Theorem log2ublem2 16067

Description: Lemma for log2ublog2 16069. (Contributed by Mario Carneiro, 17-Apr-2015.)

Hypotheses

Ref	Expression
log2ublem2.1	|- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... K ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) <_ ( 2 x. B )
log2ublem2.2	|- B e. NN0
log2ublem2.3	|- F e. NN0
log2ublem2.4	|- N e. NN0
log2ublem2.5	|- ( N - 1 ) = K
log2ublem2.6	|- ( B + F ) = G
log2ublem2.7	|- M e. NN0
log2ublem2.8	|- ( M + N ) = 3
log2ublem2.9	|- ( ( 5 x. 7 ) x. ( 9 ^ M ) ) = ( ( ( 2 x. N ) + 1 ) x. F )

Assertion

Ref	Expression
log2ublem2	|- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... N ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) <_ ( 2 x. G )

Distinct variable groups:   n, K   n, N

Allowed substitution hints:   B( n)   F( n)   G( n)   M( n)

Proof of Theorem log2ublem2

Step	Hyp	Ref	Expression
1		log2ublem2.1	. 2 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... K ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) <_ ( 2 x. B )
2		0zd 9639	. . . . 5 |- ( T. -> 0 e. ZZ )
3		log2ublem2.5	. . . . . . 7 |- ( N - 1 ) = K
4		log2ublem2.4	. . . . . . . . 9 |- N e. NN0
5	4	nn0zi 9649	. . . . . . . 8 |- N e. ZZ
6		peano2zm 9665	. . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
7	5, 6	ax-mp 5	. . . . . . 7 |- ( N - 1 ) e. ZZ
8	3, 7	eqeltrri 2312	. . . . . 6 |- K e. ZZ
9	8	a1i 9	. . . . 5 |- ( T. -> K e. ZZ )
10	2, 9	fzfigd 10851	. . . 4 |- ( T. -> ( 0 ... K ) e. Fin )
11		elfznn0 10504	. . . . . 6 |- ( n e. ( 0 ... K ) -> n e. NN0 )
12	11	adantl 277	. . . . 5 |- ( ( T. /\ n e. ( 0 ... K ) ) -> n e. NN0 )
13		2re 9357	. . . . . 6 |- 2 e. RR
14		3nn 9450	. . . . . . . 8 |- 3 e. NN
15		2nn0 9563	. . . . . . . . . 10 |- 2 e. NN0
16		nn0mulcl 9582	. . . . . . . . . 10 |- ( ( 2 e. NN0 /\ n e. NN0 ) -> ( 2 x. n ) e. NN0 )
17	15, 16	mpan 428	. . . . . . . . 9 |- ( n e. NN0 -> ( 2 x. n ) e. NN0 )
18		nn0p1nn 9585	. . . . . . . . 9 |- ( ( 2 x. n ) e. NN0 -> ( ( 2 x. n ) + 1 ) e. NN )
19	17, 18	syl 14	. . . . . . . 8 |- ( n e. NN0 -> ( ( 2 x. n ) + 1 ) e. NN )
20		nnmulcl 9308	. . . . . . . 8 |- ( ( 3 e. NN /\ ( ( 2 x. n ) + 1 ) e. NN ) -> ( 3 x. ( ( 2 x. n ) + 1 ) ) e. NN )
21	14, 19, 20	sylancr 418	. . . . . . 7 |- ( n e. NN0 -> ( 3 x. ( ( 2 x. n ) + 1 ) ) e. NN )
22		9nn 9456	. . . . . . . 8 |- 9 e. NN
23		nnexpcl 10972	. . . . . . . 8 |- ( ( 9 e. NN /\ n e. NN0 ) -> ( 9 ^ n ) e. NN )
24	22, 23	mpan 428	. . . . . . 7 |- ( n e. NN0 -> ( 9 ^ n ) e. NN )
25	21, 24	nnmulcld 9336	. . . . . 6 |- ( n e. NN0 -> ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) e. NN )
26		nndivre 9323	. . . . . 6 |- ( ( 2 e. RR /\ ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) e. NN ) -> ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. RR )
27	13, 25, 26	sylancr 418	. . . . 5 |- ( n e. NN0 -> ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. RR )
28	12, 27	syl 14	. . . 4 |- ( ( T. /\ n e. ( 0 ... K ) ) -> ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. RR )
