Theorem pntlemh 25288

Description: Lemma for pnt 25303. Bounds on the subintervals in the induction. (Contributed by Mario Carneiro, 13-Apr-2016.)

Hypotheses

Ref	Expression
pntlem1.r	|- R = ( a e. RR+ |-> ( (ψ ` a ) - a ) )
pntlem1.a	|- ( ph -> A e. RR+ )
pntlem1.b	|- ( ph -> B e. RR+ )
pntlem1.l	|- ( ph -> L e. ( 0 (,) 1 ) )
pntlem1.d	|- D = ( A + 1 )
pntlem1.f	|- F = ( ( 1 - ( 1 / D ) ) x. ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) )
pntlem1.u	|- ( ph -> U e. RR+ )
pntlem1.u2	|- ( ph -> U <_ A )
pntlem1.e	|- E = ( U / D )
pntlem1.k	|- K = ( exp ` ( B / E ) )
pntlem1.y	|- ( ph -> ( Y e. RR+ /\ 1 <_ Y ) )
pntlem1.x	|- ( ph -> ( X e. RR+ /\ Y < X ) )
pntlem1.c	|- ( ph -> C e. RR+ )
pntlem1.w	|- W = ( ( ( Y + ( 4 / ( L x. E ) ) ) ^ 2 ) + ( ( ( X x. ( K ^ 2 ) ) ^ 4 ) + ( exp ` ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) ) ) )
pntlem1.z	|- ( ph -> Z e. ( W [,) +oo ) )
pntlem1.m	|- M = ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 )
pntlem1.n	|- N = ( |_ ` ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) )

Assertion

Ref	Expression
pntlemh	|- ( ( ph /\ J e. ( M ... N ) ) -> ( X < ( K ^ J ) /\ ( K ^ J ) <_ ( sqr ` Z ) ) )

Distinct variable group:   E, a

Allowed substitution hints:   ph( a)   A( a)   B( a)   C( a)   D( a)   R( a)   U( a)   F( a)   J( a)   K( a)   L( a)   M( a)   N( a)   W( a)   X( a)   Y( a)   Z( a)

