Theorem log2ublem2 24674

Description: Lemma for log2ub 24676. (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		fzfid 12772	. . . 4 |- ( T. -> ( 0 ... K ) e. Fin )
3		elfznn0 12433	. . . . . 6 |- ( n e. ( 0 ... K ) -> n e. NN0 )
4	3	adantl 482	. . . . 5 |- ( ( T. /\ n e. ( 0 ... K ) ) -> n e. NN0 )
5		2re 11090	. . . . . 6 |- 2 e. RR
6		3nn 11186	. . . . . . . 8 |- 3 e. NN
7		2nn0 11309	. . . . . . . . . 10 |- 2 e. NN0
8		nn0mulcl 11329	. . . . . . . . . 10 |- ( ( 2 e. NN0 /\ n e. NN0 ) -> ( 2 x. n ) e. NN0 )
9	7, 8	mpan 706	. . . . . . . . 9 |- ( n e. NN0 -> ( 2 x. n ) e. NN0 )
10		nn0p1nn 11332	. . . . . . . . 9 |- ( ( 2 x. n ) e. NN0 -> ( ( 2 x. n ) + 1 ) e. NN )
11	9, 10	syl 17	. . . . . . . 8 |- ( n e. NN0 -> ( ( 2 x. n ) + 1 ) e. NN )
12		nnmulcl 11043	. . . . . . . 8 |- ( ( 3 e. NN /\ ( ( 2 x. n ) + 1 ) e. NN ) -> ( 3 x. ( ( 2 x. n ) + 1 ) ) e. NN )
13	6, 11, 12	sylancr 695	. . . . . . 7 |- ( n e. NN0 -> ( 3 x. ( ( 2 x. n ) + 1 ) ) e. NN )
14		9nn 11192	. . . . . . . 8 |- 9 e. NN
15		nnexpcl 12873	. . . . . . . 8 |- ( ( 9 e. NN /\ n e. NN0 ) -> ( 9 ^ n ) e. NN )
16	14, 15	mpan 706	. . . . . . 7 |- ( n e. NN0 -> ( 9 ^ n ) e. NN )
17	13, 16	nnmulcld 11068	. . . . . 6 |- ( n e. NN0 -> ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) e. NN )
18		nndivre 11056	. . . . . 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 )
19	5, 17, 18	sylancr 695	. . . . 5 |- ( n e. NN0 -> ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. RR )
20	4, 19	syl 17	. . . 4 |- ( ( T. /\ n e. ( 0 ... K ) ) -> ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. RR )
21	2, 20	fsumrecl 14465	. . 3 |- ( T. -> sum_ n e. ( 0 ... K ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. RR )
22	21	trud 1493	. 2 |- sum_ n e. ( 0 ... K ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. RR
23		log2ublem2.4	. . . . . 6 |- N e. NN0
24	7, 23	nn0mulcli 11331	. . . . 5 |- ( 2 x. N ) e. NN0
25		nn0p1nn 11332	. . . . 5 |- ( ( 2 x. N ) e. NN0 -> ( ( 2 x. N ) + 1 ) e. NN )
26	24, 25	ax-mp 5	. . . 4 |- ( ( 2 x. N ) + 1 ) e. NN
27	6, 26	nnmulcli 11044	. . 3 |- ( 3 x. ( ( 2 x. N ) + 1 ) ) e. NN
28		nnexpcl 12873	. . . 4 |- ( ( 9 e. NN /\ N e. NN0 ) -> ( 9 ^ N ) e. NN )
29	14, 23, 28	mp2an 708	. . 3 |- ( 9 ^ N ) e. NN
30	27, 29	nnmulcli 11044	. 2 |- ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) e. NN
31		log2ublem2.2	. . 3 |- B e. NN0
32	7, 31	nn0mulcli 11331	. 2 |- ( 2 x. B ) e. NN0
33		log2ublem2.3	. . 3 |- F e. NN0
34	7, 33	nn0mulcli 11331	. 2 |- ( 2 x. F ) e. NN0
35		nn0uz 11722	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
36	23, 35	eleqtri 2699	. . . . . 6 |- N e. ( ZZ>= ` 0 )
37	36	a1i 11	. . . . 5 |- ( T. -> N e. ( ZZ>= ` 0 ) )
38		elfznn0 12433	. . . . . . 7 |- ( n e. ( 0 ... N ) -> n e. NN0 )
39	38	adantl 482	. . . . . 6 |- ( ( T. /\ n e. ( 0 ... N ) ) -> n e. NN0 )
40	19	recnd 10068	. . . . . 6 |- ( n e. NN0 -> ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. CC )
41	39, 40	syl 17	. . . . 5 |- ( ( T. /\ n e. ( 0 ... N ) ) -> ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) e. CC )
42		oveq2 6658	. . . . . . . . 9 |- ( n = N -> ( 2 x. n ) = ( 2 x. N ) )
