Theorem log2ublem3 16085

Description: Lemma for log2ublog2 16086. In decimal, this is a proof that the first four terms of the series for log 2 is less than 5 3 0 5 6 / 7 6 5 4 5. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.)

Assertion

Ref	Expression
log2ublem3	|- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... 3 ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) <_ ;;;; 5 3 0 5 6

Proof of Theorem log2ublem3

Step	Hyp	Ref	Expression
1		0le0 9393	. . . . . . 7 |- 0 <_ 0
2		fz00m1 10451	. . . . . . . . . . 11 |- ( 0 ... ( 0 - 1 ) ) = (/)
3	2	sumeq1i 12129	. . . . . . . . . 10 |- sum_ n e. ( 0 ... ( 0 - 1 ) ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) = sum_ n e. (/) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) )
4		sum0 12155	. . . . . . . . . 10 |- sum_ n e. (/) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) = 0
5	3, 4	eqtri 2259	. . . . . . . . 9 |- sum_ n e. ( 0 ... ( 0 - 1 ) ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) = 0
6	5	oveq2i 6096	. . . . . . . 8 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... ( 0 - 1 ) ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) = ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 0 )
7		3cn 9379	. . . . . . . . . . 11 |- 3 e. CC
8		7nn0 9585	. . . . . . . . . . 11 |- 7 e. NN0
9		expcl 10994	. . . . . . . . . . 11 |- ( ( 3 e. CC /\ 7 e. NN0 ) -> ( 3 ^ 7 ) e. CC )
10	7, 8, 9	mp2an 430	. . . . . . . . . 10 |- ( 3 ^ 7 ) e. CC
11		5cn 9384	. . . . . . . . . . 11 |- 5 e. CC
12		7cn 9388	. . . . . . . . . . 11 |- 7 e. CC
13	11, 12	mulcli 8331	. . . . . . . . . 10 |- ( 5 x. 7 ) e. CC
14	10, 13	mulcli 8331	. . . . . . . . 9 |- ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) e. CC
15	14	mul01i 8718	. . . . . . . 8 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. 0 ) = 0
16	6, 15	eqtri 2259	. . . . . . 7 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... ( 0 - 1 ) ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) = 0
17		2cn 9375	. . . . . . . 8 |- 2 e. CC
18	17	mul01i 8718	. . . . . . 7 |- ( 2 x. 0 ) = 0
19	1, 16, 18	3brtr4i 4160	. . . . . 6 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... ( 0 - 1 ) ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) <_ ( 2 x. 0 )
20		0nn0 9578	. . . . . 6 |- 0 e. NN0
21		2nn0 9580	. . . . . . . . . 10 |- 2 e. NN0
22		5nn0 9583	. . . . . . . . . 10 |- 5 e. NN0
23	21, 22	deccl 9791	. . . . . . . . 9 |- ; 2 5 e. NN0