Proof of Theorem pntlemh

Step	Hyp	Ref	Expression
1		pntlem1.x	. . . . . . . . . 10 |- ( ph -> ( X e. RR+ /\ Y < X ) )
2	1	simpld 475	. . . . . . . . 9 |- ( ph -> X e. RR+ )
3	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ J e. ( M ... N ) ) -> X e. RR+ )
4	3	relogcld 24369	. . . . . . 7 |- ( ( ph /\ J e. ( M ... N ) ) -> ( log ` X ) e. RR )
5		pntlem1.r	. . . . . . . . . . . 12 |- R = ( a e. RR+ |-> ( (ψ ` a ) - a ) )
6		pntlem1.a	. . . . . . . . . . . 12 |- ( ph -> A e. RR+ )
7		pntlem1.b	. . . . . . . . . . . 12 |- ( ph -> B e. RR+ )
8		pntlem1.l	. . . . . . . . . . . 12 |- ( ph -> L e. ( 0 (,) 1 ) )
9		pntlem1.d	. . . . . . . . . . . 12 |- D = ( A + 1 )
10		pntlem1.f	. . . . . . . . . . . 12 |- F = ( ( 1 - ( 1 / D ) ) x. ( ( L / (; 3 2 x. B ) ) / ( D ^ 2 ) ) )
11		pntlem1.u	. . . . . . . . . . . 12 |- ( ph -> U e. RR+ )
12		pntlem1.u2	. . . . . . . . . . . 12 |- ( ph -> U <_ A )
13		pntlem1.e	. . . . . . . . . . . 12 |- E = ( U / D )
14		pntlem1.k	. . . . . . . . . . . 12 |- K = ( exp ` ( B / E ) )
15	5, 6, 7, 8, 9, 10, 11, 12, 13, 14	pntlemc 25284	. . . . . . . . . . 11 |- ( ph -> ( E e. RR+ /\ K e. RR+ /\ ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) ) )
16	15	simp2d 1074	. . . . . . . . . 10 |- ( ph -> K e. RR+ )
17	16	rpred 11872	. . . . . . . . 9 |- ( ph -> K e. RR )
18	15	simp3d 1075	. . . . . . . . . 10 |- ( ph -> ( E e. ( 0 (,) 1 ) /\ 1 < K /\ ( U - E ) e. RR+ ) )
19	18	simp2d 1074	. . . . . . . . 9 |- ( ph -> 1 < K )
20	17, 19	rplogcld 24375	. . . . . . . 8 |- ( ph -> ( log ` K ) e. RR+ )
21	20	adantr 481	. . . . . . 7 |- ( ( ph /\ J e. ( M ... N ) ) -> ( log ` K ) e. RR+ )
22	4, 21	rerpdivcld 11903	. . . . . 6 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( log ` X ) / ( log ` K ) ) e. RR )
23		pntlem1.y	. . . . . . . . . 10 |- ( ph -> ( Y e. RR+ /\ 1 <_ Y ) )
24		pntlem1.c	. . . . . . . . . 10 |- ( ph -> C e. RR+ )
25		pntlem1.w	. . . . . . . . . 10 |- W = ( ( ( Y + ( 4 / ( L x. E ) ) ) ^ 2 ) + ( ( ( X x. ( K ^ 2 ) ) ^ 4 ) + ( exp ` ( ( (; 3 2 x. B ) / ( ( U - E ) x. ( L x. ( E ^ 2 ) ) ) ) x. ( ( U x. 3 ) + C ) ) ) ) )
26		pntlem1.z	. . . . . . . . . 10 |- ( ph -> Z e. ( W [,) +oo ) )
27		pntlem1.m	. . . . . . . . . 10 |- M = ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 )
28		pntlem1.n	. . . . . . . . . 10 |- N = ( |_ ` ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) )
29	5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23, 1, 24, 25, 26, 27, 28	pntlemg 25287	. . . . . . . . 9 |- ( ph -> ( M e. NN /\ N e. ( ZZ>= ` M ) /\ ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) <_ ( N - M ) ) )
30	29	simp1d 1073	. . . . . . . 8 |- ( ph -> M e. NN )
31	30	adantr 481	. . . . . . 7 |- ( ( ph /\ J e. ( M ... N ) ) -> M e. NN )
32	31	nnred 11035	. . . . . 6 |- ( ( ph /\ J e. ( M ... N ) ) -> M e. RR )
33		elfzuz 12338	. . . . . . . 8 |- ( J e. ( M ... N ) -> J e. ( ZZ>= ` M ) )
