Theorem fsumvma2 24939

Description: Apply fsumvma 24938 for the common case of all numbers less than a real number A. (Contributed by Mario Carneiro, 30-Apr-2016.)

Hypotheses

Ref	Expression
fsumvma2.1	|- ( x = ( p ^ k ) -> B = C )
fsumvma2.2	|- ( ph -> A e. RR )
fsumvma2.3	|- ( ( ph /\ x e. ( 1 ... ( |_ ` A ) ) ) -> B e. CC )
fsumvma2.4	|- ( ( ph /\ ( x e. ( 1 ... ( |_ ` A ) ) /\ (Λ ` x ) = 0 ) ) -> B = 0 )

Assertion

Ref	Expression
fsumvma2	|- ( ph -> sum_ x e. ( 1 ... ( |_ ` A ) ) B = sum_ p e. ( ( 0 [,] A ) i^i Prime ) sum_ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) C )

Distinct variable groups:   k, p, x, A   x, C   ph, k, p, x   B, k, p

Allowed substitution hints:   B( x)   C( k, p)

Proof of Theorem fsumvma2

Step	Hyp	Ref	Expression
1		fsumvma2.1	. 2 |- ( x = ( p ^ k ) -> B = C )
2		fzfid 12772	. 2 |- ( ph -> ( 1 ... ( |_ ` A ) ) e. Fin )
3		elfznn 12370	. . . 4 |- ( x e. ( 1 ... ( |_ ` A ) ) -> x e. NN )
4	3	ssriv 3607	. . 3 |- ( 1 ... ( |_ ` A ) ) C_ NN
5	4	a1i 11	. 2 |- ( ph -> ( 1 ... ( |_ ` A ) ) C_ NN )
6		fsumvma2.2	. . 3 |- ( ph -> A e. RR )
7		ppifi 24832	. . 3 |- ( A e. RR -> ( ( 0 [,] A ) i^i Prime ) e. Fin )
8	6, 7	syl 17	. 2 |- ( ph -> ( ( 0 [,] A ) i^i Prime ) e. Fin )
9		elin 3796	. . . . . 6 |- ( p e. ( ( 0 [,] A ) i^i Prime ) <-> ( p e. ( 0 [,] A ) /\ p e. Prime ) )
10	9	simprbi 480	. . . . 5 |- ( p e. ( ( 0 [,] A ) i^i Prime ) -> p e. Prime )
11		elfznn 12370	. . . . 5 |- ( k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) -> k e. NN )
12	10, 11	anim12i 590	. . . 4 |- ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) -> ( p e. Prime /\ k e. NN ) )
13	12	pm4.71ri 665	. . 3 |- ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) <-> ( ( p e. Prime /\ k e. NN ) /\ ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) ) )
14	6	adantr 481	. . . . . 6 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> A e. RR )
15		prmnn 15388	. . . . . . . . 9 |- ( p e. Prime -> p e. NN )
16	15	ad2antrl 764	. . . . . . . 8 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> p e. NN )
17		nnnn0 11299	. . . . . . . . 9 |- ( k e. NN -> k e. NN0 )
18	17	ad2antll 765	. . . . . . . 8 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> k e. NN0 )
19	16, 18	nnexpcld 13030	. . . . . . 7 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ k ) e. NN )
20	19	nnzd 11481	. . . . . 6 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ k ) e. ZZ )
21		flge 12606	. . . . . 6 |- ( ( A e. RR /\ ( p ^ k ) e. ZZ ) -> ( ( p ^ k ) <_ A <-> ( p ^ k ) <_ ( |_ ` A ) ) )
22	14, 20, 21	syl2anc 693	. . . . 5 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p ^ k ) <_ A <-> ( p ^ k ) <_ ( |_ ` A ) ) )
23		simplrl 800	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> p e. Prime )
24	23, 15	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> p e. NN )
25	24	nnrpd 11870	. . . . . . . . . . 11 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> p e. RR+ )
26		simplrr 801	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> k e. NN )
27	26	nnzd 11481	. . . . . . . . . . 11 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> k e. ZZ )
28		relogexp 24342	. . . . . . . . . . 11 |- ( ( p e. RR+ /\ k e. ZZ ) -> ( log ` ( p ^ k ) ) = ( k x. ( log ` p ) ) )
29	25, 27, 28	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( log ` ( p ^ k ) ) = ( k x. ( log ` p ) ) )
30	29	breq1d 4663	. . . . . . . . 9 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( log ` ( p ^ k ) ) <_ ( log ` A ) <-> ( k x. ( log ` p ) ) <_ ( log ` A ) ) )
31	26	nnred 11035	. . . . . . . . . 10 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> k e. RR )
32	14	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> A e. RR )
33		0red 10041	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> 0 e. RR )
34	16	nnred 11035	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> p e. RR )
35	34	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> p e. RR )
36	24	nngt0d 11064	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> 0 < p )
