Theorem lgsdirnn0 15720

Description: Variation on lgsdir 15708 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 6007	. . . . . . . . 9 |- ( x = B -> ( x /L N ) = ( B /L N ) )
2	1	oveq1d 6015	. . . . . . . 8 |- ( x = B -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( B /L N ) x. ( 0 /L N ) ) )
3	2	eqeq2d 2241	. . . . . . 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 9462	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> N e. ZZ )
6		lgscl 15687	. . . . . . . . . . . . . . 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 9566	. . . . . . . . . . . . 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 8535	. . . . . . . . . . 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 6016	. . . . . . . . . . 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 2273	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
14		0z 9453	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
15	5	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> N e. ZZ )
16		lgsne0 15711	. . . . . . . . . . . . . . 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 12489	. . . . . . . . . . . . . . . . . 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 12497	. . . . . . . . . . . . . . . . . 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 2262	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 gcd N ) = N )
23	22	eqeq1d 2238	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 gcd N ) = 1 <-> N = 1 ) )
24		lgs1 15717	. . . . . . . . . . . . . . . . 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 6008	. . . . . . . . . . . . . . . . 17 |- ( N = 1 -> ( x /L N ) = ( x /L 1 ) )
27	26	eqeq1d 2238	. . . . . . . . . . . . . . . 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 6015	. . . . . . . . . . 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 15687	. . . . . . . . . . . . . 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 9566	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 0 /L N ) e. CC )
37	36	mulid2d 8161	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 1 x. ( 0 /L N ) ) = ( 0 /L N ) )
38	32, 37	eqtr2d 2263	. . . . . . . . . 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 9518	. . . . . . . . . . . 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 2411	. . . . . . . . . . 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 803	. . . . . . . . 9 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
45	44	ralrimiva 2603	. . . . . . . 8 |- ( N e. NN0 -> A. x e. ZZ ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
46	45	3ad2ant3 1044	. . . . . . 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 1022	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. ZZ )
48	3, 46, 47	rspcdva 2912	. . . . . 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 1044	. . . . . . . . 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 9566	. . . . . . 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 15687	. . . . . . . . 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 9566	. . . . . . 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 8164	. . . . 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 2265	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( 0 /L N ) = ( ( 0 /L N ) x. ( B /L N ) ) )
60		oveq1 6007	. . . . . 6 |- ( A = 0 -> ( A x. B ) = ( 0 x. B ) )
61		zcn 9447	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
62	61	3ad2ant2 1043	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. CC )
63	62	mul02d 8534	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 x. B ) = 0 )
64	60, 63	sylan9eqr 2284	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A x. B ) = 0 )
65	64	oveq1d 6015	. . . 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 6015	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A /L N ) = ( 0 /L N ) )
68	67	oveq1d 6015	. . . 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 2272	. . 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 6007	. . . . . . . 8 |- ( x = A -> ( x /L N ) = ( A /L N ) )
71	70	oveq1d 6015	. . . . . . 7 |- ( x = A -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( A /L N ) x. ( 0 /L N ) ) )
72	71	eqeq2d 2241	. . . . . 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 1021	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. ZZ )
74	72, 46, 73	rspcdva 2912	. . . . 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 6008	. . . . . 6 |- ( B = 0 -> ( A x. B ) = ( A x. 0 ) )
77	73	zcnd 9566	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. CC )
78	77	mul01d 8535	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( A x. 0 ) = 0 )
79	76, 78	sylan9eqr 2284	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( A x. B ) = 0 )
80	79	oveq1d 6015	. . . 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 6015	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( B /L N ) = ( 0 /L N ) )
83	82	oveq2d 6016	. . . 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 2272	. . 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 802	. 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 2487	. . 3 |- ( ( A =/= 0 /\ B =/= 0 ) <-> -. ( A = 0 \/ B = 0 ) )
87		lgsdir 15708	. . . 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 1321	. . 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 9518	. . . . 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 9518	. . . . 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 941	. . . 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 841	. . 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 803	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 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508  (class class class)co 6000   CCcc 7993   0cc0 7995   1c1 7996   x. cmul 8000   NN0cn0 9365   ZZcz 9442   gcd cgcd 12469   /Lclgs 15670

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-2o 6561  df-oadd 6564  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-9 9172  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-ihash 10993  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785  df-proddc 12057  df-dvds 12294  df-gcd 12470  df-prm 12625  df-phi 12728  df-pc 12803  df-lgs 15671

This theorem is referenced by: (None)