24	23, 22	deccl 9791	. . . . . . . 8 |- ;; 2 5 5 e. NN0
25		1nn0 9579	. . . . . . . 8 |- 1 e. NN0
26	24, 25	deccl 9791	. . . . . . 7 |- ;;; 2 5 5 1 e. NN0
27	26, 22	deccl 9791	. . . . . 6 |- ;;;; 2 5 5 1 5 e. NN0
28		eqid 2238	. . . . . 6 |- ( 0 - 1 ) = ( 0 - 1 )
29	27	nn0cni 9575	. . . . . . 7 |- ;;;; 2 5 5 1 5 e. CC
30	29	addlidi 8469	. . . . . 6 |- ( 0 + ;;;; 2 5 5 1 5 ) = ;;;; 2 5 5 1 5
31		3nn0 9581	. . . . . 6 |- 3 e. NN0
32	7	addridi 8468	. . . . . 6 |- ( 3 + 0 ) = 3
33	29	mullidi 8329	. . . . . . 7 |- ( 1 x. ;;;; 2 5 5 1 5 ) = ;;;; 2 5 5 1 5
34	18	oveq1i 6095	. . . . . . . . 9 |- ( ( 2 x. 0 ) + 1 ) = ( 0 + 1 )
35		0p1e1 9418	. . . . . . . . 9 |- ( 0 + 1 ) = 1
36	34, 35	eqtri 2259	. . . . . . . 8 |- ( ( 2 x. 0 ) + 1 ) = 1
37	36	oveq1i 6095	. . . . . . 7 |- ( ( ( 2 x. 0 ) + 1 ) x. ;;;; 2 5 5 1 5 ) = ( 1 x. ;;;; 2 5 5 1 5 )
38	22, 8	nn0mulcli 9601	. . . . . . . 8 |- ( 5 x. 7 ) e. NN0
39	8, 21	deccl 9791	. . . . . . . 8 |- ; 7 2 e. NN0
40		9nn0 9587	. . . . . . . 8 |- 9 e. NN0
41		2p1e3 9438	. . . . . . . . 9 |- ( 2 + 1 ) = 3
42		8nn0 9586	. . . . . . . . . 10 |- 8 e. NN0
43		1p1e2 9421	. . . . . . . . . . 11 |- ( 1 + 1 ) = 2
44		9cn 9392	. . . . . . . . . . . . . 14 |- 9 e. CC
45		exp1 10982	. . . . . . . . . . . . . 14 |- ( 9 e. CC -> ( 9 ^ 1 ) = 9 )
46	44, 45	ax-mp 5	. . . . . . . . . . . . 13 |- ( 9 ^ 1 ) = 9
47	46	oveq1i 6095	. . . . . . . . . . . 12 |- ( ( 9 ^ 1 ) x. 9 ) = ( 9 x. 9 )
48		9t9e81 9905	. . . . . . . . . . . 12 |- ( 9 x. 9 ) = ; 8 1
49	47, 48	eqtri 2259	. . . . . . . . . . 11 |- ( ( 9 ^ 1 ) x. 9 ) = ; 8 1
50	40, 25, 43, 49	numexpp1 13203	. . . . . . . . . 10 |- ( 9 ^ 2 ) = ; 8 1
51		8cn 9390	. . . . . . . . . . 11 |- 8 e. CC
52		9t8e72 9904	. . . . . . . . . . 11 |- ( 9 x. 8 ) = ; 7 2
53	44, 51, 52	mulcomli 8333	. . . . . . . . . 10 |- ( 8 x. 9 ) = ; 7 2
54	44	mullidi 8329	. . . . . . . . . 10 |- ( 1 x. 9 ) = 9
55	40, 42, 25, 50, 40, 53, 54	decmul1 9840	. . . . . . . . 9 |- ( ( 9 ^ 2 ) x. 9 ) = ;; 7 2 9
56	40, 21, 41, 55	numexpp1 13203	. . . . . . . 8 |- ( 9 ^ 3 ) = ;; 7 2 9
57	31, 25	deccl 9791	. . . . . . . 8 |- ; 3 1 e. NN0
58		eqid 2238	. . . . . . . . 9 |- ; 7 2 = ; 7 2
59		eqid 2238	. . . . . . . . 9 |- ; 3 1 = ; 3 1
60		7t5e35 9888	. . . . . . . . . . 11 |- ( 7 x. 5 ) = ; 3 5
61	12, 11, 60	mulcomli 8333	. . . . . . . . . 10 |- ( 5 x. 7 ) = ; 3 5
62		7p3e10 9851	. . . . . . . . . . 11 |- ( 7 + 3 ) = ; 1 0
63	12, 7, 62	addcomli 8471	. . . . . . . . . 10 |- ( 3 + 7 ) = ; 1 0
