Theorem efchtdvds 24885

Description: The exponentiated Chebyshev function forms a divisibility chain between any two points. (Contributed by Mario Carneiro, 22-Sep-2014.)

Assertion

Ref	Expression
efchtdvds	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( exp ` ( theta ` A ) ) || ( exp ` ( theta ` B ) ) )

Proof of Theorem efchtdvds

Dummy variables p x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		chtcl 24835	. . . . . . 7 |- ( B e. RR -> ( theta ` B ) e. RR )
2	1	3ad2ant2 1083	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( theta ` B ) e. RR )
3	2	recnd 10068	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( theta ` B ) e. CC )
4		chtcl 24835	. . . . . . 7 |- ( A e. RR -> ( theta ` A ) e. RR )
5	4	3ad2ant1 1082	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( theta ` A ) e. RR )
6	5	recnd 10068	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( theta ` A ) e. CC )
7		efsub 14830	. . . . 5 |- ( ( ( theta ` B ) e. CC /\ ( theta ` A ) e. CC ) -> ( exp ` ( ( theta ` B ) - ( theta ` A ) ) ) = ( ( exp ` ( theta ` B ) ) / ( exp ` ( theta ` A ) ) ) )
8	3, 6, 7	syl2anc 693	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( exp ` ( ( theta ` B ) - ( theta ` A ) ) ) = ( ( exp ` ( theta ` B ) ) / ( exp ` ( theta ` A ) ) ) )
9		chtfl 24875	. . . . . . . . 9 |- ( B e. RR -> ( theta ` ( |_ ` B ) ) = ( theta ` B ) )
10	9	3ad2ant2 1083	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( theta ` ( |_ ` B ) ) = ( theta ` B ) )
11		chtfl 24875	. . . . . . . . 9 |- ( A e. RR -> ( theta ` ( |_ ` A ) ) = ( theta ` A ) )
12	11	3ad2ant1 1082	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( theta ` ( |_ ` A ) ) = ( theta ` A ) )
13	10, 12	oveq12d 6668	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( ( theta ` ( |_ ` B ) ) - ( theta ` ( |_ ` A ) ) ) = ( ( theta ` B ) - ( theta ` A ) ) )
14		flword2 12614	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( |_ ` B ) e. ( ZZ>= ` ( |_ ` A ) ) )
15		chtdif 24884	. . . . . . . 8 |- ( ( |_ ` B ) e. ( ZZ>= ` ( |_ ` A ) ) -> ( ( theta ` ( |_ ` B ) ) - ( theta ` ( |_ ` A ) ) ) = sum_ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ( log ` p ) )
16	14, 15	syl 17	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( ( theta ` ( |_ ` B ) ) - ( theta ` ( |_ ` A ) ) ) = sum_ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ( log ` p ) )
17	13, 16	eqtr3d 2658	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( ( theta ` B ) - ( theta ` A ) ) = sum_ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ( log ` p ) )
18		ssrab2 3687	. . . . . . . . 9 |- { x e. RR | ( exp ` x ) e. NN } C_ RR
19		ax-resscn 9993	. . . . . . . . 9 |- RR C_ CC
20	18, 19	sstri 3612	. . . . . . . 8 |- { x e. RR | ( exp ` x ) e. NN } C_ CC
21	20	a1i 11	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> { x e. RR | ( exp ` x ) e. NN } C_ CC )
22		fveq2 6191	. . . . . . . . . . 11 |- ( x = y -> ( exp ` x ) = ( exp ` y ) )
23	22	eleq1d 2686	. . . . . . . . . 10 |- ( x = y -> ( ( exp ` x ) e. NN <-> ( exp ` y ) e. NN ) )
24	23	elrab 3363	. . . . . . . . 9 |- ( y e. { x e. RR | ( exp ` x ) e. NN } <-> ( y e. RR /\ ( exp ` y ) e. NN ) )
25		fveq2 6191	. . . . . . . . . . 11 |- ( x = z -> ( exp ` x ) = ( exp ` z ) )
26	25	eleq1d 2686	. . . . . . . . . 10 |- ( x = z -> ( ( exp ` x ) e. NN <-> ( exp ` z ) e. NN ) )
27	26	elrab 3363	. . . . . . . . 9 |- ( z e. { x e. RR | ( exp ` x ) e. NN } <-> ( z e. RR /\ ( exp ` z ) e. NN ) )
28		simpll 790	. . . . . . . . . . 11 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> y e. RR )
29		simprl 794	. . . . . . . . . . 11 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> z e. RR )
30	28, 29	readdcld 10069	. . . . . . . . . 10 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( y + z ) e. RR )