43	42	oveq1d 6665	. . . . . . . 8 |- ( n = N -> ( ( 2 x. n ) + 1 ) = ( ( 2 x. N ) + 1 ) )
44	43	oveq2d 6666	. . . . . . 7 |- ( n = N -> ( 3 x. ( ( 2 x. n ) + 1 ) ) = ( 3 x. ( ( 2 x. N ) + 1 ) ) )
45		oveq2 6658	. . . . . . 7 |- ( n = N -> ( 9 ^ n ) = ( 9 ^ N ) )
46	44, 45	oveq12d 6668	. . . . . 6 |- ( n = N -> ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) = ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) )
47	46	oveq2d 6666	. . . . 5 |- ( n = N -> ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) = ( 2 / ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) ) )
48	37, 41, 47	fsumm1 14480	. . . 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 ) ) ) ) )
49	48	trud 1493	. . 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 ) ) ) )
50		log2ublem2.5	. . . . . 6 |- ( N - 1 ) = K
51	50	oveq2i 6661	. . . . 5 |- ( 0 ... ( N - 1 ) ) = ( 0 ... K )
52	51	sumeq1i 14428	. . . 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 ) ) )
53	52	oveq1i 6660	. . 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 ) ) ) )
54	49, 53	eqtri 2644	. 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 ) ) ) )
55		2cn 11091	. . . 4 |- 2 e. CC
56	31	nn0cni 11304	. . . 4 |- B e. CC
57	33	nn0cni 11304	. . . 4 |- F e. CC
58	55, 56, 57	adddii 10050	. . 3 |- ( 2 x. ( B + F ) ) = ( ( 2 x. B ) + ( 2 x. F ) )
59		log2ublem2.6	. . . 4 |- ( B + F ) = G
60	59	oveq2i 6661	. . 3 |- ( 2 x. ( B + F ) ) = ( 2 x. G )
61	58, 60	eqtr3i 2646	. 2 |- ( ( 2 x. B ) + ( 2 x. F ) ) = ( 2 x. G )
62		7nn 11190	. . . . . . . . 9 |- 7 e. NN
63	62	nnnn0i 11300	. . . . . . . 8 |- 7 e. NN0
64		nnexpcl 12873	. . . . . . . 8 |- ( ( 3 e. NN /\ 7 e. NN0 ) -> ( 3 ^ 7 ) e. NN )
65	6, 63, 64	mp2an 708	. . . . . . 7 |- ( 3 ^ 7 ) e. NN
66		5nn 11188	. . . . . . . 8 |- 5 e. NN
67	66, 62	nnmulcli 11044	. . . . . . 7 |- ( 5 x. 7 ) e. NN
68	65, 67	nnmulcli 11044	. . . . . 6 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. NN
69	68	nnrei 11029	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. RR
70	69, 5	remulcli 10054	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 ) e. RR
71	70	leidi 10562	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 ) <_ ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 )
72	6	nnnn0i 11300	. . . . . . . . . . . 12 |- 3 e. NN0
73		nnexpcl 12873	. . . . . . . . . . . 12 |- ( ( 9 e. NN /\ 3 e. NN0 ) -> ( 9 ^ 3 ) e. NN )
74	14, 72, 73	mp2an 708	. . . . . . . . . . 11 |- ( 9 ^ 3 ) e. NN
75	74	nncni 11030	. . . . . . . . . 10 |- ( 9 ^ 3 ) e. CC
76	67	nncni 11030	. . . . . . . . . 10 |- ( 5 x. 7 ) e. CC
77	75, 76	mulcomi 10046	. . . . . . . . 9 |- ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) = ( ( 5 x. 7 ) x. ( 9 ^ 3 ) )
78		log2ublem2.8	. . . . . . . . . . . . 13 |- ( M + N ) = 3
79		log2ublem2.7	. . . . . . . . . . . . . . 15 |- M e. NN0
80	79	nn0cni 11304	. . . . . . . . . . . . . 14 |- M e. CC
81	23	nn0cni 11304	. . . . . . . . . . . . . 14 |- N e. CC
82	80, 81	addcomi 10227	. . . . . . . . . . . . 13 |- ( M + N ) = ( N + M )
83	78, 82	eqtr3i 2646	. . . . . . . . . . . 12 |- 3 = ( N + M )
