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Theorem log2ublem3 23144
Description: Lemma for log2ub 23145. 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 10626 . . . . . . 7  |-  0  <_  0
2 0re 9594 . . . . . . . . . . . . 13  |-  0  e.  RR
3 ltm1 10383 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  (
0  -  1 )  <  0 )
42, 3ax-mp 5 . . . . . . . . . . . 12  |-  ( 0  -  1 )  <  0
5 0z 10876 . . . . . . . . . . . . 13  |-  0  e.  ZZ
6 peano2zm 10908 . . . . . . . . . . . . . 14  |-  ( 0  e.  ZZ  ->  (
0  -  1 )  e.  ZZ )
75, 6ax-mp 5 . . . . . . . . . . . . 13  |-  ( 0  -  1 )  e.  ZZ
8 fzn 11706 . . . . . . . . . . . . 13  |-  ( ( 0  e.  ZZ  /\  ( 0  -  1 )  e.  ZZ )  ->  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) ) )
95, 7, 8mp2an 672 . . . . . . . . . . . 12  |-  ( ( 0  -  1 )  <  0  <->  ( 0 ... ( 0  -  1 ) )  =  (/) )
104, 9mpbi 208 . . . . . . . . . . 11  |-  ( 0 ... ( 0  -  1 ) )  =  (/)
1110sumeq1i 13494 . . . . . . . . . 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 13517 . . . . . . . . . 10  |-  sum_ n  e.  (/)  ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  =  0
1311, 12eqtri 2470 . . . . . . . . 9  |-  sum_ n  e.  ( 0 ... (
0  -  1 ) ) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  =  0
1413oveq2i 6288 . . . . . . . 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 10611 . . . . . . . . . . 11  |-  3  e.  CC
16 7nn0 10818 . . . . . . . . . . 11  |-  7  e.  NN0
17 expcl 12158 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  CC )
1815, 16, 17mp2an 672 . . . . . . . . . 10  |-  ( 3 ^ 7 )  e.  CC
19 5cn 10616 . . . . . . . . . . 11  |-  5  e.  CC
20 7cn 10620 . . . . . . . . . . 11  |-  7  e.  CC
2119, 20mulcli 9599 . . . . . . . . . 10  |-  ( 5  x.  7 )  e.  CC
2218, 21mulcli 9599 . . . . . . . . 9  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
2322mul01i 9768 . . . . . . . 8  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  0 )  =  0
2414, 23eqtri 2470 . . . . . . 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
25 2cn 10607 . . . . . . . 8  |-  2  e.  CC
2625mul01i 9768 . . . . . . 7  |-  ( 2  x.  0 )  =  0
271, 24, 263brtr4i 4461 . . . . . 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 )
28 0nn0 10811 . . . . . 6  |-  0  e.  NN0
29 2nn0 10813 . . . . . . . . . 10  |-  2  e.  NN0
30 5nn0 10816 . . . . . . . . . 10  |-  5  e.  NN0
3129, 30deccl 10993 . . . . . . . . 9  |- ; 2 5  e.  NN0
3231, 30deccl 10993 . . . . . . . 8  |- ;; 2 5 5  e.  NN0
33 1nn0 10812 . . . . . . . 8  |-  1  e.  NN0
3432, 33deccl 10993 . . . . . . 7  |- ;;; 2 5 5 1  e.  NN0
3534, 30deccl 10993 . . . . . 6  |- ;;;; 2 5 5 1 5  e.  NN0
36 eqid 2441 . . . . . 6  |-  ( 0  -  1 )  =  ( 0  -  1 )
3735nn0cni 10808 . . . . . . 7  |- ;;;; 2 5 5 1 5  e.  CC
3837addid2i 9766 . . . . . 6  |-  ( 0  + ;;;; 2 5 5 1 5 )  = ;;;; 2 5 5 1 5
39 3nn0 10814 . . . . . 6  |-  3  e.  NN0
4015addid1i 9765 . . . . . 6  |-  ( 3  +  0 )  =  3
4137mulid2i 9597 . . . . . . 7  |-  ( 1  x. ;;;; 2 5 5 1 5 )  = ;;;; 2 5 5 1 5
4226oveq1i 6287 . . . . . . . . 9  |-  ( ( 2  x.  0 )  +  1 )  =  ( 0  +  1 )
43 0p1e1 10648 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4442, 43eqtri 2470 . . . . . . . 8  |-  ( ( 2  x.  0 )  +  1 )  =  1
4544oveq1i 6287 . . . . . . 7  |-  ( ( ( 2  x.  0 )  +  1 )  x. ;;;; 2 5 5 1 5 )  =  ( 1  x. ;;;; 2 5 5 1 5 )
4630, 16nn0mulcli 10835 . . . . . . . 8  |-  ( 5  x.  7 )  e. 
