Theorem emcllem7 24728

Description: Lemma for emcl 24729 and harmonicbnd 24730. Derive bounds on gamma as F ( 1 ) and G ( 1 ). (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 9-Apr-2016.)

Hypotheses

Ref	Expression
emcl.1	|- F = ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` n ) ) )
emcl.2	|- G = ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) )
emcl.3	|- H = ( n e. NN |-> ( log ` ( 1 + ( 1 / n ) ) ) )
emcl.4	|- T = ( n e. NN |-> ( ( 1 / n ) - ( log ` ( 1 + ( 1 / n ) ) ) ) )

Assertion

Ref	Expression
emcllem7	|- ( gamma e. ( ( 1 - ( log ` 2 ) ) [,] 1 ) /\ F : NN --> ( gamma [,] 1 ) /\ G : NN --> ( ( 1 - ( log ` 2 ) ) [,] gamma ) )

Distinct variable groups:   m, H   m, n, T

Allowed substitution hints:   F( m, n)   G( m, n)   H( n)

Proof of Theorem emcllem7

Dummy variables i k x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 11723	. . . . 5 |- NN = ( ZZ>= ` 1 )
2		1zzd 11408	. . . . 5 |- ( T. -> 1 e. ZZ )
3		emcl.1	. . . . . . . 8 |- F = ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` n ) ) )
4		emcl.2	. . . . . . . 8 |- G = ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) )
5		emcl.3	. . . . . . . 8 |- H = ( n e. NN |-> ( log ` ( 1 + ( 1 / n ) ) ) )
6		emcl.4	. . . . . . . 8 |- T = ( n e. NN |-> ( ( 1 / n ) - ( log ` ( 1 + ( 1 / n ) ) ) ) )
7	3, 4, 5, 6	emcllem6 24727	. . . . . . 7 |- ( F ~~> gamma /\ G ~~> gamma )
8	7	simpri 478	. . . . . 6 |- G ~~> gamma
9	8	a1i 11	. . . . 5 |- ( T. -> G ~~> gamma )
10	3, 4	emcllem1 24722	. . . . . . . 8 |- ( F : NN --> RR /\ G : NN --> RR )
11	10	simpri 478	. . . . . . 7 |- G : NN --> RR
12	11	ffvelrni 6358	. . . . . 6 |- ( k e. NN -> ( G ` k ) e. RR )
13	12	adantl 482	. . . . 5 |- ( ( T. /\ k e. NN ) -> ( G ` k ) e. RR )
14	1, 2, 9, 13	climrecl 14314	. . . 4 |- ( T. -> gamma e. RR )
15		1nn 11031	. . . . 5 |- 1 e. NN
16		simpr 477	. . . . . . 7 |- ( ( T. /\ i e. NN ) -> i e. NN )
17	8	a1i 11	. . . . . . 7 |- ( ( T. /\ i e. NN ) -> G ~~> gamma )
18	12	adantl 482	. . . . . . 7 |- ( ( ( T. /\ i e. NN ) /\ k e. NN ) -> ( G ` k ) e. RR )
19	3, 4	emcllem2 24723	. . . . . . . . 9 |- ( k e. NN -> ( ( F ` ( k + 1 ) ) <_ ( F ` k ) /\ ( G ` k ) <_ ( G ` ( k + 1 ) ) ) )
20	19	simprd 479	. . . . . . . 8 |- ( k e. NN -> ( G ` k ) <_ ( G ` ( k + 1 ) ) )
21	20	adantl 482	. . . . . . 7 |- ( ( ( T. /\ i e. NN ) /\ k e. NN ) -> ( G ` k ) <_ ( G ` ( k + 1 ) ) )
22	1, 16, 17, 18, 21	climub 14392	. . . . . 6 |- ( ( T. /\ i e. NN ) -> ( G ` i ) <_ gamma )
23	22	ralrimiva 2966	. . . . 5 |- ( T. -> A. i e. NN ( G ` i ) <_ gamma )
