3 1376	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ ( A =/= 0 /\ B =/= 0 ) ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )
84	65, 81, 83	pm2.61da2ne 2882	1 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   NN0cn0 11292   ZZcz 11377   gcd cgcd 15216   /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:  lgsdchr  25080