Theorem lgsdirnn0 15288

Description: Variation on lgsdir 15276 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		oveq1 5929	. . . . . . . . 9 |- ( x = B -> ( x /L N ) = ( B /L N ) )
2	1	oveq1d 5937	. . . . . . . 8 |- ( x = B -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( B /L N ) x. ( 0 /L N ) ) )
3	2	eqeq2d 2208	. . . . . . 7 |- ( x = B -> ( ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) <-> ( 0 /L N ) = ( ( B /L N ) x. ( 0 /L N ) ) ) )
4		id 19	. . . . . . . . . . . . . . 15 |- ( x e. ZZ -> x e. ZZ )
5		nn0z 9346	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> N e. ZZ )
6		lgscl 15255	. . . . . . . . . . . . . . 15 |- ( ( x e. ZZ /\ N e. ZZ ) -> ( x /L N ) e. ZZ )
7	4, 5, 6	syl2anr 290	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( x /L N ) e. ZZ )
8	7	zcnd 9449	. . . . . . . . . . . . 13 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( x /L N ) e. CC )
9	8	adantr 276	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( x /L N ) e. CC )
10	9	mul01d 8419	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( ( x /L N ) x. 0 ) = 0 )
11		simpr 110	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( 0 /L N ) = 0 )
12	11	oveq2d 5938	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( x /L N ) x. 0 ) )
13	10, 12, 11	3eqtr4rd 2240	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
14		0z 9337	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
15	5	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> N e. ZZ )
16		lgsne0 15279	. . . . . . . . . . . . . . 15 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( ( 0 /L N ) =/= 0 <-> ( 0 gcd N ) = 1 ) )
17	14, 15, 16	sylancr 414	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 /L N ) =/= 0 <-> ( 0 gcd N ) = 1 ) )
18		gcdcom 12140	. . . . . . . . . . . . . . . . . 18 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 gcd N ) = ( N gcd 0 ) )
19	14, 15, 18	sylancr 414	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 gcd N ) = ( N gcd 0 ) )
20		nn0gcdid0 12148	. . . . . . . . . . . . . . . . . 18 |- ( N e. NN0 -> ( N gcd 0 ) = N )
21	20	adantr 276	. . . . . . . . . . . . . . . . 17 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( N gcd 0 ) = N )
22	19, 21	eqtrd 2229	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 gcd N ) = N )
23	22	eqeq1d 2205	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 gcd N ) = 1 <-> N = 1 ) )
24		lgs1 15285	. . . . . . . . . . . . . . . . 17 |- ( x e. ZZ -> ( x /L 1 ) = 1 )
25	24	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( x /L 1 ) = 1 )
26		oveq2 5930	. . . . . . . . . . . . . . . . 17 |- ( N = 1 -> ( x /L N ) = ( x /L 1 ) )
27	26	eqeq1d 2205	. . . . . . . . . . . . . . . 16 |- ( N = 1 -> ( ( x /L N ) = 1 <-> ( x /L 1 ) = 1 ) )
28	25, 27	syl5ibrcom 157	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( N = 1 -> ( x /L N ) = 1 ) )
29	23, 28	sylbid 150	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 gcd N ) = 1 -> ( x /L N ) = 1 ) )
30	17, 29	sylbid 150	. . . . . . . . . . . . 13 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 /L N ) =/= 0 -> ( x /L N ) = 1 ) )
31	30	imp 124	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( x /L N ) = 1 )
32	31	oveq1d 5937	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( ( x /L N ) x. ( 0 /L N ) ) = ( 1 x. ( 0 /L N ) ) )
33	5	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> N e. ZZ )
34		lgscl 15255	. . . . . . . . . . . . . 14 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 /L N ) e. ZZ )
35	14, 33, 34	sylancr 414	. . . . . . . . . . . . 13 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 0 /L N ) e. ZZ )
36	35	zcnd 9449	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 0 /L N ) e. CC )
37	36	mulid2d 8045	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 1 x. ( 0 /L N ) ) = ( 0 /L N ) )
38	32, 37	eqtr2d 2230	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
39	14, 15, 34	sylancr 414	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 /L N ) e. ZZ )
40		zdceq 9401	. . . . . . . . . . . 12 |- ( ( ( 0 /L N ) e. ZZ /\ 0 e. ZZ ) -> DECID ( 0 /L N ) = 0 )
41	39, 14, 40	sylancl 413	. . . . . . . . . . 11 |- ( ( N e. NN0 /\ x e. ZZ ) -> DECID ( 0 /L N ) = 0 )
42		dcne 2378	. . . . . . . . . . 11 |- (DECID ( 0 /L N ) = 0 <-> ( ( 0 /L N ) = 0 \/ ( 0 /L N ) =/= 0 ) )
43	41, 42	sylib 122	. . . . . . . . . 10 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 /L N ) = 0 \/ ( 0 /L N ) =/= 0 ) )
44	13, 38, 43	mpjaodan 799	. . . . . . . . 9 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
45	44	ralrimiva 2570	. . . . . . . 8 |- ( N e. NN0 -> A. x e. ZZ ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
46	45	3ad2ant3 1022	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A. x e. ZZ ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
47		simp2 1000	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. ZZ )
48	3, 46, 47	rspcdva 2873	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 /L N ) = ( ( B /L N ) x. ( 0 /L N ) ) )