24		fveq2 6191	. . . . . . . 8 |- ( i = 1 -> ( G ` i ) = ( G ` 1 ) )
25		oveq2 6658	. . . . . . . . . . . . 13 |- ( n = 1 -> ( 1 ... n ) = ( 1 ... 1 ) )
26	25	sumeq1d 14431	. . . . . . . . . . . 12 |- ( n = 1 -> sum_ m e. ( 1 ... n ) ( 1 / m ) = sum_ m e. ( 1 ... 1 ) ( 1 / m ) )
27		1z 11407	. . . . . . . . . . . . 13 |- 1 e. ZZ
28		ax-1cn 9994	. . . . . . . . . . . . 13 |- 1 e. CC
29		oveq2 6658	. . . . . . . . . . . . . . 15 |- ( m = 1 -> ( 1 / m ) = ( 1 / 1 ) )
30		1div1e1 10717	. . . . . . . . . . . . . . 15 |- ( 1 / 1 ) = 1
31	29, 30	syl6eq 2672	. . . . . . . . . . . . . 14 |- ( m = 1 -> ( 1 / m ) = 1 )
32	31	fsum1 14476	. . . . . . . . . . . . 13 |- ( ( 1 e. ZZ /\ 1 e. CC ) -> sum_ m e. ( 1 ... 1 ) ( 1 / m ) = 1 )
33	27, 28, 32	mp2an 708	. . . . . . . . . . . 12 |- sum_ m e. ( 1 ... 1 ) ( 1 / m ) = 1
34	26, 33	syl6eq 2672	. . . . . . . . . . 11 |- ( n = 1 -> sum_ m e. ( 1 ... n ) ( 1 / m ) = 1 )
35		oveq1 6657	. . . . . . . . . . . . 13 |- ( n = 1 -> ( n + 1 ) = ( 1 + 1 ) )
36		df-2 11079	. . . . . . . . . . . . 13 |- 2 = ( 1 + 1 )
37	35, 36	syl6eqr 2674	. . . . . . . . . . . 12 |- ( n = 1 -> ( n + 1 ) = 2 )
38	37	fveq2d 6195	. . . . . . . . . . 11 |- ( n = 1 -> ( log ` ( n + 1 ) ) = ( log ` 2 ) )
39	34, 38	oveq12d 6668	. . . . . . . . . 10 |- ( n = 1 -> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) = ( 1 - ( log ` 2 ) ) )
40		1re 10039	. . . . . . . . . . . 12 |- 1 e. RR
41		2rp 11837	. . . . . . . . . . . . 13 |- 2 e. RR+
42		relogcl 24322	. . . . . . . . . . . . 13 |- ( 2 e. RR+ -> ( log ` 2 ) e. RR )
43	41, 42	ax-mp 5	. . . . . . . . . . . 12 |- ( log ` 2 ) e. RR
44	40, 43	resubcli 10343	. . . . . . . . . . 11 |- ( 1 - ( log ` 2 ) ) e. RR
45	44	elexi 3213	. . . . . . . . . 10 |- ( 1 - ( log ` 2 ) ) e. _V
46	39, 4, 45	fvmpt 6282	. . . . . . . . 9 |- ( 1 e. NN -> ( G ` 1 ) = ( 1 - ( log ` 2 ) ) )
47	15, 46	ax-mp 5	. . . . . . . 8 |- ( G ` 1 ) = ( 1 - ( log ` 2 ) )
48	24, 47	syl6eq 2672	. . . . . . 7 |- ( i = 1 -> ( G ` i ) = ( 1 - ( log ` 2 ) ) )
49	48	breq1d 4663	. . . . . 6 |- ( i = 1 -> ( ( G ` i ) <_ gamma <-> ( 1 - ( log ` 2 ) ) <_ gamma ) )
50	49	rspcva 3307	. . . . 5 |- ( ( 1 e. NN /\ A. i e. NN ( G ` i ) <_ gamma ) -> ( 1 - ( log ` 2 ) ) <_ gamma )
51	15, 23, 50	sylancr 695	. . . 4 |- ( T. -> ( 1 - ( log ` 2 ) ) <_ gamma )
52		fveq2 6191	. . . . . . . . . . . 12 |- ( x = i -> ( F ` x ) = ( F ` i ) )
53	52	negeqd 10275	. . . . . . . . . . 11 |- ( x = i -> -u ( F ` x ) = -u ( F ` i ) )