NN0
4716, 29deccl 10993 . . . . . . . 8  |- ; 7 2  e.  NN0
48 9nn0 10820 . . . . . . . 8  |-  9  e.  NN0
49 2p1e3 10660 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
50 8nn0 10819 . . . . . . . . . 10  |-  8  e.  NN0
51 1p1e2 10650 . . . . . . . . . . 11  |-  ( 1  +  1 )  =  2
52 9cn 10624 . . . . . . . . . . . . . 14  |-  9  e.  CC
53 exp1 12146 . . . . . . . . . . . . . 14  |-  ( 9  e.  CC  ->  (
9 ^ 1 )  =  9 )
5452, 53ax-mp 5 . . . . . . . . . . . . 13  |-  ( 9 ^ 1 )  =  9
5554oveq1i 6287 . . . . . . . . . . . 12  |-  ( ( 9 ^ 1 )  x.  9 )  =  ( 9  x.  9 )
56 9t9e81 11081 . . . . . . . . . . . 12  |-  ( 9  x.  9 )  = ; 8
1
5755, 56eqtri 2470 . . . . . . . . . . 11  |-  ( ( 9 ^ 1 )  x.  9 )  = ; 8
1
5848, 33, 51, 57numexpp1 14436 . . . . . . . . . 10  |-  ( 9 ^ 2 )  = ; 8
1
59 8cn 10622 . . . . . . . . . . . . 13  |-  8  e.  CC
60 9t8e72 11080 . . . . . . . . . . . . 13  |-  ( 9  x.  8 )  = ; 7
2
6152, 59, 60mulcomli 9601 . . . . . . . . . . . 12  |-  ( 8  x.  9 )  = ; 7
2
6261oveq1i 6287 . . . . . . . . . . 11  |-  ( ( 8  x.  9 )  +  0 )  =  (; 7 2  +  0 )
6347nn0cni 10808 . . . . . . . . . . . 12  |- ; 7 2  e.  CC
6463addid1i 9765 . . . . . . . . . . 11  |-  (; 7 2  +  0 )  = ; 7 2
6562, 64eqtri 2470 . . . . . . . . . 10  |-  ( ( 8  x.  9 )  +  0 )  = ; 7
2
6652mulid2i 9597 . . . . . . . . . . 11  |-  ( 1  x.  9 )  =  9
6748dec0h 10995 . . . . . . . . . . 11  |-  9  = ; 0 9
6866, 67eqtri 2470 . . . . . . . . . 10  |-  ( 1  x.  9 )  = ; 0
9
6948, 50, 33, 58, 48, 28, 65, 68decmul1c 11026 . . . . . . . . 9  |-  ( ( 9 ^ 2 )  x.  9 )  = ;; 7 2 9
7048, 29, 49, 69numexpp1 14436 . . . . . . . 8  |-  ( 9 ^ 3 )  = ;; 7 2 9
7139, 33deccl 10993 . . . . . . . 8  |- ; 3 1  e.  NN0
72 eqid 2441 . . . . . . . . 9  |- ; 7 2  = ; 7 2
73 eqid 2441 . . . . . . . . 9  |- ; 3 1  = ; 3 1
74 7t5e35 11064 . . . . . . . . . . 11  |-  ( 7  x.  5 )  = ; 3
5
7520, 19, 74mulcomli 9601 . . . . . . . . . 10  |-  ( 5  x.  7 )  = ; 3
5
76 7p3e10 10682 . . . . . . . . . . . 12  |-  ( 7  +  3 )  =  10
7720, 15, 76addcomli 9770 . . . . . . . . . . 11  |-  ( 3  +  7 )  =  10
78 dec10 11009 . . . . . . . . . . 11  |-  10  = ; 1 0
7977, 78eqtri 2470 . . . . . . . . . 10  |-  ( 3  +  7 )  = ; 1
0
80 ax-1cn 9548 . . . . . . . . . . . . 13  |-  1  e.  CC
81 3p1e4 10662 . . . . . . . . . . . . 13  |-  ( 3  +  1 )  =  4
8215, 80, 81addcomli 9770 . . . . . . . . . . . 12  |-  ( 1  +  3 )  =  4