29	10, 28	fsumrecl 12151	. . 3 |- ( T. -> sum_ n e. ( 0 ... K ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. RR )
30	29	mptru 1411	. 2 |- sum_ n e. ( 0 ... K ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. RR
31	15, 4	nn0mulcli 9584	. . . . 5 |- ( 2 x. N ) e. NN0
32		nn0p1nn 9585	. . . . 5 |- ( ( 2 x. N ) e. NN0 -> ( ( 2 x. N ) + 1 ) e. NN )
33	31, 32	ax-mp 5	. . . 4 |- ( ( 2 x. N ) + 1 ) e. NN
34	14, 33	nnmulcli 9309	. . 3 |- ( 3 x. ( ( 2 x. N ) + 1 ) ) e. NN
35		nnexpcl 10972	. . . 4 |- ( ( 9 e. NN /\ N e. NN0 ) -> ( 9 ^ N ) e. NN )
36	22, 4, 35	mp2an 430	. . 3 |- ( 9 ^ N ) e. NN
37	34, 36	nnmulcli 9309	. 2 |- ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) e. NN
38		log2ublem2.2	. . 3 |- B e. NN0
39	15, 38	nn0mulcli 9584	. 2 |- ( 2 x. B ) e. NN0
40		log2ublem2.3	. . 3 |- F e. NN0
41	15, 40	nn0mulcli 9584	. 2 |- ( 2 x. F ) e. NN0
42		nn0uz 9940	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
43	4, 42	eleqtri 2313	. . . . . 6 |- N e. ( ZZ>= ` 0 )
44	43	a1i 9	. . . . 5 |- ( T. -> N e. ( ZZ>= ` 0 ) )
45		elfznn0 10504	. . . . . . 7 |- ( n e. ( 0 ... N ) -> n e. NN0 )
46	45	adantl 277	. . . . . 6 |- ( ( T. /\ n e. ( 0 ... N ) ) -> n e. NN0 )
47	27	recnd 8348	. . . . . 6 |- ( n e. NN0 -> ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. CC )
48	46, 47	syl 14	. . . . 5 |- ( ( T. /\ n e. ( 0 ... N ) ) -> ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. CC )
49		oveq2 6087	. . . . . . . . 9 |- ( n = N -> ( 2 x. n ) = ( 2 x. N ) )
50	49	oveq1d 6094	. . . . . . . 8 |- ( n = N -> ( ( 2 x. n ) + 1 ) = ( ( 2 x. N ) + 1 ) )
51	50	oveq2d 6095	. . . . . . 7 |- ( n = N -> ( 3 x. ( ( 2 x. n ) + 1 ) ) = ( 3 x. ( ( 2 x. N ) + 1 ) ) )
52		oveq2 6087	. . . . . . 7 |- ( n = N -> ( 9 ^ n ) = ( 9 ^ N ) )
53	51, 52	oveq12d 6097	. . . . . 6 |- ( n = N -> ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) = ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) )
54	53	oveq2d 6095	. . . . 5 |- ( n = N -> ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) = ( 2 / ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) ) )
55	44, 48, 54	fsumm1 12166	. . . 4 |- ( T. -> sum_ n e. ( 0 ... N ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) = ( sum_ n e. ( 0 ... ( N - 1 ) ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) + ( 2 / ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) ) ) )
56	55	mptru 1411	. . 3 |- sum_ n e. ( 0 ... N ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) = ( sum_ n e. ( 0 ... ( N - 1 ) ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) + ( 2 / ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) ) )
57	3	oveq2i 6090	. . . . 5 |- ( 0 ... ( N - 1 ) ) = ( 0 ... K )
58	57	sumeq1i 12112	. . . 4 |- sum_ n e. ( 0 ... ( N - 1 ) ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) = sum_ n e. ( 0 ... K ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) )