37		0red 10041	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> 0 e. RR )
38		nnnn0 11299	. . . . . . . . . . . . . . . . 17 |- ( p e. NN -> p e. NN0 )
39	16, 38	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> p e. NN0 )
40	39	nn0ge0d 11354	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> 0 <_ p )
41		elicc2 12238	. . . . . . . . . . . . . . . . 17 |- ( ( 0 e. RR /\ A e. RR ) -> ( p e. ( 0 [,] A ) <-> ( p e. RR /\ 0 <_ p /\ p <_ A ) ) )
42		df-3an 1039	. . . . . . . . . . . . . . . . 17 |- ( ( p e. RR /\ 0 <_ p /\ p <_ A ) <-> ( ( p e. RR /\ 0 <_ p ) /\ p <_ A ) )
43	41, 42	syl6bb 276	. . . . . . . . . . . . . . . 16 |- ( ( 0 e. RR /\ A e. RR ) -> ( p e. ( 0 [,] A ) <-> ( ( p e. RR /\ 0 <_ p ) /\ p <_ A ) ) )
44	43	baibd 948	. . . . . . . . . . . . . . 15 |- ( ( ( 0 e. RR /\ A e. RR ) /\ ( p e. RR /\ 0 <_ p ) ) -> ( p e. ( 0 [,] A ) <-> p <_ A ) )
45	37, 14, 34, 40, 44	syl22anc 1327	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p e. ( 0 [,] A ) <-> p <_ A ) )
46	45	biimpa 501	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> p <_ A )
47	33, 35, 32, 36, 46	ltletrd 10197	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> 0 < A )
48	32, 47	elrpd 11869	. . . . . . . . . . 11 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> A e. RR+ )
49	48	relogcld 24369	. . . . . . . . . 10 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( log ` A ) e. RR )
50		prmuz2 15408	. . . . . . . . . . 11 |- ( p e. Prime -> p e. ( ZZ>= ` 2 ) )
51		eluzelre 11698	. . . . . . . . . . . 12 |- ( p e. ( ZZ>= ` 2 ) -> p e. RR )
52		eluz2b2 11761	. . . . . . . . . . . . 13 |- ( p e. ( ZZ>= ` 2 ) <-> ( p e. NN /\ 1 < p ) )
53	52	simprbi 480	. . . . . . . . . . . 12 |- ( p e. ( ZZ>= ` 2 ) -> 1 < p )
54	51, 53	rplogcld 24375	. . . . . . . . . . 11 |- ( p e. ( ZZ>= ` 2 ) -> ( log ` p ) e. RR+ )
55	23, 50, 54	3syl 18	. . . . . . . . . 10 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( log ` p ) e. RR+ )
56	31, 49, 55	lemuldivd 11921	. . . . . . . . 9 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( k x. ( log ` p ) ) <_ ( log ` A ) <-> k <_ ( ( log ` A ) / ( log ` p ) ) ) )
57	49, 55	rerpdivcld 11903	. . . . . . . . . 10 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( log ` A ) / ( log ` p ) ) e. RR )
58		flge 12606	. . . . . . . . . 10 |- ( ( ( ( log ` A ) / ( log ` p ) ) e. RR /\ k e. ZZ ) -> ( k <_ ( ( log ` A ) / ( log ` p ) ) <-> k <_ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) )
59	57, 27, 58	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( k <_ ( ( log ` A ) / ( log ` p ) ) <-> k <_ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) )
60	30, 56, 59	3bitrd 294	. . . . . . . 8 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( log ` ( p ^ k ) ) <_ ( log ` A ) <-> k <_ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) )
61	19	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( p ^ k ) e. NN )
62	61	nnrpd 11870	. . . . . . . . 9 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( p ^ k ) e. RR+ )
63	62, 48	logled 24373	. . . . . . . 8 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( p ^ k ) <_ A <-> ( log ` ( p ^ k ) ) <_ ( log ` A ) ) )
64		simprr 796	. . . . . . . . . . 11 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> k e. NN )
65		nnuz 11723	. . . . . . . . . . 11 |- NN = ( ZZ>= ` 1 )
66	64, 65	syl6eleq 2711	. . . . . . . . . 10 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> k e. ( ZZ>= ` 1 ) )
67	66	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> k e. ( ZZ>= ` 1 ) )
68	57	flcld 12599	. . . . . . . . 9 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) e. ZZ )
69		elfz5 12334	. . . . . . . . 9 |- ( ( k e. ( ZZ>= ` 1 ) /\ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) e. ZZ ) -> ( k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) <-> k <_ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) )
70	67, 68, 69	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) <-> k <_ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) )