34		eluznn 11758	. . . . . . . 8 |- ( ( M e. NN /\ J e. ( ZZ>= ` M ) ) -> J e. NN )
35	30, 33, 34	syl2an 494	. . . . . . 7 |- ( ( ph /\ J e. ( M ... N ) ) -> J e. NN )
36	35	nnred 11035	. . . . . 6 |- ( ( ph /\ J e. ( M ... N ) ) -> J e. RR )
37		flltp1 12601	. . . . . . . 8 |- ( ( ( log ` X ) / ( log ` K ) ) e. RR -> ( ( log ` X ) / ( log ` K ) ) < ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 ) )
38	22, 37	syl 17	. . . . . . 7 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( log ` X ) / ( log ` K ) ) < ( ( |_ ` ( ( log ` X ) / ( log ` K ) ) ) + 1 ) )
39	38, 27	syl6breqr 4695	. . . . . 6 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( log ` X ) / ( log ` K ) ) < M )
40		elfzle1 12344	. . . . . . 7 |- ( J e. ( M ... N ) -> M <_ J )
41	40	adantl 482	. . . . . 6 |- ( ( ph /\ J e. ( M ... N ) ) -> M <_ J )
42	22, 32, 36, 39, 41	ltletrd 10197	. . . . 5 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( log ` X ) / ( log ` K ) ) < J )
43	4, 36, 21	ltdivmul2d 11924	. . . . 5 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( ( log ` X ) / ( log ` K ) ) < J <-> ( log ` X ) < ( J x. ( log ` K ) ) ) )
44	42, 43	mpbid 222	. . . 4 |- ( ( ph /\ J e. ( M ... N ) ) -> ( log ` X ) < ( J x. ( log ` K ) ) )
45	16	adantr 481	. . . . 5 |- ( ( ph /\ J e. ( M ... N ) ) -> K e. RR+ )
46		elfzelz 12342	. . . . . 6 |- ( J e. ( M ... N ) -> J e. ZZ )
47	46	adantl 482	. . . . 5 |- ( ( ph /\ J e. ( M ... N ) ) -> J e. ZZ )
48		relogexp 24342	. . . . 5 |- ( ( K e. RR+ /\ J e. ZZ ) -> ( log ` ( K ^ J ) ) = ( J x. ( log ` K ) ) )
49	45, 47, 48	syl2anc 693	. . . 4 |- ( ( ph /\ J e. ( M ... N ) ) -> ( log ` ( K ^ J ) ) = ( J x. ( log ` K ) ) )
50	44, 49	breqtrrd 4681	. . 3 |- ( ( ph /\ J e. ( M ... N ) ) -> ( log ` X ) < ( log ` ( K ^ J ) ) )
51	45, 47	rpexpcld 13032	. . . 4 |- ( ( ph /\ J e. ( M ... N ) ) -> ( K ^ J ) e. RR+ )
52		logltb 24346	. . . 4 |- ( ( X e. RR+ /\ ( K ^ J ) e. RR+ ) -> ( X < ( K ^ J ) <-> ( log ` X ) < ( log ` ( K ^ J ) ) ) )
53	3, 51, 52	syl2anc 693	. . 3 |- ( ( ph /\ J e. ( M ... N ) ) -> ( X < ( K ^ J ) <-> ( log ` X ) < ( log ` ( K ^ J ) ) ) )
54	50, 53	mpbird 247	. 2 |- ( ( ph /\ J e. ( M ... N ) ) -> X < ( K ^ J ) )
55	49	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ J e. ( M ... N ) ) -> ( 2 x. ( log ` ( K ^ J ) ) ) = ( 2 x. ( J x. ( log ` K ) ) ) )
56		2z 11409	. . . . . . . 8 |- 2 e. ZZ
57		relogexp 24342	. . . . . . . 8 |- ( ( ( K ^ J ) e. RR+ /\ 2 e. ZZ ) -> ( log ` ( ( K ^ J ) ^ 2 ) ) = ( 2 x. ( log ` ( K ^ J ) ) ) )
58	51, 56, 57	sylancl 694	. . . . . . 7 |- ( ( ph /\ J e. ( M ... N ) ) -> ( log ` ( ( K ^ J ) ^ 2 ) ) = ( 2 x. ( log ` ( K ^ J ) ) ) )
59		2cnd 11093	. . . . . . . 8 |- ( ( ph /\ J e. ( M ... N ) ) -> 2 e. CC )
60	36	recnd 10068	. . . . . . . 8 |- ( ( ph /\ J e. ( M ... N ) ) -> J e. CC )
61	45	relogcld 24369	. . . . . . . . 9 |- ( ( ph /\ J e. ( M ... N ) ) -> ( log ` K ) e. RR )
62	61	recnd 10068	. . . . . . . 8 |- ( ( ph /\ J e. ( M ... N ) ) -> ( log ` K ) e. CC )
