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Theorem log2ublem3 20244
Description: Lemma for log2ub 20245. 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.)
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
StepHypRef Expression
1 0le0 9827 . . . . . . 7  |-  0  <_  0
2 0re 8838 . . . . . . . . . . . . 13  |-  0  e.  RR
3 ltm1 9596 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  (
0  -  1 )  <  0 )
42, 3ax-mp 8 . . . . . . . . . . . 12  |-  ( 0  -  1 )  <  0
5 0z 10035 . . . . . . . . . . . . 13  |-  0  e.  ZZ
6 peano2zm 10062 . . . . . . . . . . . . . 14  |-  ( 0  e.  ZZ  ->  (
0  -  1 )  e.  ZZ )
75, 6ax-mp 8 . . . . . . . . . . . . 13  |-  ( 0  -  1 )  e.  ZZ
8 fzn 10810 . . . . . . . . . . . . 13  |-  ( ( 0  e.  ZZ  /\  ( 0  -  1 )  e.  ZZ )  ->  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) ) )
95, 7, 8mp2an 653 . . . . . . . . . . . 12  |-  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) )
104, 9mpbi 199 . . . . . . . . . . 11  |-  ( 0 ... ( 0  -  1 ) )  =  (/)
1110sumeq1i 12171 . . . . . . . . . 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 ) ) )
12 sum0 12194 . . . . . . . . . 10  |-  sum_ n  e.  (/)  ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  =  0
1311, 12eqtri 2303 . . . . . . . . 9  |-  sum_ n  e.  ( 0 ... (
0  -  1 ) ) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  =  0
1413oveq2i 5869 . . . . . . . 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 )
15 3cn 9818 . . . . . . . . . . 11  |-  3  e.  CC
16 7nn0 9987 . . . . . . . . . . 11  |-  7  e.  NN0
17 expcl 11121 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  CC )
1815, 16, 17mp2an 653 . . . . . . . . . 10  |-  ( 3 ^ 7 )  e.  CC
19 5nn 9880 . . . . . . . . . . . 12  |-  5  e.  NN
2019nncni 9756 . . . . . . . . . . 11  |-  5  e.  CC
21 7nn 9882 . . . . . . . . . . . 12  |-  7  e.  NN
2221nncni 9756 . . . . . . . . . . 11  |-  7  e.  CC
2320, 22mulcli 8842 . . . . . . . . . 10  |-  ( 5  x.  7 )  e.  CC
2418, 23mulcli 8842 . . . . . . . . 9  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
2524mul01i 9002 . . . . . . . 8  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  0 )  =  0
2614, 25eqtri 2303 . . . . . . 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
27 2cn 9816 . . . . . . . 8  |-  2  e.  CC
2827mul01i 9002 . . . . . . 7  |-  ( 2  x.  0 )  =  0
291, 26, 283brtr4i 4051 . . . . . 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 )
30 0nn0 9980 . . . . . 6  |-  0  e.  NN0
31 2nn0 9982 . . . . . . . . . 10  |-  2  e.  NN0
32 5nn0 9985 . . . . . . . . . 10  |-  5  e.  NN0
3331, 32deccl 10138 . . . . . . . . 9  |- ; 2 5  e.  NN0
3433, 32deccl 10138 . . . . . . . 8  |- ;; 2 5 5  e.  NN0
35 1nn0 9981 . . . . . . . 8  |-  1  e.  NN0
3634, 35deccl 10138 . . . . . . 7  |- ;;; 2 5 5 1  e.  NN0
3736, 32deccl 10138 . . . . . 6  |- ;;;; 2 5 5 1 5  e.  NN0
38 eqid 2283 . . . . . 6  |-  ( 0  -  1 )  =  ( 0  -  1 )
3937nn0cni 9977 . . . . . . 7  |- ;;;; 2 5 5 1 5  e.  CC
4039addid2i 9000 . . . . . 6  |-  ( 0  + ;;;; 2 5 5 1 5 )  = ;;;; 2 5 5 1 5
41 3nn0 9983 . . . . . 6  |-  3  e.  NN0
4215addid1i 8999 . . . . . 6  |-  ( 3  +  0 )  =  3
4339mulid2i 8840 . . . . . . 7  |-  ( 1  x. ;;;; 2 5 5 1 5 )  = ;;;; 2 5 5 1 5
4428oveq1i 5868 . . . . . . . . 9  |-  ( ( 2  x.  0 )  +  1 )  =  ( 0  +  1 )
45 0p1e1 9839 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4644, 45eqtri 2303 . . . . . . . 8  |-  ( ( 2  x.  0 )  +  1 )  =  1
4746oveq1i 5868 . . . . . . 7  |-  ( ( ( 2  x.  0 )  +  1 )  x. ;;;; 2 5 5 1 5 )  =  ( 1  x. ;;;; 2 5 5 1 5 )
4832, 16nn0mulcli 10002 . . . . . . . 8  |-  ( 5  x.  7 )  e. 
