Theorem 1cubr 24569

Description: The cube roots of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)

Hypothesis

Ref	Expression
1cubr.r	|- R = { 1 , ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) }

Assertion

Ref	Expression
1cubr	|- ( A e. R <-> ( A e. CC /\ ( A ^ 3 ) = 1 ) )

Proof of Theorem 1cubr

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1cubr.r	. . . . 5 |- R = { 1 , ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) }
2		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
3		neg1cn 11124	. . . . . . . . 9 |- -u 1 e. CC
4		ax-icn 9995	. . . . . . . . . 10 |- _i e. CC
5		3cn 11095	. . . . . . . . . . 11 |- 3 e. CC
6		sqrtcl 14101	. . . . . . . . . . 11 |- ( 3 e. CC -> ( sqr ` 3 ) e. CC )
7	5, 6	ax-mp 5	. . . . . . . . . 10 |- ( sqr ` 3 ) e. CC
8	4, 7	mulcli 10045	. . . . . . . . 9 |- ( _i x. ( sqr ` 3 ) ) e. CC
9	3, 8	addcli 10044	. . . . . . . 8 |- ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) e. CC
10		halfcl 11257	. . . . . . . 8 |- ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) e. CC -> ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) e. CC )
11	9, 10	ax-mp 5	. . . . . . 7 |- ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) e. CC
12	3, 8	subcli 10357	. . . . . . . 8 |- ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) e. CC
13		halfcl 11257	. . . . . . . 8 |- ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) e. CC -> ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) e. CC )
14	12, 13	ax-mp 5	. . . . . . 7 |- ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) e. CC
15	2, 11, 14	3pm3.2i 1239	. . . . . 6 |- ( 1 e. CC /\ ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) e. CC /\ ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) e. CC )
16	2	elexi 3213	. . . . . . 7 |- 1 e. _V
17		ovex 6678	. . . . . . 7 |- ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) e. _V
18		ovex 6678	. . . . . . 7 |- ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) e. _V
19	16, 17, 18	tpss 4368	. . . . . 6 |- ( ( 1 e. CC /\ ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) e. CC /\ ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) e. CC ) <-> { 1 , ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) } C_ CC )
20	15, 19	mpbi 220	. . . . 5 |- { 1 , ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) } C_ CC
21	1, 20	eqsstri 3635	. . . 4 |- R C_ CC
22	21	sseli 3599	. . 3 |- ( A e. R -> A e. CC )
23	22	pm4.71ri 665	. 2 |- ( A e. R <-> ( A e. CC /\ A e. R ) )
24		3nn 11186	. . . . 5 |- 3 e. NN
25		cxpeq 24498	. . . . 5 |- ( ( A e. CC /\ 3 e. NN /\ 1 e. CC ) -> ( ( A ^ 3 ) = 1 <-> E. n e. ( 0 ... ( 3 - 1 ) ) A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) ) )
26	24, 2, 25	mp3an23 1416	. . . 4 |- ( A e. CC -> ( ( A ^ 3 ) = 1 <-> E. n e. ( 0 ... ( 3 - 1 ) ) A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) ) )
27		eltpg 4227	. . . . 5 |- ( A e. CC -> ( A e. { 1 , ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) } <-> ( A = 1 \/ A = ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) \/ A = ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) ) ) )
28	1	eleq2i 2693	. . . . 5 |- ( A e. R <-> A e. { 1 , ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) , ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) } )
29		3m1e2 11137	. . . . . . . . . 10 |- ( 3 - 1 ) = 2
30		2cn 11091	. . . . . . . . . . 11 |- 2 e. CC
31	30	addid2i 10224	. . . . . . . . . 10 |- ( 0 + 2 ) = 2
32	29, 31	eqtr4i 2647	. . . . . . . . 9 |- ( 3 - 1 ) = ( 0 + 2 )