71	60, 63, 70	3bitr4d 300	. . . . . . 7 |- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( p ^ k ) <_ A <-> k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) )
72	71	pm5.32da 673	. . . . . 6 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p e. ( 0 [,] A ) /\ ( p ^ k ) <_ A ) <-> ( p e. ( 0 [,] A ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) ) )
73	16	nncnd 11036	. . . . . . . . . . 11 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> p e. CC )
74	73	exp1d 13003	. . . . . . . . . 10 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ 1 ) = p )
75	16	nnge1d 11063	. . . . . . . . . . 11 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> 1 <_ p )
76	34, 75, 66	leexp2ad 13041	. . . . . . . . . 10 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ 1 ) <_ ( p ^ k ) )
77	74, 76	eqbrtrrd 4677	. . . . . . . . 9 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> p <_ ( p ^ k ) )
78	19	nnred 11035	. . . . . . . . . 10 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ k ) e. RR )
79		letr 10131	. . . . . . . . . 10 |- ( ( p e. RR /\ ( p ^ k ) e. RR /\ A e. RR ) -> ( ( p <_ ( p ^ k ) /\ ( p ^ k ) <_ A ) -> p <_ A ) )
80	34, 78, 14, 79	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p <_ ( p ^ k ) /\ ( p ^ k ) <_ A ) -> p <_ A ) )
81	77, 80	mpand 711	. . . . . . . 8 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p ^ k ) <_ A -> p <_ A ) )
82	81, 45	sylibrd 249	. . . . . . 7 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p ^ k ) <_ A -> p e. ( 0 [,] A ) ) )
83	82	pm4.71rd 667	. . . . . 6 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p ^ k ) <_ A <-> ( p e. ( 0 [,] A ) /\ ( p ^ k ) <_ A ) ) )
84	9	rbaib 947	. . . . . . . 8 |- ( p e. Prime -> ( p e. ( ( 0 [,] A ) i^i Prime ) <-> p e. ( 0 [,] A ) ) )
85	84	ad2antrl 764	. . . . . . 7 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p e. ( ( 0 [,] A ) i^i Prime ) <-> p e. ( 0 [,] A ) ) )
86	85	anbi1d 741	. . . . . 6 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) <-> ( p e. ( 0 [,] A ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) ) )
87	72, 83, 86	3bitr4rd 301	. . . . 5 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) <-> ( p ^ k ) <_ A ) )
88	19, 65	syl6eleq 2711	. . . . . 6 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ k ) e. ( ZZ>= ` 1 ) )
89	14	flcld 12599	. . . . . 6 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( |_ ` A ) e. ZZ )
90		elfz5 12334	. . . . . 6 |- ( ( ( p ^ k ) e. ( ZZ>= ` 1 ) /\ ( |_ ` A ) e. ZZ ) -> ( ( p ^ k ) e. ( 1 ... ( |_ ` A ) ) <-> ( p ^ k ) <_ ( |_ ` A ) ) )
91	88, 89, 90	syl2anc 693	. . . . 5 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p ^ k ) e. ( 1 ... ( |_ ` A ) ) <-> ( p ^ k ) <_ ( |_ ` A ) ) )
92	22, 87, 91	3bitr4d 300	. . . 4 |- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) <-> ( p ^ k ) e. ( 1 ... ( |_ ` A ) ) ) )
93	92	pm5.32da 673	. . 3 |- ( ph -> ( ( ( p e. Prime /\ k e. NN ) /\ ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) ) <-> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. ( 1 ... ( |_ ` A ) ) ) ) )
94	13, 93	syl5bb 272	. 2 |- ( ph -> ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) <-> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. ( 1 ... ( |_ ` A ) ) ) ) )
95		fsumvma2.3	. 2 |- ( ( ph /\ x e. ( 1 ... ( |_ ` A ) ) ) -> B e. CC )
96		fsumvma2.4	. 2 |- ( ( ph /\ ( x e. ( 1 ... ( |_ ` A ) ) /\ (Λ ` x ) = 0 ) ) -> B = 0 )
97	1, 2, 5, 8, 94, 95, 96	fsumvma 24938	1 |- ( ph -> sum_ x e. ( 1 ... ( |_ ` A ) ) B = sum_ p e. ( ( 0 [,] A ) i^i Prime ) sum_ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   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   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   [,]cicc 12178   ...cfz 12326   |_cfl 12591   ^cexp 12860   sum_csu 14416   Primecprime 15385   logclog 24301  Λcvma 24818

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-gcd 15217  df-prm 15386  df-pc 15542  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-vma 24824

This theorem is referenced by:  chpval2  24943  rplogsumlem2  25174  rpvmasumlem  25176