NN0
4916, 31deccl 10138 . . . . . . . 8  |- ; 7 2  e.  NN0
50 9nn0 9989 . . . . . . . 8  |-  9  e.  NN0
51 2p1e3 9847 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
52 8nn0 9988 . . . . . . . . . 10  |-  8  e.  NN0
53 1p1e2 9840 . . . . . . . . . . 11  |-  ( 1  +  1 )  =  2
54 9nn 9884 . . . . . . . . . . . . . . 15  |-  9  e.  NN
5554nncni 9756 . . . . . . . . . . . . . 14  |-  9  e.  CC
56 exp1 11109 . . . . . . . . . . . . . 14  |-  ( 9  e.  CC  ->  (
9 ^ 1 )  =  9 )
5755, 56ax-mp 8 . . . . . . . . . . . . 13  |-  ( 9 ^ 1 )  =  9
5857oveq1i 5868 . . . . . . . . . . . 12  |-  ( ( 9 ^ 1 )  x.  9 )  =  ( 9  x.  9 )
59 9t9e81 10226 . . . . . . . . . . . 12  |-  ( 9  x.  9 )  = ; 8
1
6058, 59eqtri 2303 . . . . . . . . . . 11  |-  ( ( 9 ^ 1 )  x.  9 )  = ; 8
1
6150, 35, 53, 60numexpp1 13093 . . . . . . . . . 10  |-  ( 9 ^ 2 )  = ; 8
1
62 8nn 9883 . . . . . . . . . . . . . 14  |-  8  e.  NN
6362nncni 9756 . . . . . . . . . . . . 13  |-  8  e.  CC
64 9t8e72 10225 . . . . . . . . . . . . 13  |-  ( 9  x.  8 )  = ; 7
2
6555, 63, 64mulcomli 8844 . . . . . . . . . . . 12  |-  ( 8  x.  9 )  = ; 7
2
6665oveq1i 5868 . . . . . . . . . . 11  |-  ( ( 8  x.  9 )  +  0 )  =  (; 7 2  +  0 )
6749nn0cni 9977 . . . . . . . . . . . 12  |- ; 7 2  e.  CC
6867addid1i 8999 . . . . . . . . . . 11  |-  (; 7 2  +  0 )  = ; 7 2
6966, 68eqtri 2303 . . . . . . . . . 10  |-  ( ( 8  x.  9 )  +  0 )  = ; 7
2
7055mulid2i 8840 . . . . . . . . . . 11  |-  ( 1  x.  9 )  =  9
7150dec0h 10140 . . . . . . . . . . 11  |-  9  = ; 0 9
7270, 71eqtri 2303 . . . . . . . . . 10  |-  ( 1  x.  9 )  = ; 0
9
7350, 52, 35, 61, 50, 30, 69, 72decmul1c 10171 . . . . . . . . 9  |-  ( ( 9 ^ 2 )  x.  9 )  = ;; 7 2 9
7450, 31, 51, 73numexpp1 13093 . . . . . . . 8  |-  ( 9 ^ 3 )  = ;; 7 2 9
7541, 35deccl 10138 . . . . . . . 8  |- ; 3 1  e.  NN0
76 eqid 2283 . . . . . . . . 9  |- ; 7 2  = ; 7 2
77 eqid 2283 . . . . . . . . 9  |- ; 3 1  = ; 3 1
78 7t5e35 10209 . . . . . . . . . . 11  |-  ( 7  x.  5 )  = ; 3
5
7922, 20, 78mulcomli 8844 . . . . . . . . . 10  |-  ( 5  x.  7 )  = ; 3
5
80 7p3e10 9868 . . . . . . . . . . . 12  |-  ( 7  +  3 )  =  10
8122, 15, 80addcomli 9004 . . . . . . . . . . 11  |-  ( 3  +  7 )  =  10
82 dec10 10154 . . . . . . . . . . 11  |-  10  = ; 1 0
8381, 82eqtri 2303 . . . . . . . . . 10  |-  ( 3  +  7 )  = ; 1
0
84 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
85 3p1e4 9848 . . . . . . . . . . . . 13  |-  ( 3  +  1 )  =  4
8615, 84, 85addcomli 9004 . . . . . . . . . . . 12  |-  ( 1  +  3 )  =  4
8786oveq2i 5869 . . . . . . . . . . 11  |-  ( ( 3  x.  7 )  +  ( 1  +  3 ) )  =  ( ( 3  x.  7 )  +  4 )