33	32	oveq2i 6661	. . . . . . . 8 |- ( 0 ... ( 3 - 1 ) ) = ( 0 ... ( 0 + 2 ) )
34		0z 11388	. . . . . . . . 9 |- 0 e. ZZ
35		fztp 12397	. . . . . . . . 9 |- ( 0 e. ZZ -> ( 0 ... ( 0 + 2 ) ) = { 0 , ( 0 + 1 ) , ( 0 + 2 ) } )
36	34, 35	ax-mp 5	. . . . . . . 8 |- ( 0 ... ( 0 + 2 ) ) = { 0 , ( 0 + 1 ) , ( 0 + 2 ) }
37	33, 36	eqtri 2644	. . . . . . 7 |- ( 0 ... ( 3 - 1 ) ) = { 0 , ( 0 + 1 ) , ( 0 + 2 ) }
38	37	rexeqi 3143	. . . . . 6 |- ( E. n e. ( 0 ... ( 3 - 1 ) ) A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> E. n e. { 0 , ( 0 + 1 ) , ( 0 + 2 ) } A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) )
39		3ne0 11115	. . . . . . . . . . 11 |- 3 =/= 0
40	5, 39	reccli 10755	. . . . . . . . . 10 |- ( 1 / 3 ) e. CC
41		1cxp 24418	. . . . . . . . . 10 |- ( ( 1 / 3 ) e. CC -> ( 1 ^c ( 1 / 3 ) ) = 1 )
42	40, 41	ax-mp 5	. . . . . . . . 9 |- ( 1 ^c ( 1 / 3 ) ) = 1
43	42	oveq1i 6660	. . . . . . . 8 |- ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) )
44	43	eqeq2i 2634	. . . . . . 7 |- ( A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) )
45	44	rexbii 3041	. . . . . 6 |- ( E. n e. { 0 , ( 0 + 1 ) , ( 0 + 2 ) } A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> E. n e. { 0 , ( 0 + 1 ) , ( 0 + 2 ) } A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) )
46	34	elexi 3213	. . . . . . 7 |- 0 e. _V
47		ovex 6678	. . . . . . 7 |- ( 0 + 1 ) e. _V
48		ovex 6678	. . . . . . 7 |- ( 0 + 2 ) e. _V
49		oveq2 6658	. . . . . . . . . . 11 |- ( n = 0 -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) = ( ( -u 1 ^c ( 2 / 3 ) ) ^ 0 ) )
50	30, 5, 39	divcli 10767	. . . . . . . . . . . . 13 |- ( 2 / 3 ) e. CC
51		cxpcl 24420	. . . . . . . . . . . . 13 |- ( ( -u 1 e. CC /\ ( 2 / 3 ) e. CC ) -> ( -u 1 ^c ( 2 / 3 ) ) e. CC )
52	3, 50, 51	mp2an 708	. . . . . . . . . . . 12 |- ( -u 1 ^c ( 2 / 3 ) ) e. CC
53		exp0 12864	. . . . . . . . . . . 12 |- ( ( -u 1 ^c ( 2 / 3 ) ) e. CC -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ 0 ) = 1 )
54	52, 53	ax-mp 5	. . . . . . . . . . 11 |- ( ( -u 1 ^c ( 2 / 3 ) ) ^ 0 ) = 1
55	49, 54	syl6eq 2672	. . . . . . . . . 10 |- ( n = 0 -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) = 1 )
56	55	oveq2d 6666	. . . . . . . . 9 |- ( n = 0 -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( 1 x. 1 ) )
57		1t1e1 11175	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
58	56, 57	syl6eq 2672	. . . . . . . 8 |- ( n = 0 -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = 1 )
59	58	eqeq2d 2632	. . . . . . 7 |- ( n = 0 -> ( A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> A = 1 ) )
60		id 22	. . . . . . . . . . . . 13 |- ( n = ( 0 + 1 ) -> n = ( 0 + 1 ) )
61	2	addid2i 10224	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
62	60, 61	syl6eq 2672	. . . . . . . . . . . 12 |- ( n = ( 0 + 1 ) -> n = 1 )
63	62	oveq2d 6666	. . . . . . . . . . 11 |- ( n = ( 0 + 1 ) -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) = ( ( -u 1 ^c ( 2 / 3 ) ) ^ 1 ) )
64		exp1 12866	. . . . . . . . . . . 12 |- ( ( -u 1 ^c ( 2 / 3 ) ) e. CC -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ 1 ) = ( -u 1 ^c ( 2 / 3 ) ) )
65	52, 64	ax-mp 5	. . . . . . . . . . 11 |- ( ( -u 1 ^c ( 2 / 3 ) ) ^ 1 ) = ( -u 1 ^c ( 2 / 3 ) )
66	63, 65	syl6eq 2672	. . . . . . . . . 10 |- ( n = ( 0 + 1 ) -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) = ( -u 1 ^c ( 2 / 3 ) ) )
67	66	oveq2d 6666	. . . . . . . . 9 |- ( n = ( 0 + 1 ) -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( 1 x. ( -u 1 ^c ( 2 / 3 ) ) ) )