59	58	oveq1i 6089	. . 3 |- ( sum_ n e. ( 0 ... ( N - 1 ) ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) + ( 2 / ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) ) ) = ( sum_ n e. ( 0 ... K ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) + ( 2 / ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) ) )
60	56, 59	eqtri 2259	. 2 |- sum_ n e. ( 0 ... N ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) = ( sum_ n e. ( 0 ... K ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) + ( 2 / ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) ) )
61		2cn 9358	. . . 4 |- 2 e. CC
62	38	nn0cni 9558	. . . 4 |- B e. CC
63	40	nn0cni 9558	. . . 4 |- F e. CC
64	61, 62, 63	adddii 8330	. . 3 |- ( 2 x. ( B + F ) ) = ( ( 2 x. B ) + ( 2 x. F ) )
65		log2ublem2.6	. . . 4 |- ( B + F ) = G
66	65	oveq2i 6090	. . 3 |- ( 2 x. ( B + F ) ) = ( 2 x. G )
67	64, 66	eqtr3i 2261	. 2 |- ( ( 2 x. B ) + ( 2 x. F ) ) = ( 2 x. G )
68		7nn 9454	. . . . . . . . 9 |- 7 e. NN
69	68	nnnn0i 9554	. . . . . . . 8 |- 7 e. NN0
70		nnexpcl 10972	. . . . . . . 8 |- ( ( 3 e. NN /\ 7 e. NN0 ) -> ( 3 ^ 7 ) e. NN )
71	14, 69, 70	mp2an 430	. . . . . . 7 |- ( 3 ^ 7 ) e. NN
72		5nn 9452	. . . . . . . 8 |- 5 e. NN
73	72, 68	nnmulcli 9309	. . . . . . 7 |- ( 5 x. 7 ) e. NN
74	71, 73	nnmulcli 9309	. . . . . 6 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN
75	74	nnrei 9296	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. RR
76	75, 13	remulcli 8334	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 ) e. RR
77	76	leidi 8807	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 ) <_ ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 )
78	14	nnnn0i 9554	. . . . . . . . . . . 12 |- 3 e. NN0
79		nnexpcl 10972	. . . . . . . . . . . 12 |- ( ( 9 e. NN /\ 3 e. NN0 ) -> ( 9 ^ 3 ) e. NN )
80	22, 78, 79	mp2an 430	. . . . . . . . . . 11 |- ( 9 ^ 3 ) e. NN
81	80	nncni 9297	. . . . . . . . . 10 |- ( 9 ^ 3 ) e. CC
82	73	nncni 9297	. . . . . . . . . 10 |- ( 5 x. 7 ) e. CC
83	81, 82	mulcomi 8326	. . . . . . . . 9 |- ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) = ( ( 5 x. 7 ) x. ( 9 ^ 3 ) )
84		log2ublem2.8	. . . . . . . . . . . . 13 |- ( M + N ) = 3
85		log2ublem2.7	. . . . . . . . . . . . . . 15 |- M e. NN0
86	85	nn0cni 9558	. . . . . . . . . . . . . 14 |- M e. CC
87	4	nn0cni 9558	. . . . . . . . . . . . . 14 |- N e. CC
88	86, 87	addcomi 8464	. . . . . . . . . . . . 13 |- ( M + N ) = ( N + M )
89	84, 88	eqtr3i 2261	. . . . . . . . . . . 12 |- 3 = ( N + M )
90	89	oveq2i 6090	. . . . . . . . . . 11 |- ( 9 ^ 3 ) = ( 9 ^ ( N + M ) )
91	22	nncni 9297	. . . . . . . . . . . 12 |- 9 e. CC
92		expadd 11001	. . . . . . . . . . . 12 |- ( ( 9 e. CC /\ N e. NN0 /\ M e. NN0 ) -> ( 9 ^ ( N + M ) ) = ( ( 9 ^ N ) x. ( 9 ^ M ) ) )
93	91, 4, 85, 92	mp3an 1378	. . . . . . . . . . 11 |- ( 9 ^ ( N + M ) ) = ( ( 9 ^ N ) x. ( 9 ^ M ) )
94	90, 93	eqtri 2259	. . . . . . . . . 10 |- ( 9 ^ 3 ) = ( ( 9 ^ N ) x. ( 9 ^ M ) )
95	94	oveq2i 6090	. . . . . . . . 9 |- ( ( 5 x. 7 ) x. ( 9 ^ 3 ) ) = ( ( 5 x. 7 ) x. ( ( 9 ^ N ) x. ( 9 ^ M ) ) )