64		ax-1cn 8272	. . . . . . . . . . . . 13 |- 1 e. CC
65		3p1e4 9440	. . . . . . . . . . . . 13 |- ( 3 + 1 ) = 4
66	7, 64, 65	addcomli 8471	. . . . . . . . . . . 12 |- ( 1 + 3 ) = 4
67	66	oveq2i 6096	. . . . . . . . . . 11 |- ( ( 3 x. 7 ) + ( 1 + 3 ) ) = ( ( 3 x. 7 ) + 4 )
68		4nn0 9582	. . . . . . . . . . . 12 |- 4 e. NN0
69		7t3e21 9886	. . . . . . . . . . . . 13 |- ( 7 x. 3 ) = ; 2 1
70	12, 7, 69	mulcomli 8333	. . . . . . . . . . . 12 |- ( 3 x. 7 ) = ; 2 1
71		4cn 9382	. . . . . . . . . . . . 13 |- 4 e. CC
72		4p1e5 9441	. . . . . . . . . . . . 13 |- ( 4 + 1 ) = 5
73	71, 64, 72	addcomli 8471	. . . . . . . . . . . 12 |- ( 1 + 4 ) = 5
74	21, 25, 68, 70, 73	decaddi 9836	. . . . . . . . . . 11 |- ( ( 3 x. 7 ) + 4 ) = ; 2 5
75	67, 74	eqtri 2259	. . . . . . . . . 10 |- ( ( 3 x. 7 ) + ( 1 + 3 ) ) = ; 2 5
76	61	oveq1i 6095	. . . . . . . . . . 11 |- ( ( 5 x. 7 ) + 0 ) = (; 3 5 + 0 )
77	31, 22	deccl 9791	. . . . . . . . . . . . 13 |- ; 3 5 e. NN0
78	77	nn0cni 9575	. . . . . . . . . . . 12 |- ; 3 5 e. CC
79	78	addridi 8468	. . . . . . . . . . 11 |- (; 3 5 + 0 ) = ; 3 5
80	76, 79	eqtri 2259	. . . . . . . . . 10 |- ( ( 5 x. 7 ) + 0 ) = ; 3 5
81	31, 22, 25, 20, 61, 63, 8, 22, 31, 75, 80	decmac 9828	. . . . . . . . 9 |- ( ( ( 5 x. 7 ) x. 7 ) + ( 3 + 7 ) ) = ;; 2 5 5
82	25	dec0h 9798	. . . . . . . . . 10 |- 1 = ; 0 1
83		3t2e6 9461	. . . . . . . . . . . 12 |- ( 3 x. 2 ) = 6
84	83, 35	oveq12i 6097	. . . . . . . . . . 11 |- ( ( 3 x. 2 ) + ( 0 + 1 ) ) = ( 6 + 1 )
85		6p1e7 9443	. . . . . . . . . . 11 |- ( 6 + 1 ) = 7
86	84, 85	eqtri 2259	. . . . . . . . . 10 |- ( ( 3 x. 2 ) + ( 0 + 1 ) ) = 7
87		5t2e10 9876	. . . . . . . . . . 11 |- ( 5 x. 2 ) = ; 1 0
88	25, 20, 35, 87	decsuc 9807	. . . . . . . . . 10 |- ( ( 5 x. 2 ) + 1 ) = ; 1 1
89	31, 22, 20, 25, 61, 82, 21, 25, 25, 86, 88	decmac 9828	. . . . . . . . 9 |- ( ( ( 5 x. 7 ) x. 2 ) + 1 ) = ; 7 1
90	8, 21, 31, 25, 58, 59, 38, 25, 8, 81, 89	decma2c 9829	. . . . . . . 8 |- ( ( ( 5 x. 7 ) x. ; 7 2 ) + ; 3 1 ) = ;;; 2 5 5 1
91		9t3e27 9899	. . . . . . . . . . 11 |- ( 9 x. 3 ) = ; 2 7
92	44, 7, 91	mulcomli 8333	. . . . . . . . . 10 |- ( 3 x. 9 ) = ; 2 7
93		7p4e11 9852	. . . . . . . . . 10 |- ( 7 + 4 ) = ; 1 1
94	21, 8, 68, 92, 41, 25, 93	decaddci 9837	. . . . . . . . 9 |- ( ( 3 x. 9 ) + 4 ) = ; 3 1
95		9t5e45 9901	. . . . . . . . . 10 |- ( 9 x. 5 ) = ; 4 5
96	44, 11, 95	mulcomli 8333	. . . . . . . . 9 |- ( 5 x. 9 ) = ; 4 5
97	40, 31, 22, 61, 22, 68, 94, 96	decmul1c 9841	. . . . . . . 8 |- ( ( 5 x. 7 ) x. 9 ) = ;; 3 1 5