84	83	oveq2i 6661	. . . . . . . . . . 11 |- ( 9 ^ 3 ) = ( 9 ^ ( N + M ) )
85	14	nncni 11030	. . . . . . . . . . . 12 |- 9 e. CC
86		expadd 12902	. . . . . . . . . . . 12 |- ( ( 9 e. CC /\ N e. NN0 /\ M e. NN0 ) -> ( 9 ^ ( N + M ) ) = ( ( 9 ^ N ) x. ( 9 ^ M ) ) )
87	85, 23, 79, 86	mp3an 1424	. . . . . . . . . . 11 |- ( 9 ^ ( N + M ) ) = ( ( 9 ^ N ) x. ( 9 ^ M ) )
88	84, 87	eqtri 2644	. . . . . . . . . 10 |- ( 9 ^ 3 ) = ( ( 9 ^ N ) x. ( 9 ^ M ) )
89	88	oveq2i 6661	. . . . . . . . 9 |- ( ( 5 x. 7 ) x. ( 9 ^ 3 ) ) = ( ( 5 x. 7 ) x. ( ( 9 ^ N ) x. ( 9 ^ M ) ) )
90	29	nncni 11030	. . . . . . . . . 10 |- ( 9 ^ N ) e. CC
91		nnexpcl 12873	. . . . . . . . . . . 12 |- ( ( 9 e. NN /\ M e. NN0 ) -> ( 9 ^ M ) e. NN )
92	14, 79, 91	mp2an 708	. . . . . . . . . . 11 |- ( 9 ^ M ) e. NN
93	92	nncni 11030	. . . . . . . . . 10 |- ( 9 ^ M ) e. CC
94	76, 90, 93	mul12i 10231	. . . . . . . . 9 |- ( ( 5 x. 7 ) x. ( ( 9 ^ N ) x. ( 9 ^ M ) ) ) = ( ( 9 ^ N ) x. ( ( 5 x. 7 ) x. ( 9 ^ M ) ) )
95	77, 89, 94	3eqtri 2648	. . . . . . . 8 |- ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) = ( ( 9 ^ N ) x. ( ( 5 x. 7 ) x. ( 9 ^ M ) ) )
96		log2ublem2.9	. . . . . . . . 9 |- ( ( 5 x. 7 ) x. ( 9 ^ M ) ) = ( ( ( 2 x. N ) + 1 ) x. F )
97	96	oveq2i 6661	. . . . . . . 8 |- ( ( 9 ^ N ) x. ( ( 5 x. 7 ) x. ( 9 ^ M ) ) ) = ( ( 9 ^ N ) x. ( ( ( 2 x. N ) + 1 ) x. F ) )
98	95, 97	eqtri 2644	. . . . . . 7 |- ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) = ( ( 9 ^ N ) x. ( ( ( 2 x. N ) + 1 ) x. F ) )
99	98	oveq2i 6661	. . . . . 6 |- ( 3 x. ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) ) = ( 3 x. ( ( 9 ^ N ) x. ( ( ( 2 x. N ) + 1 ) x. F ) ) )
100		df-7 11084	. . . . . . . . . 10 |- 7 = ( 6 + 1 )
101	100	oveq2i 6661	. . . . . . . . 9 |- ( 3 ^ 7 ) = ( 3 ^ ( 6 + 1 ) )
102		3cn 11095	. . . . . . . . . . 11 |- 3 e. CC
103		6nn0 11313	. . . . . . . . . . 11 |- 6 e. NN0
104		expp1 12867	. . . . . . . . . . 11 |- ( ( 3 e. CC /\ 6 e. NN0 ) -> ( 3 ^ ( 6 + 1 ) ) = ( ( 3 ^ 6 ) x. 3 ) )
105	102, 103, 104	mp2an 708	. . . . . . . . . 10 |- ( 3 ^ ( 6 + 1 ) ) = ( ( 3 ^ 6 ) x. 3 )
106		expmul 12905	. . . . . . . . . . . . 13 |- ( ( 3 e. CC /\ 2 e. NN0 /\ 3 e. NN0 ) -> ( 3 ^ ( 2 x. 3 ) ) = ( ( 3 ^ 2 ) ^ 3 ) )
107	102, 7, 72, 106	mp3an 1424	. . . . . . . . . . . 12 |- ( 3 ^ ( 2 x. 3 ) ) = ( ( 3 ^ 2 ) ^ 3 )
108	55, 102	mulcomi 10046	. . . . . . . . . . . . . 14 |- ( 2 x. 3 ) = ( 3 x. 2 )
109		3t2e6 11179	. . . . . . . . . . . . . 14 |- ( 3 x. 2 ) = 6
110	108, 109	eqtri 2644	. . . . . . . . . . . . 13 |- ( 2 x. 3 ) = 6
111	110	oveq2i 6661	. . . . . . . . . . . 12 |- ( 3 ^ ( 2 x. 3 ) ) = ( 3 ^ 6 )
112		sq3 12961	. . . . . . . . . . . . 13 |- ( 3 ^ 2 ) = 9
113	112	oveq1i 6660	. . . . . . . . . . . 12 |- ( ( 3 ^ 2 ) ^ 3 ) = ( 9 ^ 3 )
114	107, 111, 113	3eqtr3i 2652	. . . . . . . . . . 11 |- ( 3 ^ 6 ) = ( 9 ^ 3 )
115	114	oveq1i 6660	. . . . . . . . . 10 |- ( ( 3 ^ 6 ) x. 3 ) = ( ( 9 ^ 3 ) x. 3 )
116	105, 115	eqtri 2644	. . . . . . . . 9 |- ( 3 ^ ( 6 + 1 ) ) = ( ( 9 ^ 3 ) x. 3 )
117	75, 102	mulcomi 10046	. . . . . . . . 9 |- ( ( 9 ^ 3 ) x. 3 ) = ( 3 x. ( 9 ^ 3 ) )