88 4nn0 9984 . . . . . . . . . . . 12  |-  4  e.  NN0
89 7t3e21 10207 . . . . . . . . . . . . 13  |-  ( 7  x.  3 )  = ; 2
1
9022, 15, 89mulcomli 8844 . . . . . . . . . . . 12  |-  ( 3  x.  7 )  = ; 2
1
91 4cn 9820 . . . . . . . . . . . . 13  |-  4  e.  CC
92 4p1e5 9849 . . . . . . . . . . . . 13  |-  ( 4  +  1 )  =  5
9391, 84, 92addcomli 9004 . . . . . . . . . . . 12  |-  ( 1  +  4 )  =  5
9431, 35, 88, 90, 93decaddi 10168 . . . . . . . . . . 11  |-  ( ( 3  x.  7 )  +  4 )  = ; 2
5
9587, 94eqtri 2303 . . . . . . . . . 10  |-  ( ( 3  x.  7 )  +  ( 1  +  3 ) )  = ; 2
5
9679oveq1i 5868 . . . . . . . . . . 11  |-  ( ( 5  x.  7 )  +  0 )  =  (; 3 5  +  0 )
9741, 32deccl 10138 . . . . . . . . . . . . 13  |- ; 3 5  e.  NN0
9897nn0cni 9977 . . . . . . . . . . . 12  |- ; 3 5  e.  CC
9998addid1i 8999 . . . . . . . . . . 11  |-  (; 3 5  +  0 )  = ; 3 5
10096, 99eqtri 2303 . . . . . . . . . 10  |-  ( ( 5  x.  7 )  +  0 )  = ; 3
5
10141, 32, 35, 30, 79, 83, 16, 32, 41, 95, 100decmac 10163 . . . . . . . . 9  |-  ( ( ( 5  x.  7 )  x.  7 )  +  ( 3  +  7 ) )  = ;; 2 5 5
10235dec0h 10140 . . . . . . . . . 10  |-  1  = ; 0 1
103 3t2e6 9872 . . . . . . . . . . . 12  |-  ( 3  x.  2 )  =  6
104103, 45oveq12i 5870 . . . . . . . . . . 11  |-  ( ( 3  x.  2 )  +  ( 0  +  1 ) )  =  ( 6  +  1 )
105 6p1e7 9851 . . . . . . . . . . 11  |-  ( 6  +  1 )  =  7
106104, 105eqtri 2303 . . . . . . . . . 10  |-  ( ( 3  x.  2 )  +  ( 0  +  1 ) )  =  7
107 5t2e10 9875 . . . . . . . . . . . 12  |-  ( 5  x.  2 )  =  10
108107, 82eqtri 2303 . . . . . . . . . . 11  |-  ( 5  x.  2 )  = ; 1
0
10935, 30, 45, 108decsuc 10147 . . . . . . . . . 10  |-  ( ( 5  x.  2 )  +  1 )  = ; 1
1
11041, 32, 30, 35, 79, 102, 31, 35, 35, 106, 109decmac 10163 . . . . . . . . 9  |-  ( ( ( 5  x.  7 )  x.  2 )  +  1 )  = ; 7
1
11116, 31, 41, 35, 76, 77, 48, 35, 16, 101, 110decma2c 10164 . . . . . . . 8  |-  ( ( ( 5  x.  7 )  x. ; 7 2 )  + ; 3
1 )  = ;;; 2 5 5 1
112 9t3e27 10220 . . . . . . . . . . 11  |-  ( 9  x.  3 )  = ; 2
7
11355, 15, 112mulcomli 8844 . . . . . . . . . 10  |-  ( 3  x.  9 )  = ; 2
7
114 7p4e11 10176 . . . . . . . . . 10  |-  ( 7  +  4 )  = ; 1
1
11531, 16, 88, 113, 51, 35, 114decaddci 10169 . . . . . . . . 9  |-  ( ( 3  x.  9 )  +  4 )  = ; 3
1
116 9t5e45 10222 . . . . . . . . . 10  |-  ( 9  x.  5 )  = ; 4
5
11755, 20, 116mulcomli 8844 . . . . . . . . 9  |-  ( 5  x.  9 )  = ; 4
5
11850, 41, 32, 79, 32, 88, 115, 117decmul1c 10171 . . . . . . . 8  |-  ( ( 5  x.  7 )  x.  9 )  = ;; 3 1 5
11948, 49, 50, 74, 32, 75, 111, 118decmul2c 10172 . . . . . . 7  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 3 ) )  = ;;;; 2 5 5 1 5