98	38, 39, 40, 56, 22, 57, 90, 97	decmul2c 9842	. . . . . . 7 |- ( ( 5 x. 7 ) x. ( 9 ^ 3 ) ) = ;;;; 2 5 5 1 5
99	33, 37, 98	3eqtr4ri 2270	. . . . . 6 |- ( ( 5 x. 7 ) x. ( 9 ^ 3 ) ) = ( ( ( 2 x. 0 ) + 1 ) x. ;;;; 2 5 5 1 5 )
100	19, 20, 27, 20, 28, 30, 31, 32, 99	log2ublem2 16084	. . . . 5 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... 0 ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) <_ ( 2 x. ;;;; 2 5 5 1 5 )
101	40, 68	deccl 9791	. . . . . 6 |- ; 9 4 e. NN0
102	101, 22	deccl 9791	. . . . 5 |- ;; 9 4 5 e. NN0
103		1m1e0 9373	. . . . 5 |- ( 1 - 1 ) = 0
104		eqid 2238	. . . . . 6 |- ;;;; 2 5 5 1 5 = ;;;; 2 5 5 1 5
105		eqid 2238	. . . . . 6 |- ;; 9 4 5 = ;; 9 4 5
106		6nn0 9584	. . . . . . . . 9 |- 6 e. NN0
107	21, 106	deccl 9791	. . . . . . . 8 |- ; 2 6 e. NN0
108	107, 68	deccl 9791	. . . . . . 7 |- ;; 2 6 4 e. NN0
109		5p1e6 9442	. . . . . . 7 |- ( 5 + 1 ) = 6
110		eqid 2238	. . . . . . . 8 |- ;;; 2 5 5 1 = ;;; 2 5 5 1
111		eqid 2238	. . . . . . . 8 |- ; 9 4 = ; 9 4
112		eqid 2238	. . . . . . . . 9 |- ;; 2 5 5 = ;; 2 5 5
113		eqid 2238	. . . . . . . . . 10 |- ; 2 5 = ; 2 5
114	21, 22, 109, 113	decsuc 9807	. . . . . . . . 9 |- (; 2 5 + 1 ) = ; 2 6
115		9p5e14 9866	. . . . . . . . . 10 |- ( 9 + 5 ) = ; 1 4
116	44, 11, 115	addcomli 8471	. . . . . . . . 9 |- ( 5 + 9 ) = ; 1 4
117	23, 22, 40, 112, 114, 68, 116	decaddci 9837	. . . . . . . 8 |- (;; 2 5 5 + 9 ) = ;; 2 6 4
118	24, 25, 40, 68, 110, 111, 117, 73	decadd 9830	. . . . . . 7 |- (;;; 2 5 5 1 + ; 9 4 ) = ;;; 2 6 4 5
119	108, 22, 109, 118	decsuc 9807	. . . . . 6 |- ( (;;; 2 5 5 1 + ; 9 4 ) + 1 ) = ;;; 2 6 4 6
120		5p5e10 9847	. . . . . 6 |- ( 5 + 5 ) = ; 1 0
121	26, 22, 101, 22, 104, 105, 119, 120	decaddc2 9832	. . . . 5 |- (;;;; 2 5 5 1 5 + ;; 9 4 5 ) = ;;;; 2 6 4 6 0
122	44	sqvali 11056	. . . . . . . 8 |- ( 9 ^ 2 ) = ( 9 x. 9 )
123		3t3e9 9462	. . . . . . . . 9 |- ( 3 x. 3 ) = 9
124	123	oveq1i 6095	. . . . . . . 8 |- ( ( 3 x. 3 ) x. 9 ) = ( 9 x. 9 )
125	7, 7, 44	mulassi 8335	. . . . . . . 8 |- ( ( 3 x. 3 ) x. 9 ) = ( 3 x. ( 3 x. 9 ) )
126	122, 124, 125	3eqtr2i 2265	. . . . . . 7 |- ( 9 ^ 2 ) = ( 3 x. ( 3 x. 9 ) )
127	126	oveq2i 6096	. . . . . 6 |- ( ( 5 x. 7 ) x. ( 9 ^ 2 ) ) = ( ( 5 x. 7 ) x. ( 3 x. ( 3 x. 9 ) ) )
128	7, 44	mulcli 8331	. . . . . . . 8 |- ( 3 x. 9 ) e. CC
129	13, 7, 128	mul12i 8472	. . . . . . 7 |- ( ( 5 x. 7 ) x. ( 3 x. ( 3 x. 9 ) ) ) = ( 3 x. ( ( 5 x. 7 ) x. ( 3 x. 9 ) ) )