54		eqid 2622	. . . . . . . . . . 11 |- ( x e. NN |-> -u ( F ` x ) ) = ( x e. NN |-> -u ( F ` x ) )
55		negex 10279	. . . . . . . . . . 11 |- -u ( F ` i ) e. _V
56	53, 54, 55	fvmpt 6282	. . . . . . . . . 10 |- ( i e. NN -> ( ( x e. NN |-> -u ( F ` x ) ) ` i ) = -u ( F ` i ) )
57	56	adantl 482	. . . . . . . . 9 |- ( ( T. /\ i e. NN ) -> ( ( x e. NN |-> -u ( F ` x ) ) ` i ) = -u ( F ` i ) )
58	7	simpli 474	. . . . . . . . . . . . 13 |- F ~~> gamma
59	58	a1i 11	. . . . . . . . . . . 12 |- ( T. -> F ~~> gamma )
60		0cnd 10033	. . . . . . . . . . . 12 |- ( T. -> 0 e. CC )
61		nnex 11026	. . . . . . . . . . . . . 14 |- NN e. _V
62	61	mptex 6486	. . . . . . . . . . . . 13 |- ( x e. NN |-> -u ( F ` x ) ) e. _V
63	62	a1i 11	. . . . . . . . . . . 12 |- ( T. -> ( x e. NN |-> -u ( F ` x ) ) e. _V )
64	10	simpli 474	. . . . . . . . . . . . . . 15 |- F : NN --> RR
65	64	ffvelrni 6358	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( F ` k ) e. RR )
66	65	adantl 482	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( F ` k ) e. RR )
67	66	recnd 10068	. . . . . . . . . . . 12 |- ( ( T. /\ k e. NN ) -> ( F ` k ) e. CC )
68		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( x = k -> ( F ` x ) = ( F ` k ) )
69	68	negeqd 10275	. . . . . . . . . . . . . . 15 |- ( x = k -> -u ( F ` x ) = -u ( F ` k ) )
70		negex 10279	. . . . . . . . . . . . . . 15 |- -u ( F ` k ) e. _V
71	69, 54, 70	fvmpt 6282	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( ( x e. NN |-> -u ( F ` x ) ) ` k ) = -u ( F ` k ) )
72	71	adantl 482	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( ( x e. NN |-> -u ( F ` x ) ) ` k ) = -u ( F ` k ) )
73		df-neg 10269	. . . . . . . . . . . . 13 |- -u ( F ` k ) = ( 0 - ( F ` k ) )
74	72, 73	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( T. /\ k e. NN ) -> ( ( x e. NN |-> -u ( F ` x ) ) ` k ) = ( 0 - ( F ` k ) ) )
75	1, 2, 59, 60, 63, 67, 74	climsubc2 14369	. . . . . . . . . . 11 |- ( T. -> ( x e. NN |-> -u ( F ` x ) ) ~~> ( 0 - gamma ) )
76	75	adantr 481	. . . . . . . . . 10 |- ( ( T. /\ i e. NN ) -> ( x e. NN |-> -u ( F ` x ) ) ~~> ( 0 - gamma ) )
77	66	renegcld 10457	. . . . . . . . . . . 12 |- ( ( T. /\ k e. NN ) -> -u ( F ` k ) e. RR )
78	72, 77	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( T. /\ k e. NN ) -> ( ( x e. NN |-> -u ( F ` x ) ) ` k ) e. RR )
79	78	adantlr 751	. . . . . . . . . 10 |- ( ( ( T. /\ i e. NN ) /\ k e. NN ) -> ( ( x e. NN |-> -u ( F ` x ) ) ` k ) e. RR )
80	19	simpld 475	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( F ` ( k + 1 ) ) <_ ( F ` k ) )