63	59, 60, 62	mulassd 10063	. . . . . . 7 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( 2 x. J ) x. ( log ` K ) ) = ( 2 x. ( J x. ( log ` K ) ) ) )
64	55, 58, 63	3eqtr4d 2666	. . . . . 6 |- ( ( ph /\ J e. ( M ... N ) ) -> ( log ` ( ( K ^ J ) ^ 2 ) ) = ( ( 2 x. J ) x. ( log ` K ) ) )
65		elfzle2 12345	. . . . . . . . . . 11 |- ( J e. ( M ... N ) -> J <_ N )
66	65	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ J e. ( M ... N ) ) -> J <_ N )
67	66, 28	syl6breq 4694	. . . . . . . . 9 |- ( ( ph /\ J e. ( M ... N ) ) -> J <_ ( |_ ` ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) ) )
68	5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23, 1, 24, 25, 26	pntlemb 25286	. . . . . . . . . . . . . . 15 |- ( ph -> ( Z e. RR+ /\ ( 1 < Z /\ _e <_ ( sqr ` Z ) /\ ( sqr ` Z ) <_ ( Z / Y ) ) /\ ( ( 4 / ( L x. E ) ) <_ ( sqr ` Z ) /\ ( ( ( log ` X ) / ( log ` K ) ) + 2 ) <_ ( ( ( log ` Z ) / ( log ` K ) ) / 4 ) /\ ( ( U x. 3 ) + C ) <_ ( ( ( U - E ) x. ( ( L x. ( E ^ 2 ) ) / (; 3 2 x. B ) ) ) x. ( log ` Z ) ) ) ) )
69	68	simp1d 1073	. . . . . . . . . . . . . 14 |- ( ph -> Z e. RR+ )
70	69	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ J e. ( M ... N ) ) -> Z e. RR+ )
71	70	relogcld 24369	. . . . . . . . . . . 12 |- ( ( ph /\ J e. ( M ... N ) ) -> ( log ` Z ) e. RR )
72	71, 21	rerpdivcld 11903	. . . . . . . . . . 11 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( log ` Z ) / ( log ` K ) ) e. RR )
73	72	rehalfcld 11279	. . . . . . . . . 10 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) e. RR )
74		flge 12606	. . . . . . . . . 10 |- ( ( ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) e. RR /\ J e. ZZ ) -> ( J <_ ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) <-> J <_ ( |_ ` ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) ) ) )
75	73, 47, 74	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ J e. ( M ... N ) ) -> ( J <_ ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) <-> J <_ ( |_ ` ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) ) ) )
76	67, 75	mpbird 247	. . . . . . . 8 |- ( ( ph /\ J e. ( M ... N ) ) -> J <_ ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) )
77		2re 11090	. . . . . . . . . 10 |- 2 e. RR
78	77	a1i 11	. . . . . . . . 9 |- ( ( ph /\ J e. ( M ... N ) ) -> 2 e. RR )
79		2pos 11112	. . . . . . . . . 10 |- 0 < 2
80	79	a1i 11	. . . . . . . . 9 |- ( ( ph /\ J e. ( M ... N ) ) -> 0 < 2 )
81		lemuldiv2 10904	. . . . . . . . 9 |- ( ( J e. RR /\ ( ( log ` Z ) / ( log ` K ) ) e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( 2 x. J ) <_ ( ( log ` Z ) / ( log ` K ) ) <-> J <_ ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) ) )
82	36, 72, 78, 80, 81	syl112anc 1330	. . . . . . . 8 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( 2 x. J ) <_ ( ( log ` Z ) / ( log ` K ) ) <-> J <_ ( ( ( log ` Z ) / ( log ` K ) ) / 2 ) ) )