130	21, 68	deccl 9791	. . . . . . . . 9 |- ; 2 4 e. NN0
131		eqid 2238	. . . . . . . . . 10 |- ; 2 4 = ; 2 4
132	83, 41	oveq12i 6097	. . . . . . . . . . 11 |- ( ( 3 x. 2 ) + ( 2 + 1 ) ) = ( 6 + 3 )
133		6p3e9 9455	. . . . . . . . . . 11 |- ( 6 + 3 ) = 9
134	132, 133	eqtri 2259	. . . . . . . . . 10 |- ( ( 3 x. 2 ) + ( 2 + 1 ) ) = 9
135	71	addlidi 8469	. . . . . . . . . . 11 |- ( 0 + 4 ) = 4
136	25, 20, 68, 87, 135	decaddi 9836	. . . . . . . . . 10 |- ( ( 5 x. 2 ) + 4 ) = ; 1 4
137	31, 22, 21, 68, 61, 131, 21, 68, 25, 134, 136	decmac 9828	. . . . . . . . 9 |- ( ( ( 5 x. 7 ) x. 2 ) + ; 2 4 ) = ; 9 4
138	21, 25, 31, 70, 66	decaddi 9836	. . . . . . . . . 10 |- ( ( 3 x. 7 ) + 3 ) = ; 2 4
139	8, 31, 22, 61, 22, 31, 138, 61	decmul1c 9841	. . . . . . . . 9 |- ( ( 5 x. 7 ) x. 7 ) = ;; 2 4 5
140	38, 21, 8, 92, 22, 130, 137, 139	decmul2c 9842	. . . . . . . 8 |- ( ( 5 x. 7 ) x. ( 3 x. 9 ) ) = ;; 9 4 5
141	140	oveq2i 6096	. . . . . . 7 |- ( 3 x. ( ( 5 x. 7 ) x. ( 3 x. 9 ) ) ) = ( 3 x. ;; 9 4 5 )
142	129, 141	eqtri 2259	. . . . . 6 |- ( ( 5 x. 7 ) x. ( 3 x. ( 3 x. 9 ) ) ) = ( 3 x. ;; 9 4 5 )
143		df-3 9364	. . . . . . . 8 |- 3 = ( 2 + 1 )
144	17	mulridi 8328	. . . . . . . . 9 |- ( 2 x. 1 ) = 2
145	144	oveq1i 6095	. . . . . . . 8 |- ( ( 2 x. 1 ) + 1 ) = ( 2 + 1 )
146	143, 145	eqtr4i 2262	. . . . . . 7 |- 3 = ( ( 2 x. 1 ) + 1 )
147	146	oveq1i 6095	. . . . . 6 |- ( 3 x. ;; 9 4 5 ) = ( ( ( 2 x. 1 ) + 1 ) x. ;; 9 4 5 )
148	127, 142, 147	3eqtri 2263	. . . . 5 |- ( ( 5 x. 7 ) x. ( 9 ^ 2 ) ) = ( ( ( 2 x. 1 ) + 1 ) x. ;; 9 4 5 )
149	100, 27, 102, 25, 103, 121, 21, 41, 148	log2ublem2 16084	. . . 4 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... 1 ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) <_ ( 2 x. ;;;; 2 6 4 6 0 )
150	108, 106	deccl 9791	. . . . 5 |- ;;; 2 6 4 6 e. NN0
151	150, 20	deccl 9791	. . . 4 |- ;;;; 2 6 4 6 0 e. NN0
152	106, 31	deccl 9791	. . . 4 |- ; 6 3 e. NN0
153		2m1e1 9422	. . . 4 |- ( 2 - 1 ) = 1
154		eqid 2238	. . . . 5 |- ;;;; 2 6 4 6 0 = ;;;; 2 6 4 6 0
155		eqid 2238	. . . . 5 |- ; 6 3 = ; 6 3
156		eqid 2238	. . . . . 6 |- ;;; 2 6 4 6 = ;;; 2 6 4 6
157		eqid 2238	. . . . . . 7 |- ;; 2 6 4 = ;; 2 6 4
158	107, 68, 72, 157	decsuc 9807	. . . . . 6 |- (;; 2 6 4 + 1 ) = ;; 2 6 5
159		6p6e12 9850	. . . . . 6 |- ( 6 + 6 ) = ; 1 2
160	108, 106, 106, 156, 158, 21, 159	decaddci 9837	. . . . 5 |- (;;; 2 6 4 6 + 6 ) = ;;; 2 6 5 2
161	7	addlidi 8469	. . . . 5 |- ( 0 + 3 ) = 3