12043, 47, 1193eqtr4ri 2314 . . . . . 6  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 3 ) )  =  ( ( ( 2  x.  0 )  +  1 )  x. ;;;; 2 5 5 1 5 )
12129, 30, 37, 30, 38, 40, 41, 42, 120log2ublem2 20243 . . . . 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 )
12250, 88deccl 10138 . . . . . 6  |- ; 9 4  e.  NN0
123122, 32deccl 10138 . . . . 5  |- ;; 9 4 5  e.  NN0
124 1m1e0 9814 . . . . 5  |-  ( 1  -  1 )  =  0
125 eqid 2283 . . . . . 6  |- ;;;; 2 5 5 1 5  = ;;;; 2 5 5 1 5
126 eqid 2283 . . . . . 6  |- ;; 9 4 5  = ;; 9 4 5
127 6nn0 9986 . . . . . . . . 9  |-  6  e.  NN0
12831, 127deccl 10138 . . . . . . . 8  |- ; 2 6  e.  NN0
129128, 88deccl 10138 . . . . . . 7  |- ;; 2 6 4  e.  NN0
130 5p1e6 9850 . . . . . . 7  |-  ( 5  +  1 )  =  6
131 eqid 2283 . . . . . . . 8  |- ;;; 2 5 5 1  = ;;; 2 5 5 1
132 eqid 2283 . . . . . . . 8  |- ; 9 4  = ; 9 4
133 eqid 2283 . . . . . . . . 9  |- ;; 2 5 5  = ;; 2 5 5
134 eqid 2283 . . . . . . . . . 10  |- ; 2 5  = ; 2 5
13531, 32, 130, 134decsuc 10147 . . . . . . . . 9  |-  (; 2 5  +  1 )  = ; 2 6
136 9p5e14 10189 . . . . . . . . . 10  |-  ( 9  +  5 )  = ; 1
4
13755, 20, 136addcomli 9004 . . . . . . . . 9  |-  ( 5  +  9 )  = ; 1
4
13833, 32, 50, 133, 135, 88, 137decaddci 10169 . . . . . . . 8  |-  (;; 2 5 5  +  9 )  = ;; 2 6 4
13934, 35, 50, 88, 131, 132, 138, 93decadd 10165 . . . . . . 7  |-  (;;; 2 5 5 1  + ; 9 4 )  = ;;; 2 6 4 5
140129, 32, 130, 139decsuc 10147 . . . . . 6  |-  ( (;;; 2 5 5 1  + ; 9
4 )  +  1 )  = ;;; 2 6 4 6
141 5p5e10 9863 . . . . . 6  |-  ( 5  +  5 )  =  10
14236, 32, 122, 32, 125, 126, 140, 141decaddc2 10167 . . . . 5  |-  (;;;; 2 5 5 1 5  + ;; 9 4 5 )  = ;;;; 2 6 4 6 0
14355sqvali 11183 . . . . . . . . 9  |-  ( 9 ^ 2 )  =  ( 9  x.  9 )
144 3t3e9 9873 . . . . . . . . . 10  |-  ( 3  x.  3 )  =  9
145144oveq1i 5868 . . . . . . . . 9  |-  ( ( 3  x.  3 )  x.  9 )  =  ( 9  x.  9 )
146143, 145eqtr4i 2306 . . . . . . . 8  |-  ( 9 ^ 2 )  =  ( ( 3  x.  3 )  x.  9 )
14715, 15, 55mulassi 8846 . . . . . . . 8  |-  ( ( 3  x.  3 )  x.  9 )  =  ( 3  x.  (
3  x.  9 ) )
148146, 147eqtri 2303 . . . . . . 7  |-  ( 9 ^ 2 )  =  ( 3  x.  (
3  x.  9 ) )
149148oveq2i 5869 . . . . . 6  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 2 ) )  =  ( ( 5  x.  7 )  x.  (
3  x.  ( 3  x.  9 ) ) )
15015, 55mulcli 8842 . . . . . . . 8  |-  ( 3  x.  9 )  e.  CC
15123, 15, 150mul12i 9007 . . . . . . 7  |-  ( ( 5  x.  7 )  x.  ( 3  x.  ( 3  x.  9 ) ) )  =  ( 3  x.  (
( 5  x.  7 )  x.  ( 3  x.  9 ) ) )
15231, 88deccl 10138 . . . . . . . . 9  |- ; 2 4  e.  NN0