31	28	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> y e. CC )
32	29	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> z e. CC )
33		efadd 14824	. . . . . . . . . . . 12 |- ( ( y e. CC /\ z e. CC ) -> ( exp ` ( y + z ) ) = ( ( exp ` y ) x. ( exp ` z ) ) )
34	31, 32, 33	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( exp ` ( y + z ) ) = ( ( exp ` y ) x. ( exp ` z ) ) )
35		nnmulcl 11043	. . . . . . . . . . . 12 |- ( ( ( exp ` y ) e. NN /\ ( exp ` z ) e. NN ) -> ( ( exp ` y ) x. ( exp ` z ) ) e. NN )
36	35	ad2ant2l 782	. . . . . . . . . . 11 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( ( exp ` y ) x. ( exp ` z ) ) e. NN )
37	34, 36	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( exp ` ( y + z ) ) e. NN )
38		fveq2 6191	. . . . . . . . . . . 12 |- ( x = ( y + z ) -> ( exp ` x ) = ( exp ` ( y + z ) ) )
39	38	eleq1d 2686	. . . . . . . . . . 11 |- ( x = ( y + z ) -> ( ( exp ` x ) e. NN <-> ( exp ` ( y + z ) ) e. NN ) )
40	39	elrab 3363	. . . . . . . . . 10 |- ( ( y + z ) e. { x e. RR | ( exp ` x ) e. NN } <-> ( ( y + z ) e. RR /\ ( exp ` ( y + z ) ) e. NN ) )
41	30, 37, 40	sylanbrc 698	. . . . . . . . 9 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( y + z ) e. { x e. RR | ( exp ` x ) e. NN } )
42	24, 27, 41	syl2anb 496	. . . . . . . 8 |- ( ( y e. { x e. RR | ( exp ` x ) e. NN } /\ z e. { x e. RR | ( exp ` x ) e. NN } ) -> ( y + z ) e. { x e. RR | ( exp ` x ) e. NN } )
43	42	adantl 482	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ ( y e. { x e. RR | ( exp ` x ) e. NN } /\ z e. { x e. RR | ( exp ` x ) e. NN } ) ) -> ( y + z ) e. { x e. RR | ( exp ` x ) e. NN } )
44		fzfid 12772	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) e. Fin )
45		inss1 3833	. . . . . . . 8 |- ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) C_ ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) )
46		ssfi 8180	. . . . . . . 8 |- ( ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) e. Fin /\ ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) C_ ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) ) -> ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) e. Fin )
47	44, 45, 46	sylancl 694	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) e. Fin )
48		inss2 3834	. . . . . . . . . . . 12 |- ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) C_ Prime
49		simpr 477	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ) -> p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) )
50	48, 49	sseldi 3601	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ) -> p e. Prime )
51		prmnn 15388	. . . . . . . . . . 11 |- ( p e. Prime -> p e. NN )
52	50, 51	syl 17	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ) -> p e. NN )
53	52	nnrpd 11870	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ) -> p e. RR+ )
54	53	relogcld 24369	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ) -> ( log ` p ) e. RR )
55	53	reeflogd 24370	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ) -> ( exp ` ( log ` p ) ) = p )
56	55, 52	eqeltrd 2701	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ) -> ( exp ` ( log ` p ) ) e. NN )
57		fveq2 6191	. . . . . . . . . 10 |- ( x = ( log ` p ) -> ( exp ` x ) = ( exp ` ( log ` p ) ) )
58	57	eleq1d 2686	. . . . . . . . 9 |- ( x = ( log ` p ) -> ( ( exp ` x ) e. NN <-> ( exp ` ( log ` p ) ) e. NN ) )
59	58	elrab 3363	. . . . . . . 8 |- ( ( log ` p ) e. { x e. RR | ( exp ` x ) e. NN } <-> ( ( log ` p ) e. RR /\ ( exp ` ( log ` p ) ) e. NN ) )
60	54, 56, 59	sylanbrc 698	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ) -> ( log ` p ) e. { x e. RR | ( exp ` x ) e. NN } )
61		0re 10040	. . . . . . . . 9 |- 0 e. RR
62		1nn 11031	. . . . . . . . 9 |- 1 e. NN