162	150, 20, 106, 31, 154, 155, 160, 161	decadd 9830	. . . 4 |- (;;;; 2 6 4 6 0 + ; 6 3 ) = ;;;; 2 6 5 2 3
163		1p2e3 9439	. . . 4 |- ( 1 + 2 ) = 3
164	46	oveq2i 6096	. . . . 5 |- ( ( 5 x. 7 ) x. ( 9 ^ 1 ) ) = ( ( 5 x. 7 ) x. 9 )
165	11, 12, 44	mulassi 8335	. . . . . 6 |- ( ( 5 x. 7 ) x. 9 ) = ( 5 x. ( 7 x. 9 ) )
166		9t7e63 9903	. . . . . . . 8 |- ( 9 x. 7 ) = ; 6 3
167	44, 12, 166	mulcomli 8333	. . . . . . 7 |- ( 7 x. 9 ) = ; 6 3
168	167	oveq2i 6096	. . . . . 6 |- ( 5 x. ( 7 x. 9 ) ) = ( 5 x. ; 6 3 )
169	165, 168	eqtri 2259	. . . . 5 |- ( ( 5 x. 7 ) x. 9 ) = ( 5 x. ; 6 3 )
170		df-5 9366	. . . . . . 7 |- 5 = ( 4 + 1 )
171		2t2e4 9459	. . . . . . . 8 |- ( 2 x. 2 ) = 4
172	171	oveq1i 6095	. . . . . . 7 |- ( ( 2 x. 2 ) + 1 ) = ( 4 + 1 )
173	170, 172	eqtr4i 2262	. . . . . 6 |- 5 = ( ( 2 x. 2 ) + 1 )
174	173	oveq1i 6095	. . . . 5 |- ( 5 x. ; 6 3 ) = ( ( ( 2 x. 2 ) + 1 ) x. ; 6 3 )
175	164, 169, 174	3eqtri 2263	. . . 4 |- ( ( 5 x. 7 ) x. ( 9 ^ 1 ) ) = ( ( ( 2 x. 2 ) + 1 ) x. ; 6 3 )
176	149, 151, 152, 21, 153, 162, 25, 163, 175	log2ublem2 16084	. . 3 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... 2 ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) <_ ( 2 x. ;;;; 2 6 5 2 3 )
177	107, 22	deccl 9791	. . . . 5 |- ;; 2 6 5 e. NN0
178	177, 21	deccl 9791	. . . 4 |- ;;; 2 6 5 2 e. NN0
179	178, 31	deccl 9791	. . 3 |- ;;;; 2 6 5 2 3 e. NN0
180		3m1e2 9424	. . 3 |- ( 3 - 1 ) = 2
181		eqid 2238	. . . 4 |- ;;;; 2 6 5 2 3 = ;;;; 2 6 5 2 3
182		5p3e8 9452	. . . . 5 |- ( 5 + 3 ) = 8
183	11, 7, 182	addcomli 8471	. . . 4 |- ( 3 + 5 ) = 8
184	178, 31, 22, 181, 183	decaddi 9836	. . 3 |- (;;;; 2 6 5 2 3 + 5 ) = ;;;; 2 6 5 2 8
185	12, 11	mulcli 8331	. . . . 5 |- ( 7 x. 5 ) e. CC
186	185	mulridi 8328	. . . 4 |- ( ( 7 x. 5 ) x. 1 ) = ( 7 x. 5 )
187	11, 12	mulcomi 8332	. . . . 5 |- ( 5 x. 7 ) = ( 7 x. 5 )
188		exp0 10980	. . . . . 6 |- ( 9 e. CC -> ( 9 ^ 0 ) = 1 )
189	44, 188	ax-mp 5	. . . . 5 |- ( 9 ^ 0 ) = 1
190	187, 189	oveq12i 6097	. . . 4 |- ( ( 5 x. 7 ) x. ( 9 ^ 0 ) ) = ( ( 7 x. 5 ) x. 1 )
191	7, 17, 83	mulcomli 8333	. . . . . . 7 |- ( 2 x. 3 ) = 6
192	191	oveq1i 6095	. . . . . 6 |- ( ( 2 x. 3 ) + 1 ) = ( 6 + 1 )
193		df-7 9368	. . . . . 6 |- 7 = ( 6 + 1 )
194	192, 193	eqtr4i 2262	. . . . 5 |- ( ( 2 x. 3 ) + 1 ) = 7
195	194	oveq1i 6095	. . . 4 |- ( ( ( 2 x. 3 ) + 1 ) x. 5 ) = ( 7 x. 5 )
196	186, 190, 195	3eqtr4i 2269	. . 3 |- ( ( 5 x. 7 ) x. ( 9 ^ 0 ) ) = ( ( ( 2 x. 3 ) + 1 ) x. 5 )