118	101, 116, 117	3eqtri 2648	. . . . . . . 8 |- ( 3 ^ 7 ) = ( 3 x. ( 9 ^ 3 ) )
119	118	oveq1i 6660	. . . . . . 7 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) = ( ( 3 x. ( 9 ^ 3 ) ) x. ( 5 x. 7 ) )
120	102, 75, 76	mulassi 10049	. . . . . . 7 |- ( ( 3 x. ( 9 ^ 3 ) ) x. ( 5 x. 7 ) ) = ( 3 x. ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) )
121	119, 120	eqtri 2644	. . . . . 6 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) = ( 3 x. ( ( 9 ^ 3 ) x. ( 5 x. 7 ) ) )
122	26	nncni 11030	. . . . . . . . 9 |- ( ( 2 x. N ) + 1 ) e. CC
123	102, 122, 90	mul32i 10232	. . . . . . . 8 |- ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) = ( ( 3 x. ( 9 ^ N ) ) x. ( ( 2 x. N ) + 1 ) )
124	123	oveq1i 6660	. . . . . . 7 |- ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. F ) = ( ( ( 3 x. ( 9 ^ N ) ) x. ( ( 2 x. N ) + 1 ) ) x. F )
125	102, 90	mulcli 10045	. . . . . . . 8 |- ( 3 x. ( 9 ^ N ) ) e. CC
126	125, 122, 57	mulassi 10049	. . . . . . 7 |- ( ( ( 3 x. ( 9 ^ N ) ) x. ( ( 2 x. N ) + 1 ) ) x. F ) = ( ( 3 x. ( 9 ^ N ) ) x. ( ( ( 2 x. N ) + 1 ) x. F ) )
127	122, 57	mulcli 10045	. . . . . . . 8 |- ( ( ( 2 x. N ) + 1 ) x. F ) e. CC
128	102, 90, 127	mulassi 10049	. . . . . . 7 |- ( ( 3 x. ( 9 ^ N ) ) x. ( ( ( 2 x. N ) + 1 ) x. F ) ) = ( 3 x. ( ( 9 ^ N ) x. ( ( ( 2 x. N ) + 1 ) x. F ) ) )
129	124, 126, 128	3eqtri 2648	. . . . . 6 |- ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. F ) = ( 3 x. ( ( 9 ^ N ) x. ( ( ( 2 x. N ) + 1 ) x. F ) ) )
130	99, 121, 129	3eqtr4i 2654	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) = ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. F )
131	130	oveq2i 6661	. . . 4 |- ( 2 x. ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) ) = ( 2 x. ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. F ) )
132	65	nncni 11030	. . . . . 6 |- ( 3 ^ 7 ) e. CC
133	132, 76	mulcli 10045	. . . . 5 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. CC
134	133, 55	mulcomi 10046	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 ) = ( 2 x. ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) )
135	30	nncni 11030	. . . . 5 |- ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) e. CC
136	135, 55, 57	mul12i 10231	. . . 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 ) )
137	131, 134, 136	3eqtr4i 2654	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 ) = ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. ( 2 x. F ) )
138	71, 137	breqtri 4678	. 2 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 2 ) <_ ( ( ( 3 x. ( ( 2 x. N ) + 1 ) ) x. ( 9 ^ N ) ) x. ( 2 x. F ) )
139	1, 22, 7, 30, 32, 34, 54, 61, 138	log2ublem1 24673	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 384   = wceq 1483   T. wtru 1484   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   5c5 11073   6c6 11074   7c7 11075   9c9 11077   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   sum_csu 14416

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

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  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-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  log2ublem3  24675