81	80	adantl 482	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( F ` ( k + 1 ) ) <_ ( F ` k ) )
82		peano2nn 11032	. . . . . . . . . . . . . . . 16 |- ( k e. NN -> ( k + 1 ) e. NN )
83	82	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( T. /\ k e. NN ) -> ( k + 1 ) e. NN )
84	64	ffvelrni 6358	. . . . . . . . . . . . . . 15 |- ( ( k + 1 ) e. NN -> ( F ` ( k + 1 ) ) e. RR )
85	83, 84	syl 17	. . . . . . . . . . . . . 14 |- ( ( T. /\ k e. NN ) -> ( F ` ( k + 1 ) ) e. RR )
86	85, 66	lenegd 10606	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( ( F ` ( k + 1 ) ) <_ ( F ` k ) <-> -u ( F ` k ) <_ -u ( F ` ( k + 1 ) ) ) )
87	81, 86	mpbid 222	. . . . . . . . . . . 12 |- ( ( T. /\ k e. NN ) -> -u ( F ` k ) <_ -u ( F ` ( k + 1 ) ) )
88		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( x = ( k + 1 ) -> ( F ` x ) = ( F ` ( k + 1 ) ) )
89	88	negeqd 10275	. . . . . . . . . . . . . 14 |- ( x = ( k + 1 ) -> -u ( F ` x ) = -u ( F ` ( k + 1 ) ) )
90		negex 10279	. . . . . . . . . . . . . 14 |- -u ( F ` ( k + 1 ) ) e. _V
91	89, 54, 90	fvmpt 6282	. . . . . . . . . . . . 13 |- ( ( k + 1 ) e. NN -> ( ( x e. NN |-> -u ( F ` x ) ) ` ( k + 1 ) ) = -u ( F ` ( k + 1 ) ) )
92	83, 91	syl 17	. . . . . . . . . . . 12 |- ( ( T. /\ k e. NN ) -> ( ( x e. NN |-> -u ( F ` x ) ) ` ( k + 1 ) ) = -u ( F ` ( k + 1 ) ) )
93	87, 72, 92	3brtr4d 4685	. . . . . . . . . . 11 |- ( ( T. /\ k e. NN ) -> ( ( x e. NN |-> -u ( F ` x ) ) ` k ) <_ ( ( x e. NN |-> -u ( F ` x ) ) ` ( k + 1 ) ) )
94	93	adantlr 751	. . . . . . . . . 10 |- ( ( ( T. /\ i e. NN ) /\ k e. NN ) -> ( ( x e. NN |-> -u ( F ` x ) ) ` k ) <_ ( ( x e. NN |-> -u ( F ` x ) ) ` ( k + 1 ) ) )
95	1, 16, 76, 79, 94	climub 14392	. . . . . . . . 9 |- ( ( T. /\ i e. NN ) -> ( ( x e. NN |-> -u ( F ` x ) ) ` i ) <_ ( 0 - gamma ) )
96	57, 95	eqbrtrrd 4677	. . . . . . . 8 |- ( ( T. /\ i e. NN ) -> -u ( F ` i ) <_ ( 0 - gamma ) )
97		df-neg 10269	. . . . . . . 8 |- -u gamma = ( 0 - gamma )
98	96, 97	syl6breqr 4695	. . . . . . 7 |- ( ( T. /\ i e. NN ) -> -u ( F ` i ) <_ -u gamma )
99	14	trud 1493	. . . . . . . 8 |- gamma e. RR
100	64	ffvelrni 6358	. . . . . . . . 9 |- ( i e. NN -> ( F ` i ) e. RR )
101	100	adantl 482	. . . . . . . 8 |- ( ( T. /\ i e. NN ) -> ( F ` i ) e. RR )
102		leneg 10531	. . . . . . . 8 |- ( ( gamma e. RR /\ ( F ` i ) e. RR ) -> ( gamma <_ ( F ` i ) <-> -u ( F ` i ) <_ -u gamma ) )
103	99, 101, 102	sylancr 695	. . . . . . 7 |- ( ( T. /\ i e. NN ) -> ( gamma <_ ( F ` i ) <-> -u ( F ` i ) <_ -u gamma ) )