197	176, 179, 22, 31, 180, 184, 20, 161, 196	log2ublem2 16084	. 2 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... 3 ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) <_ ( 2 x. ;;;; 2 6 5 2 8 )
198		eqid 2238	. . 3 |- ;;;; 2 6 5 2 8 = ;;;; 2 6 5 2 8
199		eqid 2238	. . . 4 |- ;;; 2 6 5 2 = ;;; 2 6 5 2
200		eqid 2238	. . . . 5 |- ;; 2 6 5 = ;; 2 6 5
201		00id 8467	. . . . . 6 |- ( 0 + 0 ) = 0
202	20	dec0h 9798	. . . . . 6 |- 0 = ; 0 0
203	201, 202	eqtri 2259	. . . . 5 |- ( 0 + 0 ) = ; 0 0
204		eqid 2238	. . . . . 6 |- ; 2 6 = ; 2 6
205	35, 82	eqtri 2259	. . . . . 6 |- ( 0 + 1 ) = ; 0 1
206	171, 35	oveq12i 6097	. . . . . . 7 |- ( ( 2 x. 2 ) + ( 0 + 1 ) ) = ( 4 + 1 )
207	206, 72	eqtri 2259	. . . . . 6 |- ( ( 2 x. 2 ) + ( 0 + 1 ) ) = 5
208		6cn 9386	. . . . . . . 8 |- 6 e. CC
209		6t2e12 9880	. . . . . . . 8 |- ( 6 x. 2 ) = ; 1 2
210	208, 17, 209	mulcomli 8333	. . . . . . 7 |- ( 2 x. 6 ) = ; 1 2
211	25, 21, 41, 210	decsuc 9807	. . . . . 6 |- ( ( 2 x. 6 ) + 1 ) = ; 1 3
212	21, 106, 20, 25, 204, 205, 21, 31, 25, 207, 211	decma2c 9829	. . . . 5 |- ( ( 2 x. ; 2 6 ) + ( 0 + 1 ) ) = ; 5 3
213	11, 17, 87	mulcomli 8333	. . . . . . 7 |- ( 2 x. 5 ) = ; 1 0
214	213	oveq1i 6095	. . . . . 6 |- ( ( 2 x. 5 ) + 0 ) = (; 1 0 + 0 )
215		dec10p 9819	. . . . . 6 |- (; 1 0 + 0 ) = ; 1 0
216	214, 215	eqtri 2259	. . . . 5 |- ( ( 2 x. 5 ) + 0 ) = ; 1 0
217	107, 22, 20, 20, 200, 203, 21, 20, 25, 212, 216	decma2c 9829	. . . 4 |- ( ( 2 x. ;; 2 6 5 ) + ( 0 + 0 ) ) = ;; 5 3 0
218	22	dec0h 9798	. . . . 5 |- 5 = ; 0 5
219	172, 72, 218	3eqtri 2263	. . . 4 |- ( ( 2 x. 2 ) + 1 ) = ; 0 5
220	177, 21, 20, 25, 199, 82, 21, 22, 20, 217, 219	decma2c 9829	. . 3 |- ( ( 2 x. ;;; 2 6 5 2 ) + 1 ) = ;;; 5 3 0 5
221		8t2e16 9891	. . . 4 |- ( 8 x. 2 ) = ; 1 6
222	51, 17, 221	mulcomli 8333	. . 3 |- ( 2 x. 8 ) = ; 1 6
223	21, 178, 42, 198, 106, 25, 220, 222	decmul2c 9842	. 2 |- ( 2 x. ;;;; 2 6 5 2 8 ) = ;;;; 5 3 0 5 6
224	197, 223	breqtri 4155	1 |- ( ( ( 3 ^ 7 ) x. ( 5 x. 7 ) ) x. sum_ n e. ( 0 ... 3 ) ( 2 / ( ( 3 x. ( ( 2 x. n ) + 1 ) ) x. ( 9 ^ n ) ) ) ) <_ ;;;; 5 3 0 5 6

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209   (/)c0 3520   class class class wbr 4130  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   <_ cle 8361   - cmin 8497   / cdiv 9002   2c2 9355   3c3 9356   4c4 9357   5c5 9358   6c6 9359   7c7 9360   8c8 9361   9c9 9362   NN0cn0 9563  ;cdc 9777   ...cfz 10411   ^cexp 10975   sum_csu 12119

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120

This theorem is used by:  log2ublog2  16086