153 eqid 2283 . . . . . . . . . 10  |- ; 2 4  = ; 2 4
154103, 51oveq12i 5870 . . . . . . . . . . 11  |-  ( ( 3  x.  2 )  +  ( 2  +  1 ) )  =  ( 6  +  3 )
155 6p3e9 9865 . . . . . . . . . . 11  |-  ( 6  +  3 )  =  9
156154, 155eqtri 2303 . . . . . . . . . 10  |-  ( ( 3  x.  2 )  +  ( 2  +  1 ) )  =  9
15791addid2i 9000 . . . . . . . . . . 11  |-  ( 0  +  4 )  =  4
15835, 30, 88, 108, 157decaddi 10168 . . . . . . . . . 10  |-  ( ( 5  x.  2 )  +  4 )  = ; 1
4
15941, 32, 31, 88, 79, 153, 31, 88, 35, 156, 158decmac 10163 . . . . . . . . 9  |-  ( ( ( 5  x.  7 )  x.  2 )  + ; 2 4 )  = ; 9
4
16031, 35, 41, 90, 86decaddi 10168 . . . . . . . . . 10  |-  ( ( 3  x.  7 )  +  3 )  = ; 2
4
16116, 41, 32, 79, 32, 41, 160, 79decmul1c 10171 . . . . . . . . 9  |-  ( ( 5  x.  7 )  x.  7 )  = ;; 2 4 5
16248, 31, 16, 113, 32, 152, 159, 161decmul2c 10172 . . . . . . . 8  |-  ( ( 5  x.  7 )  x.  ( 3  x.  9 ) )  = ;; 9 4 5
163162oveq2i 5869 . . . . . . 7  |-  ( 3  x.  ( ( 5  x.  7 )  x.  ( 3  x.  9 ) ) )  =  ( 3  x. ;; 9 4 5 )
164151, 163eqtri 2303 . . . . . 6  |-  ( ( 5  x.  7 )  x.  ( 3  x.  ( 3  x.  9 ) ) )  =  ( 3  x. ;; 9 4 5 )
165 df-3 9805 . . . . . . . 8  |-  3  =  ( 2  +  1 )
16627mulid1i 8839 . . . . . . . . 9  |-  ( 2  x.  1 )  =  2
167166oveq1i 5868 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
168165, 167eqtr4i 2306 . . . . . . 7  |-  3  =  ( ( 2  x.  1 )  +  1 )
169168oveq1i 5868 . . . . . 6  |-  ( 3  x. ;; 9 4 5 )  =  ( ( ( 2  x.  1 )  +  1 )  x. ;; 9 4 5 )
170149, 164, 1693eqtri 2307 . . . . 5  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 2 ) )  =  ( ( ( 2  x.  1 )  +  1 )  x. ;; 9 4 5 )
171121, 37, 123, 35, 124, 142, 31, 51, 170log2ublem2 20243 . . . 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 )
172129, 127deccl 10138 . . . . 5  |- ;;; 2 6 4 6  e.  NN0
173172, 30deccl 10138 . . . 4  |- ;;;; 2 6 4 6 0  e.  NN0
174127, 41deccl 10138 . . . 4  |- ; 6 3  e.  NN0
17527, 84, 84, 53subaddrii 9135 . . . 4  |-  ( 2  -  1 )  =  1
176 eqid 2283 . . . . 5  |- ;;;; 2 6 4 6 0  = ;;;; 2 6 4 6 0
177 eqid 2283 . . . . 5  |- ; 6 3  = ; 6 3
178 eqid 2283 . . . . . 6  |- ;;; 2 6 4 6  = ;;; 2 6 4 6
179 eqid 2283 . . . . . . 7  |- ;; 2 6 4  = ;; 2 6 4
180128, 88, 92, 179decsuc 10147 . . . . . 6  |-  (;; 2 6 4  +  1 )  = ;; 2 6 5
181 6p6e12 10175 . . . . . 6  |-  ( 6  +  6 )  = ; 1
2
182129, 127, 127, 178, 180, 31, 181decaddci 10169 . . . . 5  |-  (;;; 2 6 4 6  +  6 )  = ;;; 2 6 5 2
18315addid2i 9000 . . . . 5  |-  ( 0  +  3 )  =  3
184172, 30, 127, 41, 176, 177, 182, 183decadd 10165 . . . 4  |-  (;;;; 2 6 4 6 0  + ; 6 3 )  = ;;;; 2 6 5 2 3