104	98, 103	mpbird 247	. . . . . 6 |- ( ( T. /\ i e. NN ) -> gamma <_ ( F ` i ) )
105	104	ralrimiva 2966	. . . . 5 |- ( T. -> A. i e. NN gamma <_ ( F ` i ) )
106		fveq2 6191	. . . . . . . 8 |- ( i = 1 -> ( F ` i ) = ( F ` 1 ) )
107		fveq2 6191	. . . . . . . . . . . . 13 |- ( n = 1 -> ( log ` n ) = ( log ` 1 ) )
108		log1 24332	. . . . . . . . . . . . 13 |- ( log ` 1 ) = 0
109	107, 108	syl6eq 2672	. . . . . . . . . . . 12 |- ( n = 1 -> ( log ` n ) = 0 )
110	34, 109	oveq12d 6668	. . . . . . . . . . 11 |- ( n = 1 -> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` n ) ) = ( 1 - 0 ) )
111		1m0e1 11131	. . . . . . . . . . 11 |- ( 1 - 0 ) = 1
112	110, 111	syl6eq 2672	. . . . . . . . . 10 |- ( n = 1 -> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` n ) ) = 1 )
113	40	elexi 3213	. . . . . . . . . 10 |- 1 e. _V
114	112, 3, 113	fvmpt 6282	. . . . . . . . 9 |- ( 1 e. NN -> ( F ` 1 ) = 1 )
115	15, 114	ax-mp 5	. . . . . . . 8 |- ( F ` 1 ) = 1
116	106, 115	syl6eq 2672	. . . . . . 7 |- ( i = 1 -> ( F ` i ) = 1 )
117	116	breq2d 4665	. . . . . 6 |- ( i = 1 -> ( gamma <_ ( F ` i ) <-> gamma <_ 1 ) )
118	117	rspcva 3307	. . . . 5 |- ( ( 1 e. NN /\ A. i e. NN gamma <_ ( F ` i ) ) -> gamma <_ 1 )
119	15, 105, 118	sylancr 695	. . . 4 |- ( T. -> gamma <_ 1 )
120	44, 40	elicc2i 12239	. . . 4 |- ( gamma e. ( ( 1 - ( log ` 2 ) ) [,] 1 ) <-> ( gamma e. RR /\ ( 1 - ( log ` 2 ) ) <_ gamma /\ gamma <_ 1 ) )
121	14, 51, 119, 120	syl3anbrc 1246	. . 3 |- ( T. -> gamma e. ( ( 1 - ( log ` 2 ) ) [,] 1 ) )
122		ffn 6045	. . . . 5 |- ( F : NN --> RR -> F Fn NN )
123	64, 122	mp1i 13	. . . 4 |- ( T. -> F Fn NN )
124	16, 1	syl6eleq 2711	. . . . . . . 8 |- ( ( T. /\ i e. NN ) -> i e. ( ZZ>= ` 1 ) )
125		elfznn 12370	. . . . . . . . . 10 |- ( k e. ( 1 ... i ) -> k e. NN )
126	125	adantl 482	. . . . . . . . 9 |- ( ( ( T. /\ i e. NN ) /\ k e. ( 1 ... i ) ) -> k e. NN )
127	126, 65	syl 17	. . . . . . . 8 |- ( ( ( T. /\ i e. NN ) /\ k e. ( 1 ... i ) ) -> ( F ` k ) e. RR )
128		elfznn 12370	. . . . . . . . . 10 |- ( k e. ( 1 ... ( i - 1 ) ) -> k e. NN )
129	128	adantl 482	. . . . . . . . 9 |- ( ( ( T. /\ i e. NN ) /\ k e. ( 1 ... ( i - 1 ) ) ) -> k e. NN )
130	129, 80	syl 17	. . . . . . . 8 |- ( ( ( T. /\ i e. NN ) /\ k e. ( 1 ... ( i - 1 ) ) ) -> ( F ` ( k + 1 ) ) <_ ( F ` k ) )
131	124, 127, 130	monoord2 12832	. . . . . . 7 |- ( ( T. /\ i e. NN ) -> ( F ` i ) <_ ( F ` 1 ) )
132	131, 115	syl6breq 4694	. . . . . 6 |- ( ( T. /\ i e. NN ) -> ( F ` i ) <_ 1 )