49	48	adantr 276	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( 0 /L N ) = ( ( B /L N ) x. ( 0 /L N ) ) )
50	5	3ad2ant3 1022	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> N e. ZZ )
51	14, 50, 34	sylancr 414	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 /L N ) e. ZZ )
52	51	zcnd 9449	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 /L N ) e. CC )
53	52	adantr 276	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( 0 /L N ) e. CC )
54		lgscl 15255	. . . . . . . . 9 |- ( ( B e. ZZ /\ N e. ZZ ) -> ( B /L N ) e. ZZ )
55	47, 50, 54	syl2anc 411	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( B /L N ) e. ZZ )
56	55	zcnd 9449	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( B /L N ) e. CC )
57	56	adantr 276	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( B /L N ) e. CC )
58	53, 57	mulcomd 8048	. . . . 5 |- ( ( ( 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 ) ) )
59	49, 58	eqtr4d 2232	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( 0 /L N ) = ( ( 0 /L N ) x. ( B /L N ) ) )
60		oveq1 5929	. . . . . 6 |- ( A = 0 -> ( A x. B ) = ( 0 x. B ) )
61		zcn 9331	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
62	61	3ad2ant2 1021	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. CC )
63	62	mul02d 8418	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 x. B ) = 0 )
64	60, 63	sylan9eqr 2251	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A x. B ) = 0 )
65	64	oveq1d 5937	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( ( A x. B ) /L N ) = ( 0 /L N ) )
66		simpr 110	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> A = 0 )
67	66	oveq1d 5937	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A /L N ) = ( 0 /L N ) )
68	67	oveq1d 5937	. . . 4 |- ( ( ( 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 ) ) )
69	59, 65, 68	3eqtr4d 2239	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )
70		oveq1 5929	. . . . . . . 8 |- ( x = A -> ( x /L N ) = ( A /L N ) )
71	70	oveq1d 5937	. . . . . . 7 |- ( x = A -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( A /L N ) x. ( 0 /L N ) ) )
72	71	eqeq2d 2208	. . . . . 6 |- ( x = A -> ( ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) <-> ( 0 /L N ) = ( ( A /L N ) x. ( 0 /L N ) ) ) )
73		simp1 999	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. ZZ )
74	72, 46, 73	rspcdva 2873	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 /L N ) = ( ( A /L N ) x. ( 0 /L N ) ) )
75	74	adantr 276	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( 0 /L N ) = ( ( A /L N ) x. ( 0 /L N ) ) )
76		oveq2 5930	. . . . . 6 |- ( B = 0 -> ( A x. B ) = ( A x. 0 ) )
77	73	zcnd 9449	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. CC )
78	77	mul01d 8419	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( A x. 0 ) = 0 )
79	76, 78	sylan9eqr 2251	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( A x. B ) = 0 )
80	79	oveq1d 5937	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( ( A x. B ) /L N ) = ( 0 /L N ) )
81		simpr 110	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> B = 0 )
82	81	oveq1d 5937	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( B /L N ) = ( 0 /L N ) )
83	82	oveq2d 5938	. . . 4 |- ( ( ( 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 ) ) )
84	75, 80, 83	3eqtr4d 2239	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )
85	69, 84	jaodan 798	. 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 ) ) )
86		neanior 2454	. . 3 |- ( ( A =/= 0 /\ B =/= 0 ) <-> -. ( A = 0 \/ B = 0 ) )
87		lgsdir 15276	. . . 4 |- ( ( ( 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 ) ) )
88	5, 87	syl3anl3 1299	. . 3 |- ( ( ( 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 ) ) )
89	86, 88	sylan2br 288	. 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 ) ) )
90		zdceq 9401	. . . . 5 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A = 0 )
91	73, 14, 90	sylancl 413	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> DECID A = 0 )
92		zdceq 9401	. . . . 5 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> DECID B = 0 )
93	47, 14, 92	sylancl 413	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> DECID B = 0 )
94		dcor 937	. . . 4 |- (DECID A = 0 -> (DECID B = 0 -> DECID ( A = 0 \/ B = 0 ) ) )
95	91, 93, 94	sylc 62	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> DECID ( A = 0 \/ B = 0 ) )
96		exmiddc 837	. . 3 |- (DECID ( A = 0 \/ B = 0 ) -> ( ( A = 0 \/ B = 0 ) \/ -. ( A = 0 \/ B = 0 ) ) )
97	95, 96	syl 14	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( ( A = 0 \/ B = 0 ) \/ -. ( A = 0 \/ B = 0 ) ) )
98	85, 89, 97	mpjaodan 799	1 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( ( A x. B ) /L N ) = ( ( A /L N ) x. ( B /L N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   A.wral 2475  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   x. cmul 7884   NN0cn0 9249   ZZcz 9326   gcd cgcd 12120   /Lclgs 15238

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-proddc 11716  df-dvds 11953  df-gcd 12121  df-prm 12276  df-phi 12379  df-pc 12454  df-lgs 15239

This theorem is referenced by: (None)