Theorem lgsdirnn0 15920

Description: Variation on lgsdir 15908 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 6057	. . . . . . . . 9 |- ( x = B -> ( x /L N ) = ( B /L N ) )
2	1	oveq1d 6065	. . . . . . . 8 |- ( x = B -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( B /L N ) x. ( 0 /L N ) ) )
3	2	eqeq2d 2244	. . . . . . 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 9597	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> N e. ZZ )
6		lgscl 15887	. . . . . . . . . . . . . . 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 9701	. . . . . . . . . . . . 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 8666	. . . . . . . . . . 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 6066	. . . . . . . . . . 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 2276	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) = 0 ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
14		0z 9588	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
15	5	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> N e. ZZ )
16		lgsne0 15911	. . . . . . . . . . . . . . 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 12669	. . . . . . . . . . . . . . . . . 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 12677	. . . . . . . . . . . . . . . . . 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 2265	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 gcd N ) = N )
23	22	eqeq1d 2241	. . . . . . . . . . . . . . 15 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( 0 gcd N ) = 1 <-> N = 1 ) )
24		lgs1 15917	. . . . . . . . . . . . . . . . 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 6058	. . . . . . . . . . . . . . . . 17 |- ( N = 1 -> ( x /L N ) = ( x /L 1 ) )
27	26	eqeq1d 2241	. . . . . . . . . . . . . . . 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 6065	. . . . . . . . . . 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 15887	. . . . . . . . . . . . . 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 9701	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 0 /L N ) e. CC )
37	36	mullidd 8292	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ x e. ZZ ) /\ ( 0 /L N ) =/= 0 ) -> ( 1 x. ( 0 /L N ) ) = ( 0 /L N ) )
38	32, 37	eqtr2d 2266	. . . . . . . . . 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 9653	. . . . . . . . . . . 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 2423	. . . . . . . . . . 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 806	. . . . . . . . 9 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
45	44	ralrimiva 2615	. . . . . . . 8 |- ( N e. NN0 -> A. x e. ZZ ( 0 /L N ) = ( ( x /L N ) x. ( 0 /L N ) ) )
46	45	3ad2ant3 1047	. . . . . . 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 1025	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. ZZ )
48	3, 46, 47	rspcdva 2926	. . . . . 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 1047	. . . . . . . . 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 9701	. . . . . . 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 15887	. . . . . . . . 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 9701	. . . . . . 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 8295	. . . . 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 2268	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( 0 /L N ) = ( ( 0 /L N ) x. ( B /L N ) ) )
60		oveq1 6057	. . . . . 6 |- ( A = 0 -> ( A x. B ) = ( 0 x. B ) )
61		zcn 9582	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
62	61	3ad2ant2 1046	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> B e. CC )
63	62	mul02d 8665	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( 0 x. B ) = 0 )
64	60, 63	sylan9eqr 2287	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A x. B ) = 0 )
65	64	oveq1d 6065	. . . 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 6065	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ A = 0 ) -> ( A /L N ) = ( 0 /L N ) )
68	67	oveq1d 6065	. . . 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 2275	. . 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 6057	. . . . . . . 8 |- ( x = A -> ( x /L N ) = ( A /L N ) )
71	70	oveq1d 6065	. . . . . . 7 |- ( x = A -> ( ( x /L N ) x. ( 0 /L N ) ) = ( ( A /L N ) x. ( 0 /L N ) ) )
72	71	eqeq2d 2244	. . . . . 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 1024	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. ZZ )
74	72, 46, 73	rspcdva 2926	. . . . 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 6058	. . . . . 6 |- ( B = 0 -> ( A x. B ) = ( A x. 0 ) )
77	73	zcnd 9701	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> A e. CC )
78	77	mul01d 8666	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) -> ( A x. 0 ) = 0 )
79	76, 78	sylan9eqr 2287	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( A x. B ) = 0 )
80	79	oveq1d 6065	. . . 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 6065	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN0 ) /\ B = 0 ) -> ( B /L N ) = ( 0 /L N ) )
83	82	oveq2d 6066	. . . 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 2275	. . 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 805	. 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 2499	. . 3 |- ( ( A =/= 0 /\ B =/= 0 ) <-> -. ( A = 0 \/ B = 0 ) )
87		lgsdir 15908	. . . 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 1324	. . 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 9653	. . . . 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 9653	. . . . 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 944	. . . 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 844	. . 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 806	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 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   A.wral 2520  (class class class)co 6050   CCcc 8125   0cc0 8127   1c1 8128   x. cmul 8132   NN0cn0 9496   ZZcz 9577   gcd cgcd 12649   /Lclgs 15870

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-2o 6648  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-proddc 12237  df-dvds 12474  df-gcd 12650  df-prm 12805  df-phi 12908  df-pc 12983  df-lgs 15871

This theorem is referenced by: (None)