Theorem lgsdirnn0 15775

Description: Variation on lgsdir 15763 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 6024	. . . . . . . . 9 |- ( x = B -> ( x /L N ) = ( B /L N ) )
2	1	oveq1d 6032	. . . . . . . 8 |- ( x = B -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( B /L N ) x. ( 0 /L N ) ) )
3	2	eqeq2d 2243	. . . . . . 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 9498	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> N e. ZZ )
6		lgscl 15742	. . . . . . . . . . . . . . 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 9602	. . . . . . . . . . . . 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 8571	. . . . . . . . . . 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 6033	. . . . . . . . . . 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 2275	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
14		0z 9489	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
15	5	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> N e. ZZ )
16		lgsne0 15766	. . . . . . . . . . . . . . 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 12543	. . . . . . . . . . . . . . . . . 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 12551	. . . . . . . . . . . . . . . . . 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 2264	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 gcd N ) = N )
23	22	eqeq1d 2240	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 gcd N ) = 1 <-> N = 1 ) )
24		lgs1 15772	. . . . . . . . . . . . . . . . 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 6025	. . . . . . . . . . . . . . . . 17 |- ( N = 1 -> ( x /L N ) = ( x /L 1 ) )
27	26	eqeq1d 2240	. . . . . . . . . . . . . . . 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 6032	. . . . . . . . . . 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 15742	. . . . . . . . . . . . . 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 9602	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 0 /L N ) e. CC )
37	36	mulid2d 8197	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 1 x. ( 0 /L N ) ) = ( 0 /L N ) )
38	32, 37	eqtr2d 2265	. . . . . . . . . 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 9554	. . . . . . . . . . . 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 2413	. . . . . . . . . . 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 805	. . . . . . . . 9 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
45	44	ralrimiva 2605	. . . . . . . 8 |- ( N e. NN0 -> A. x e. ZZ ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
46	45	3ad2ant3 1046	. . . . . . 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 1024	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. ZZ )
48	3, 46, 47	rspcdva 2915	. . . . . 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 1046	. . . . . . . . 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 9602	. . . . . . 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 15742	. . . . . . . . 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 9602	. . . . . . 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 8200	. . . . 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 2267	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( 0 /L N ) = ( ( 0 /L N ) x. ( B /L N ) ) )
60		oveq1 6024	. . . . . 6 |- ( A = 0 -> ( A x. B ) = ( 0 x. B ) )
61		zcn 9483	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
62	61	3ad2ant2 1045	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. CC )
63	62	mul02d 8570	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 x. B ) = 0 )
64	60, 63	sylan9eqr 2286	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A x. B ) = 0 )
65	64	oveq1d 6032	. . . 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 6032	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A /L N ) = ( 0 /L N ) )
68	67	oveq1d 6032	. . . 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 2274	. . 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 6024	. . . . . . . 8 |- ( x = A -> ( x /L N ) = ( A /L N ) )
71	70	oveq1d 6032	. . . . . . 7 |- ( x = A -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( A /L N ) x. ( 0 /L N ) ) )
72	71	eqeq2d 2243	. . . . . 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 1023	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. ZZ )
74	72, 46, 73	rspcdva 2915	. . . . 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 6025	. . . . . 6 |- ( B = 0 -> ( A x. B ) = ( A x. 0 ) )
77	73	zcnd 9602	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. CC )
78	77	mul01d 8571	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( A x. 0 ) = 0 )
79	76, 78	sylan9eqr 2286	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( A x. B ) = 0 )
80	79	oveq1d 6032	. . . 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 6032	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( B /L N ) = ( 0 /L N ) )
83	82	oveq2d 6033	. . . 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 2274	. . 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 804	. 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 2489	. . 3 |- ( ( A =/= 0 /\ B =/= 0 ) <-> -. ( A = 0 \/ B = 0 ) )
87		lgsdir 15763	. . . 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 1323	. . 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 9554	. . . . 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 9554	. . . . 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 943	. . . 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 843	. . 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 805	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 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   A.wral 2510  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   x. cmul 8036   NN0cn0 9401   ZZcz 9478   gcd cgcd 12523   /Lclgs 15725

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-2o 6582  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-proddc 12111  df-dvds 12348  df-gcd 12524  df-prm 12679  df-phi 12782  df-pc 12857  df-lgs 15726

This theorem is referenced by: (None)