18584, 27addcomi 9003 . . . . 5  |-  ( 1  +  2 )  =  ( 2  +  1 )
186185, 165eqtr4i 2306 . . . 4  |-  ( 1  +  2 )  =  3
18757oveq2i 5869 . . . . 5  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 1 ) )  =  ( ( 5  x.  7 )  x.  9 )
18820, 22, 55mulassi 8846 . . . . . 6  |-  ( ( 5  x.  7 )  x.  9 )  =  ( 5  x.  (
7  x.  9 ) )
189 9t7e63 10224 . . . . . . . 8  |-  ( 9  x.  7 )  = ; 6
3
19055, 22, 189mulcomli 8844 . . . . . . 7  |-  ( 7  x.  9 )  = ; 6
3
191190oveq2i 5869 . . . . . 6  |-  ( 5  x.  ( 7  x.  9 ) )  =  ( 5  x. ; 6 3 )
192188, 191eqtri 2303 . . . . 5  |-  ( ( 5  x.  7 )  x.  9 )  =  ( 5  x. ; 6 3 )
193 df-5 9807 . . . . . . 7  |-  5  =  ( 4  +  1 )
194 2t2e4 9871 . . . . . . . 8  |-  ( 2  x.  2 )  =  4
195194oveq1i 5868 . . . . . . 7  |-  ( ( 2  x.  2 )  +  1 )  =  ( 4  +  1 )
196193, 195eqtr4i 2306 . . . . . 6  |-  5  =  ( ( 2  x.  2 )  +  1 )
197196oveq1i 5868 . . . . 5  |-  ( 5  x. ; 6 3 )  =  ( ( ( 2  x.  2 )  +  1 )  x. ; 6 3 )
198187, 192, 1973eqtri 2307 . . . 4  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 1 ) )  =  ( ( ( 2  x.  2 )  +  1 )  x. ; 6 3 )
199171, 173, 174, 31, 175, 184, 35, 186, 198log2ublem2 20243 . . 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 )
200128, 32deccl 10138 . . . . 5  |- ;; 2 6 5  e.  NN0
201200, 31deccl 10138 . . . 4  |- ;;; 2 6 5 2  e.  NN0
202201, 41deccl 10138 . . 3  |- ;;;; 2 6 5 2 3  e.  NN0
20315, 84, 27, 186subaddrii 9135 . . 3  |-  ( 3  -  1 )  =  2
204 eqid 2283 . . . 4  |- ;;;; 2 6 5 2 3  = ;;;; 2 6 5 2 3
205 5p3e8 9861 . . . . 5  |-  ( 5  +  3 )  =  8
20620, 15, 205addcomli 9004 . . . 4  |-  ( 3  +  5 )  =  8
207201, 41, 32, 204, 206decaddi 10168 . . 3  |-  (;;;; 2 6 5 2 3  +  5 )  = ;;;; 2 6 5 2 8
20822, 20mulcli 8842 . . . . 5  |-  ( 7  x.  5 )  e.  CC
209208mulid1i 8839 . . . 4  |-  ( ( 7  x.  5 )  x.  1 )  =  ( 7  x.  5 )
21020, 22mulcomi 8843 . . . . 5  |-  ( 5  x.  7 )  =  ( 7  x.  5 )
211 exp0 11108 . . . . . 6  |-  ( 9  e.  CC  ->  (
9 ^ 0 )  =  1 )
21255, 211ax-mp 8 . . . . 5  |-  ( 9 ^ 0 )  =  1
213210, 212oveq12i 5870 . . . 4  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 0 ) )  =  ( ( 7  x.  5 )  x.  1 )
21415, 27, 103mulcomli 8844 . . . . . . 7  |-  ( 2  x.  3 )  =  6
215214oveq1i 5868 . . . . . 6  |-  ( ( 2  x.  3 )  +  1 )  =  ( 6  +  1 )
216 df-7 9809 . . . . . 6  |-  7  =  ( 6  +  1 )
217215, 216eqtr4i 2306 . . . . 5  |-  ( ( 2  x.  3 )  +  1 )  =  7
218217oveq1i 5868 . . . 4  |-  ( ( ( 2  x.  3 )  +  1 )  x.  5 )  =  ( 7  x.  5 )
219209, 213, 2183eqtr4i 2313 . . 3  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 0 ) )  =  ( ( ( 2  x.  3 )  +  1 )  x.  5 )