133	99, 40	elicc2i 12239	. . . . . 6 |- ( ( F ` i ) e. ( gamma [,] 1 ) <-> ( ( F ` i ) e. RR /\ gamma <_ ( F ` i ) /\ ( F ` i ) <_ 1 ) )
134	101, 104, 132, 133	syl3anbrc 1246	. . . . 5 |- ( ( T. /\ i e. NN ) -> ( F ` i ) e. ( gamma [,] 1 ) )
135	134	ralrimiva 2966	. . . 4 |- ( T. -> A. i e. NN ( F ` i ) e. ( gamma [,] 1 ) )
136		ffnfv 6388	. . . 4 |- ( F : NN --> ( gamma [,] 1 ) <-> ( F Fn NN /\ A. i e. NN ( F ` i ) e. ( gamma [,] 1 ) ) )
137	123, 135, 136	sylanbrc 698	. . 3 |- ( T. -> F : NN --> ( gamma [,] 1 ) )
138		ffn 6045	. . . . 5 |- ( G : NN --> RR -> G Fn NN )
139	11, 138	mp1i 13	. . . 4 |- ( T. -> G Fn NN )
140	11	ffvelrni 6358	. . . . . . 7 |- ( i e. NN -> ( G ` i ) e. RR )
141	140	adantl 482	. . . . . 6 |- ( ( T. /\ i e. NN ) -> ( G ` i ) e. RR )
142	126, 12	syl 17	. . . . . . . 8 |- ( ( ( T. /\ i e. NN ) /\ k e. ( 1 ... i ) ) -> ( G ` k ) e. RR )
143	129, 20	syl 17	. . . . . . . 8 |- ( ( ( T. /\ i e. NN ) /\ k e. ( 1 ... ( i - 1 ) ) ) -> ( G ` k ) <_ ( G ` ( k + 1 ) ) )
144	124, 142, 143	monoord 12831	. . . . . . 7 |- ( ( T. /\ i e. NN ) -> ( G ` 1 ) <_ ( G ` i ) )
145	47, 144	syl5eqbrr 4689	. . . . . 6 |- ( ( T. /\ i e. NN ) -> ( 1 - ( log ` 2 ) ) <_ ( G ` i ) )
146	44, 99	elicc2i 12239	. . . . . 6 |- ( ( G ` i ) e. ( ( 1 - ( log ` 2 ) ) [,] gamma ) <-> ( ( G ` i ) e. RR /\ ( 1 - ( log ` 2 ) ) <_ ( G ` i ) /\ ( G ` i ) <_ gamma ) )
147	141, 145, 22, 146	syl3anbrc 1246	. . . . 5 |- ( ( T. /\ i e. NN ) -> ( G ` i ) e. ( ( 1 - ( log ` 2 ) ) [,] gamma ) )
148	147	ralrimiva 2966	. . . 4 |- ( T. -> A. i e. NN ( G ` i ) e. ( ( 1 - ( log ` 2 ) ) [,] gamma ) )
149		ffnfv 6388	. . . 4 |- ( G : NN --> ( ( 1 - ( log ` 2 ) ) [,] gamma ) <-> ( G Fn NN /\ A. i e. NN ( G ` i ) e. ( ( 1 - ( log ` 2 ) ) [,] gamma ) ) )
150	139, 148, 149	sylanbrc 698	. . 3 |- ( T. -> G : NN --> ( ( 1 - ( log ` 2 ) ) [,] gamma ) )
151	121, 137, 150	3jca 1242	. 2 |- ( T. -> ( gamma e. ( ( 1 - ( log ` 2 ) ) [,] 1 ) /\ F : NN --> ( gamma [,] 1 ) /\ G : NN --> ( ( 1 - ( log ` 2 ) ) [,] gamma ) ) )
152	151	trud 1493	1 |- ( gamma e. ( ( 1 - ( log ` 2 ) ) [,] 1 ) /\ F : NN --> ( gamma [,] 1 ) /\ G : NN --> ( ( 1 - ( log ` 2 ) ) [,] gamma ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   A.wral 2912   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   [,]cicc 12178   ...cfz 12326   ~~> cli 14215   sum_csu 14416   logclog 24301   gammacem 24718

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-em 24719

This theorem is referenced by:  emcl  24729  harmonicbnd  24730  harmonicbnd2  24731