63		fveq2 6191	. . . . . . . . . . . 12 |- ( x = 0 -> ( exp ` x ) = ( exp ` 0 ) )
64		ef0 14821	. . . . . . . . . . . 12 |- ( exp ` 0 ) = 1
65	63, 64	syl6eq 2672	. . . . . . . . . . 11 |- ( x = 0 -> ( exp ` x ) = 1 )
66	65	eleq1d 2686	. . . . . . . . . 10 |- ( x = 0 -> ( ( exp ` x ) e. NN <-> 1 e. NN ) )
67	66	elrab 3363	. . . . . . . . 9 |- ( 0 e. { x e. RR | ( exp ` x ) e. NN } <-> ( 0 e. RR /\ 1 e. NN ) )
68	61, 62, 67	mpbir2an 955	. . . . . . . 8 |- 0 e. { x e. RR | ( exp ` x ) e. NN }
69	68	a1i 11	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> 0 e. { x e. RR | ( exp ` x ) e. NN } )
70	21, 43, 47, 60, 69	fsumcllem 14463	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> sum_ p e. ( ( ( ( |_ ` A ) + 1 ) ... ( |_ ` B ) ) i^i Prime ) ( log ` p ) e. { x e. RR | ( exp ` x ) e. NN } )
71	17, 70	eqeltrd 2701	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( ( theta ` B ) - ( theta ` A ) ) e. { x e. RR | ( exp ` x ) e. NN } )
72		fveq2 6191	. . . . . . . 8 |- ( x = ( ( theta ` B ) - ( theta ` A ) ) -> ( exp ` x ) = ( exp ` ( ( theta ` B ) - ( theta ` A ) ) ) )
73	72	eleq1d 2686	. . . . . . 7 |- ( x = ( ( theta ` B ) - ( theta ` A ) ) -> ( ( exp ` x ) e. NN <-> ( exp ` ( ( theta ` B ) - ( theta ` A ) ) ) e. NN ) )
74	73	elrab 3363	. . . . . 6 |- ( ( ( theta ` B ) - ( theta ` A ) ) e. { x e. RR | ( exp ` x ) e. NN } <-> ( ( ( theta ` B ) - ( theta ` A ) ) e. RR /\ ( exp ` ( ( theta ` B ) - ( theta ` A ) ) ) e. NN ) )
75	74	simprbi 480	. . . . 5 |- ( ( ( theta ` B ) - ( theta ` A ) ) e. { x e. RR | ( exp ` x ) e. NN } -> ( exp ` ( ( theta ` B ) - ( theta ` A ) ) ) e. NN )
76	71, 75	syl 17	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( exp ` ( ( theta ` B ) - ( theta ` A ) ) ) e. NN )
77	8, 76	eqeltrrd 2702	. . 3 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( ( exp ` ( theta ` B ) ) / ( exp ` ( theta ` A ) ) ) e. NN )
78	77	nnzd 11481	. 2 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( ( exp ` ( theta ` B ) ) / ( exp ` ( theta ` A ) ) ) e. ZZ )
79		efchtcl 24837	. . . . 5 |- ( A e. RR -> ( exp ` ( theta ` A ) ) e. NN )
80	79	3ad2ant1 1082	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( exp ` ( theta ` A ) ) e. NN )
81	80	nnzd 11481	. . 3 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( exp ` ( theta ` A ) ) e. ZZ )
82	80	nnne0d 11065	. . 3 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( exp ` ( theta ` A ) ) =/= 0 )
83		efchtcl 24837	. . . . 5 |- ( B e. RR -> ( exp ` ( theta ` B ) ) e. NN )
84	83	3ad2ant2 1083	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( exp ` ( theta ` B ) ) e. NN )
85	84	nnzd 11481	. . 3 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( exp ` ( theta ` B ) ) e. ZZ )
86		dvdsval2 14986	. . 3 |- ( ( ( exp ` ( theta ` A ) ) e. ZZ /\ ( exp ` ( theta ` A ) ) =/= 0 /\ ( exp ` ( theta ` B ) ) e. ZZ ) -> ( ( exp ` ( theta ` A ) ) || ( exp ` ( theta ` B ) ) <-> ( ( exp ` ( theta ` B ) ) / ( exp ` ( theta ` A ) ) ) e. ZZ ) )
87	81, 82, 85, 86	syl3anc 1326	. 2 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( ( exp ` ( theta ` A ) ) || ( exp ` ( theta ` B ) ) <-> ( ( exp ` ( theta ` B ) ) / ( exp ` ( theta ` A ) ) ) e. ZZ ) )
88	78, 87	mpbird 247	1 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( exp ` ( theta ` A ) ) || ( exp ` ( theta ` B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   i^i cin 3573   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   |_cfl 12591   sum_csu 14416   expce 14792   || cdvds 14983   Primecprime 15385   logclog 24301   thetaccht 24817

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-dvds 14984  df-prm 15386  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-cht 24823

This theorem is referenced by:  bposlem6  25014