68	52	mulid2i 10043	. . . . . . . . . 10 |- ( 1 x. ( -u 1 ^c ( 2 / 3 ) ) ) = ( -u 1 ^c ( 2 / 3 ) )
69		1cubrlem 24568	. . . . . . . . . . 11 |- ( ( -u 1 ^c ( 2 / 3 ) ) = ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) /\ ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) = ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) )
70	69	simpli 474	. . . . . . . . . 10 |- ( -u 1 ^c ( 2 / 3 ) ) = ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 )
71	68, 70	eqtri 2644	. . . . . . . . 9 |- ( 1 x. ( -u 1 ^c ( 2 / 3 ) ) ) = ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 )
72	67, 71	syl6eq 2672	. . . . . . . 8 |- ( n = ( 0 + 1 ) -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) )
73	72	eqeq2d 2632	. . . . . . 7 |- ( n = ( 0 + 1 ) -> ( A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> A = ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) ) )
74		id 22	. . . . . . . . . . . 12 |- ( n = ( 0 + 2 ) -> n = ( 0 + 2 ) )
75	74, 31	syl6eq 2672	. . . . . . . . . . 11 |- ( n = ( 0 + 2 ) -> n = 2 )
76	75	oveq2d 6666	. . . . . . . . . 10 |- ( n = ( 0 + 2 ) -> ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) = ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) )
77	76	oveq2d 6666	. . . . . . . . 9 |- ( n = ( 0 + 2 ) -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) ) )
78	52	sqcli 12944	. . . . . . . . . . 11 |- ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) e. CC
79	78	mulid2i 10043	. . . . . . . . . 10 |- ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) ) = ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 )
80	69	simpri 478	. . . . . . . . . 10 |- ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) = ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 )
81	79, 80	eqtri 2644	. . . . . . . . 9 |- ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ 2 ) ) = ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 )
82	77, 81	syl6eq 2672	. . . . . . . 8 |- ( n = ( 0 + 2 ) -> ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) = ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) )
83	82	eqeq2d 2632	. . . . . . 7 |- ( n = ( 0 + 2 ) -> ( A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> A = ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) ) )
84	46, 47, 48, 59, 73, 83	rextp 4241	. . . . . 6 |- ( E. n e. { 0 , ( 0 + 1 ) , ( 0 + 2 ) } A = ( 1 x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> ( A = 1 \/ A = ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) \/ A = ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) ) )
85	38, 45, 84	3bitri 286	. . . . 5 |- ( E. n e. ( 0 ... ( 3 - 1 ) ) A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) <-> ( A = 1 \/ A = ( ( -u 1 + ( _i x. ( sqr ` 3 ) ) ) / 2 ) \/ A = ( ( -u 1 - ( _i x. ( sqr ` 3 ) ) ) / 2 ) ) )
86	27, 28, 85	3bitr4g 303	. . . 4 |- ( A e. CC -> ( A e. R <-> E. n e. ( 0 ... ( 3 - 1 ) ) A = ( ( 1 ^c ( 1 / 3 ) ) x. ( ( -u 1 ^c ( 2 / 3 ) ) ^ n ) ) ) )
87	26, 86	bitr4d 271	. . 3 |- ( A e. CC -> ( ( A ^ 3 ) = 1 <-> A e. R ) )
88	87	pm5.32i 669	. 2 |- ( ( A e. CC /\ ( A ^ 3 ) = 1 ) <-> ( A e. CC /\ A e. R ) )
89	23, 88	bitr4i 267	1 |- ( A e. R <-> ( A e. CC /\ ( A ^ 3 ) = 1 ) )

Syntax hints:   <-> wb 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   {ctp 4181   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   ZZcz 11377   ...cfz 12326   ^cexp 12860   sqrcsqrt 13973   ^c ccxp 24302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by:  cubic  24576