Theorem lgsdirnn0 14017

Description: Variation on lgsdir 14005 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 5872	. . . . . . . . 9 |- ( x = B -> ( x /L N ) = ( B /L N ) )
2	1	oveq1d 5880	. . . . . . . 8 |- ( x = B -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( B /L N ) x. ( 0 /L N ) ) )
3	2	eqeq2d 2187	. . . . . . 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 9244	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> N e. ZZ )
6		lgscl 13984	. . . . . . . . . . . . . . 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 9347	. . . . . . . . . . . . 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 8324	. . . . . . . . . . 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 5881	. . . . . . . . . . 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 2219	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
14		0z 9235	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
15	5	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> N e. ZZ )
16		lgsne0 14008	. . . . . . . . . . . . . . 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 11939	. . . . . . . . . . . . . . . . . 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 11947	. . . . . . . . . . . . . . . . . 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 2208	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 gcd N ) = N )
23	22	eqeq1d 2184	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 gcd N ) = 1 <-> N = 1 ) )
24		lgs1 14014	. . . . . . . . . . . . . . . . 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 5873	. . . . . . . . . . . . . . . . 17 |- ( N = 1 -> ( x /L N ) = ( x /L 1 ) )
27	26	eqeq1d 2184	. . . . . . . . . . . . . . . 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 5880	. . . . . . . . . . 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 13984	. . . . . . . . . . . . . 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 9347	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 0 /L N ) e. CC )
37	36	mulid2d 7950	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 1 x. ( 0 /L N ) ) = ( 0 /L N ) )
38	32, 37	eqtr2d 2209	. . . . . . . . . 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 9299	. . . . . . . . . . . 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 2356	. . . . . . . . . . 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 798	. . . . . . . . 9 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
45	44	ralrimiva 2548	. . . . . . . 8 |- ( N e. NN0 -> A. x e. ZZ ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
46	45	3ad2ant3 1020	. . . . . . 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 998	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. ZZ )
48	3, 46, 47	rspcdva 2844	. . . . . 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 1020	. . . . . . . . 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 9347	. . . . . . 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 13984	. . . . . . . . 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 9347	. . . . . . 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 7953	. . . . 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 2211	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( 0 /L N ) = ( ( 0 /L N ) x. ( B /L N ) ) )
60		oveq1 5872	. . . . . 6 |- ( A = 0 -> ( A x. B ) = ( 0 x. B ) )
61		zcn 9229	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
62	61	3ad2ant2 1019	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. CC )
63	62	mul02d 8323	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 x. B ) = 0 )
64	60, 63	sylan9eqr 2230	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A x. B ) = 0 )
65	64	oveq1d 5880	. . . 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 5880	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A /L N ) = ( 0 /L N ) )
68	67	oveq1d 5880	. . . 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 2218	. . 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 5872	. . . . . . . 8 |- ( x = A -> ( x /L N ) = ( A /L N ) )
71	70	oveq1d 5880	. . . . . . 7 |- ( x = A -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( A /L N ) x. ( 0 /L N ) ) )
72	71	eqeq2d 2187	. . . . . 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 997	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. ZZ )
74	72, 46, 73	rspcdva 2844	. . . . 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 5873	. . . . . 6 |- ( B = 0 -> ( A x. B ) = ( A x. 0 ) )
77	73	zcnd 9347	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. CC )
78	77	mul01d 8324	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( A x. 0 ) = 0 )
79	76, 78	sylan9eqr 2230	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( A x. B ) = 0 )
80	79	oveq1d 5880	. . . 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 5880	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( B /L N ) = ( 0 /L N ) )
83	82	oveq2d 5881	. . . 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 2218	. . 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 797	. 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 2432	. . 3 |- ( ( A =/= 0 /\ B =/= 0 ) <-> -. ( A = 0 \/ B = 0 ) )
87		lgsdir 14005	. . . 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 1288	. . 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 9299	. . . . 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 9299	. . . . 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 935	. . . 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 836	. . 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 798	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 708  DECID wdc 834   /\ w3a 978   = wceq 1353   e. wcel 2146   =/= wne 2345   A.wral 2453  (class class class)co 5865   CCcc 7784   0cc0 7786   1c1 7787   x. cmul 7791   NN0cn0 9147   ZZcz 9224   gcd cgcd 11908   /Lclgs 13967

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-frec 6382  df-1o 6407  df-2o 6408  df-oadd 6411  df-er 6525  df-en 6731  df-dom 6732  df-fin 6733  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-5 8952  df-6 8953  df-7 8954  df-8 8955  df-9 8956  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-fl 10238  df-mod 10291  df-seqfrec 10414  df-exp 10488  df-ihash 10722  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-clim 11253  df-proddc 11525  df-dvds 11761  df-gcd 11909  df-prm 12073  df-phi 12176  df-pc 12250  df-lgs 13968

This theorem is referenced by: (None)