220199, 202, 32, 41, 203, 207, 30, 183, 219log2ublem2 20243 . 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 )
221 eqid 2283 . . 3  |- ;;;; 2 6 5 2 8  = ;;;; 2 6 5 2 8
222 eqid 2283 . . . 4  |- ;;; 2 6 5 2  = ;;; 2 6 5 2
223 eqid 2283 . . . . 5  |- ;; 2 6 5  = ;; 2 6 5
224 00id 8987 . . . . . 6  |-  ( 0  +  0 )  =  0
22530dec0h 10140 . . . . . 6  |-  0  = ; 0 0
226224, 225eqtri 2303 . . . . 5  |-  ( 0  +  0 )  = ; 0
0
227 eqid 2283 . . . . . 6  |- ; 2 6  = ; 2 6
22845, 102eqtri 2303 . . . . . 6  |-  ( 0  +  1 )  = ; 0
1
229194, 45oveq12i 5870 . . . . . . 7  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
230229, 92eqtri 2303 . . . . . 6  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  5
231 6nn 9881 . . . . . . . . 9  |-  6  e.  NN
232231nncni 9756 . . . . . . . 8  |-  6  e.  CC
233 6t2e12 10201 . . . . . . . 8  |-  ( 6  x.  2 )  = ; 1
2
234232, 27, 233mulcomli 8844 . . . . . . 7  |-  ( 2  x.  6 )  = ; 1
2
23535, 31, 51, 234decsuc 10147 . . . . . 6  |-  ( ( 2  x.  6 )  +  1 )  = ; 1
3
23631, 127, 30, 35, 227, 228, 31, 41, 35, 230, 235decma2c 10164 . . . . 5  |-  ( ( 2  x. ; 2 6 )  +  ( 0  +  1 ) )  = ; 5 3
23720, 27, 107mulcomli 8844 . . . . . . 7  |-  ( 2  x.  5 )  =  10
238237oveq1i 5868 . . . . . 6  |-  ( ( 2  x.  5 )  +  0 )  =  ( 10  +  0 )
239 dec10p 10153 . . . . . 6  |-  ( 10  +  0 )  = ; 1
0
240238, 239eqtri 2303 . . . . 5  |-  ( ( 2  x.  5 )  +  0 )  = ; 1
0
241128, 32, 30, 30, 223, 226, 31, 30, 35, 236, 240decma2c 10164 . . . 4  |-  ( ( 2  x. ;; 2 6 5 )  +  ( 0  +  0 ) )  = ;; 5 3 0
24232dec0h 10140 . . . . 5  |-  5  = ; 0 5
243195, 92, 2423eqtri 2307 . . . 4  |-  ( ( 2  x.  2 )  +  1 )  = ; 0
5
244200, 31, 30, 35, 222, 102, 31, 32, 30, 241, 243decma2c 10164 . . 3  |-  ( ( 2  x. ;;; 2 6 5 2 )  +  1 )  = ;;; 5 3 0 5
245 8t2e16 10212 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
24663, 27, 245mulcomli 8844 . . 3  |-  ( 2  x.  8 )  = ; 1
6
24731, 201, 52, 221, 127, 35, 244, 246decmul2c 10172 . 2  |-  ( 2  x. ;;;; 2 6 5 2 8 )  = ;;;; 5 3 0 5 6
248220, 247breqtri 4046 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
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   (/)c0 3455   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   3c3 9796   4c4 9797   5c5 9798   6c6 9799   7c7 9800   8c8 9801   9c9 9802   10c10 9803   NN0cn0 9965   ZZcz 10024  ;cdc 10124   ...cfz 10782   ^cexp 11104   sum_csu 12158
This theorem is referenced by:  log2ub  20245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159
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