83	76, 82	mpbird 247	. . . . . . 7 |- ( ( ph /\ J e. ( M ... N ) ) -> ( 2 x. J ) <_ ( ( log ` Z ) / ( log ` K ) ) )
84		remulcl 10021	. . . . . . . . 9 |- ( ( 2 e. RR /\ J e. RR ) -> ( 2 x. J ) e. RR )
85	77, 36, 84	sylancr 695	. . . . . . . 8 |- ( ( ph /\ J e. ( M ... N ) ) -> ( 2 x. J ) e. RR )
86	85, 71, 21	lemuldivd 11921	. . . . . . 7 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( ( 2 x. J ) x. ( log ` K ) ) <_ ( log ` Z ) <-> ( 2 x. J ) <_ ( ( log ` Z ) / ( log ` K ) ) ) )
87	83, 86	mpbird 247	. . . . . 6 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( 2 x. J ) x. ( log ` K ) ) <_ ( log ` Z ) )
88	64, 87	eqbrtrd 4675	. . . . 5 |- ( ( ph /\ J e. ( M ... N ) ) -> ( log ` ( ( K ^ J ) ^ 2 ) ) <_ ( log ` Z ) )
89		rpexpcl 12879	. . . . . . 7 |- ( ( ( K ^ J ) e. RR+ /\ 2 e. ZZ ) -> ( ( K ^ J ) ^ 2 ) e. RR+ )
90	51, 56, 89	sylancl 694	. . . . . 6 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( K ^ J ) ^ 2 ) e. RR+ )
91	90, 70	logled 24373	. . . . 5 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( ( K ^ J ) ^ 2 ) <_ Z <-> ( log ` ( ( K ^ J ) ^ 2 ) ) <_ ( log ` Z ) ) )
92	88, 91	mpbird 247	. . . 4 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( K ^ J ) ^ 2 ) <_ Z )
93	70	rprege0d 11879	. . . . 5 |- ( ( ph /\ J e. ( M ... N ) ) -> ( Z e. RR /\ 0 <_ Z ) )
94		resqrtth 13996	. . . . 5 |- ( ( Z e. RR /\ 0 <_ Z ) -> ( ( sqr ` Z ) ^ 2 ) = Z )
95	93, 94	syl 17	. . . 4 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( sqr ` Z ) ^ 2 ) = Z )
96	92, 95	breqtrrd 4681	. . 3 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( K ^ J ) ^ 2 ) <_ ( ( sqr ` Z ) ^ 2 ) )
97	51	rprege0d 11879	. . . 4 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( K ^ J ) e. RR /\ 0 <_ ( K ^ J ) ) )
98	70	rpsqrtcld 14150	. . . . 5 |- ( ( ph /\ J e. ( M ... N ) ) -> ( sqr ` Z ) e. RR+ )
99	98	rprege0d 11879	. . . 4 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( sqr ` Z ) e. RR /\ 0 <_ ( sqr ` Z ) ) )
100		le2sq 12938	. . . 4 |- ( ( ( ( K ^ J ) e. RR /\ 0 <_ ( K ^ J ) ) /\ ( ( sqr ` Z ) e. RR /\ 0 <_ ( sqr ` Z ) ) ) -> ( ( K ^ J ) <_ ( sqr ` Z ) <-> ( ( K ^ J ) ^ 2 ) <_ ( ( sqr ` Z ) ^ 2 ) ) )
101	97, 99, 100	syl2anc 693	. . 3 |- ( ( ph /\ J e. ( M ... N ) ) -> ( ( K ^ J ) <_ ( sqr ` Z ) <-> ( ( K ^ J ) ^ 2 ) <_ ( ( sqr ` Z ) ^ 2 ) ) )
102	96, 101	mpbird 247	. 2 |- ( ( ph /\ J e. ( M ... N ) ) -> ( K ^ J ) <_ ( sqr ` Z ) )
103	54, 102	jca 554	1 |- ( ( ph /\ J e. ( M ... N ) ) -> ( X < ( K ^ J ) /\ ( K ^ J ) <_ ( sqr ` Z ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   ZZcz 11377  ;cdc 11493   ZZ>=cuz 11687   RR+crp 11832   (,)cioo 12175   [,)cico 12177   ...cfz 12326   |_cfl 12591   ^cexp 12860   sqrcsqrt 13973   expce 14792   _eceu 14793   logclog 24301  ψcchp 24819

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-e 14799  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

This theorem is referenced by:  pntlemr  25291  pntlemj  25292  pntlemi  25293  pntlemf  25294