96	36	nncni 9297	. . . . . . . . . 10 |- ( 9 ^ N ) e. CC
97		nnexpcl 10972	. . . . . . . . . . . 12 |- ( ( 9 e. NN /\ M e. NN0 ) -> ( 9 ^ M ) e. NN )
98	22, 85, 97	mp2an 430	. . . . . . . . . . 11 |- ( 9 ^ M ) e. NN
99	98	nncni 9297	. . . . . . . . . 10 |- ( 9 ^ M ) e. CC
100	82, 96, 99	mul12i 8466	. . . . . . . . 9 |- ( ( 5 x. 7 ) x. ( ( 9 ^ N ) x. ( 9 ^ M ) ) ) = ( ( 9 ^ N ) x. ( ( 5 x. 7 ) x. ( 9 ^ M ) ) )
101	83, 95, 100	3eqtri 2263	. . . . . . . 8 |- ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) = ( ( 9 ^ N ) x. ( ( 5 x. 7 ) x. ( 9 ^ M ) ) )
102		log2ublem2.9	. . . . . . . . 9 |- ( ( 5 x. 7 ) x. ( 9 ^ M ) ) = ( ( ( 2 x. N ) + 1 ) x. F )
103	102	oveq2i 6090	. . . . . . . 8 |- ( ( 9 ^ N ) x. ( ( 5 x. 7 ) x. ( 9 ^ M ) ) ) = ( ( 9 ^ N ) x. ( ( ( 2 x. N ) + 1 ) x. F ) )
104	101, 103	eqtri 2259	. . . . . . 7 |- ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) = ( ( 9 ^ N ) x. ( ( ( 2 x. N ) + 1 ) x. F ) )
105	104	oveq2i 6090	. . . . . 6 |- ( 3 x. ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) ) = ( 3 x. ( ( 9 ^ N ) x. ( ( ( 2 x. N ) + 1 ) x. F ) ) )
106		df-7 9351	. . . . . . . . . 10 |- 7 = ( 6 + 1 )
107	106	oveq2i 6090	. . . . . . . . 9 |- ( 3 ^ 7 ) = ( 3 ^ ( 6 + 1 ) )
108		3cn 9362	. . . . . . . . . . 11 |- 3 e. CC
109		6nn0 9567	. . . . . . . . . . 11 |- 6 e. NN0
110		expp1 10966	. . . . . . . . . . 11 |- ( ( 3 e. CC /\ 6 e. NN0 ) -> ( 3 ^ ( 6 + 1 ) ) = ( ( 3 ^ 6 ) x. 3 ) )
111	108, 109, 110	mp2an 430	. . . . . . . . . 10 |- ( 3 ^ ( 6 + 1 ) ) = ( ( 3 ^ 6 ) x. 3 )
112		expmul 11004	. . . . . . . . . . . . 13 |- ( ( 3 e. CC /\ 2 e. NN0 /\ 3 e. NN0 ) -> ( 3 ^ ( 2 x. 3 ) ) = ( ( 3 ^ 2 ) ^ 3 ) )
113	108, 15, 78, 112	mp3an 1378	. . . . . . . . . . . 12 |- ( 3 ^ ( 2 x. 3 ) ) = ( ( 3 ^ 2 ) ^ 3 )
114	61, 108	mulcomi 8326	. . . . . . . . . . . . . 14 |- ( 2 x. 3 ) = ( 3 x. 2 )
115		3t2e6 9444	. . . . . . . . . . . . . 14 |- ( 3 x. 2 ) = 6
116	114, 115	eqtri 2259	. . . . . . . . . . . . 13 |- ( 2 x. 3 ) = 6
117	116	oveq2i 6090	. . . . . . . . . . . 12 |- ( 3 ^ ( 2 x. 3 ) ) = ( 3 ^ 6 )
118		sq3 11056	. . . . . . . . . . . . 13 |- ( 3 ^ 2 ) = 9
119	118	oveq1i 6089	. . . . . . . . . . . 12 |- ( ( 3 ^ 2 ) ^ 3 ) = ( 9 ^ 3 )
120	113, 117, 119	3eqtr3i 2267	. . . . . . . . . . 11 |- ( 3 ^ 6 ) = ( 9 ^ 3 )
121	120	oveq1i 6089	. . . . . . . . . 10 |- ( ( 3 ^ 6 ) x. 3 ) = ( ( 9 ^ 3 ) x. 3 )
122	111, 121	eqtri 2259	. . . . . . . . 9 |- ( 3 ^ ( 6 + 1 ) ) = ( ( 9 ^ 3 ) x. 3 )
123	81, 108	mulcomi 8326	. . . . . . . . 9 |- ( ( 9 ^ 3 ) x. 3 ) = ( 3 x. ( 9 ^ 3 ) )
124	107, 122, 123	3eqtri 2263	. . . . . . . 8 |- ( 3 ^ 7 ) = ( 3 x. ( 9 ^ 3 ) )
125	124	oveq1i 6089	. . . . . . 7 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) = ( ( 3 x. ( 9 ^ 3 ) ) x. ( 5 x. 7 ) )
126	108, 81, 82	mulassi 8329	. . . . . . 7 |- ( ( 3 x. ( 9 ^ 3 ) ) x. ( 5 x. 7 ) ) = ( 3 x. ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) )
127	125, 126	eqtri 2259	. . . . . 6 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) = ( 3 x. ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) )
128	33	nncni 9297	. . . . . . . . 9 |- ( ( 2 x. N ) + 1 ) e. CC
129	108, 128, 96	mul32i 8467	. . . . . . . 8 |- ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) = ( ( 3 x. ( 9 ^ N ) ) x. ( ( 2 x. N ) + 1 ) )
130	129	oveq1i 6089	. . . . . . 7 |- ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. F ) = ( ( ( 3 x. ( 9 ^ N ) ) x. ( ( 2 x. N ) + 1 ) ) x. F )
131	108, 96	mulcli 8325	. . . . . . . 8 |- ( 3 x. ( 9 ^ N ) ) e. CC
132	131, 128, 63	mulassi 8329	. . . . . . 7 |- ( ( ( 3 x. ( 9 ^ N ) ) x. ( ( 2 x. N ) + 1 ) ) x. F ) = ( ( 3 x. ( 9 ^ N ) ) x. ( ( ( 2 x. N ) + 1 ) x. F ) )
133	128, 63	mulcli 8325	. . . . . . . 8 |- ( ( ( 2 x. N ) + 1 ) x. F ) e. CC
134	108, 96, 133	mulassi 8329	. . . . . . 7 |- ( ( 3 x. ( 9 ^ N ) ) x. ( ( ( 2 x. N ) + 1 ) x. F ) ) = ( 3 x. ( ( 9 ^ N ) x. ( ( ( 2 x. N ) + 1 ) x. F ) ) )
135	130, 132, 134	3eqtri 2263	. . . . . 6 |- ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. F ) = ( 3 x. ( ( 9 ^ N ) x. ( ( ( 2 x. N ) + 1 ) x. F ) ) )
136	105, 127, 135	3eqtr4i 2269	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) = ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. F )
137	136	oveq2i 6090	. . . 4 |- ( 2 x. ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) ) = ( 2 x. ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. F ) )
138	71	nncni 9297	. . . . . 6 |- ( 3 ^ 7 ) e. CC
139	138, 82	mulcli 8325	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. CC
140	139, 61	mulcomi 8326	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 ) = ( 2 x. ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) )
141	37	nncni 9297	. . . . 5 |- ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) e. CC
142	141, 61, 63	mul12i 8466	. . . 4 |- ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. ( 2 x. F ) ) = ( 2 x. ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. F ) )
143	137, 140, 142	3eqtr4i 2269	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 ) = ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. ( 2 x. F ) )
144	77, 143	breqtri 4153	. 2 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 ) <_ ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. ( 2 x. F ) )
145	1, 30, 15, 37, 39, 41, 60, 67, 144	log2ublem1 16066	1 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... N ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) <_ ( 2 x. G )

Syntax hints:   /\ wa 104   = wceq 1402   T. wtru 1403   e. wcel 2209   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   CCcc 8171   RRcr 8172   0cc0 8173   1c1 8174   + caddc 8176   x. cmul 8178   <_ cle 8355   - cmin 8491   / cdiv 8996   NNcn 9287   2c2 9338   3c3 9339   5c5 9341   6c6 9342   7c7 9343   9c9 9345   NN0cn0 9546   ZZcz 9627   ZZ>=cuz 9904   ...cfz 10394   ^cexp 10958   sum_csu 12102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-sumdc 12103

This theorem is referenced by:  log2ublem3  16068