8382oveq2i 6288 . . . . . . . . . . 11  |-  ( ( 3  x.  7 )  +  ( 1  +  3 ) )  =  ( ( 3  x.  7 )  +  4 )
84 4nn0 10815 . . . . . . . . . . . 12  |-  4  e.  NN0
85 7t3e21 11062 . . . . . . . . . . . . 13  |-  ( 7  x.  3 )  = ; 2
1
8620, 15, 85mulcomli 9601 . . . . . . . . . . . 12  |-  ( 3  x.  7 )  = ; 2
1
87 4cn 10614 . . . . . . . . . . . . 13  |-  4  e.  CC
88 4p1e5 10663 . . . . . . . . . . . . 13  |-  ( 4  +  1 )  =  5
8987, 80, 88addcomli 9770 . . . . . . . . . . . 12  |-  ( 1  +  4 )  =  5
9029, 33, 84, 86, 89decaddi 11023 . . . . . . . . . . 11  |-  ( ( 3  x.  7 )  +  4 )  = ; 2
5
9183, 90eqtri 2470 . . . . . . . . . 10  |-  ( ( 3  x.  7 )  +  ( 1  +  3 ) )  = ; 2
5
9275oveq1i 6287 . . . . . . . . . . 11  |-  ( ( 5  x.  7 )  +  0 )  =  (; 3 5  +  0 )
9339, 30deccl 10993 . . . . . . . . . . . . 13  |- ; 3 5  e.  NN0
9493nn0cni 10808 . . . . . . . . . . . 12  |- ; 3 5  e.  CC
9594addid1i 9765 . . . . . . . . . . 11  |-  (; 3 5  +  0 )  = ; 3 5
9692, 95eqtri 2470 . . . . . . . . . 10  |-  ( ( 5  x.  7 )  +  0 )  = ; 3
5
9739, 30, 33, 28, 75, 79, 16, 30, 39, 91, 96decmac 11018 . . . . . . . . 9  |-  ( ( ( 5  x.  7 )  x.  7 )  +  ( 3  +  7 ) )  = ;; 2 5 5
9833dec0h 10995 . . . . . . . . . 10  |-  1  = ; 0 1
99 3t2e6 10688 . . . . . . . . . . . 12  |-  ( 3  x.  2 )  =  6
10099, 43oveq12i 6289 . . . . . . . . . . 11  |-  ( ( 3  x.  2 )  +  ( 0  +  1 ) )  =  ( 6  +  1 )
101 6p1e7 10665 . . . . . . . . . . 11  |-  ( 6  +  1 )  =  7
102100, 101eqtri 2470 . . . . . . . . . 10  |-  ( ( 3  x.  2 )  +  ( 0  +  1 ) )  =  7
103 5t2e10 10691 . . . . . . . . . . . 12  |-  ( 5  x.  2 )  =  10
104103, 78eqtri 2470 . . . . . . . . . . 11  |-  ( 5  x.  2 )  = ; 1
0
10533, 28, 43, 104decsuc 11002 . . . . . . . . . 10  |-  ( ( 5  x.  2 )  +  1 )  = ; 1
1
10639, 30, 28, 33, 75, 98, 29, 33, 33, 102, 105decmac 11018 . . . . . . . . 9  |-  ( ( ( 5  x.  7 )  x.  2 )  +  1 )  = ; 7
1
10716, 29, 39, 33, 72, 73, 46, 33, 16, 97, 106decma2c 11019 . . . . . . . 8  |-  ( ( ( 5  x.  7 )  x. ; 7 2 )  + ; 3
1 )  = ;;; 2 5 5 1
108 9t3e27 11075 . . . . . . . . . . 11  |-  ( 9  x.  3 )  = ; 2
7
10952, 15, 108mulcomli 9601 . . . . . . . . . 10  |-  ( 3  x.  9 )  = ; 2
7
110 7p4e11 11031 . . . . . . . . . 10  |-  ( 7  +  4 )  = ; 1
1
11129, 16, 84, 109, 49, 33, 110decaddci 11024 . . . . . . . . 9  |-  ( ( 3  x.  9 )  +  4 )  = ; 3
1
112 9t5e45 11077 . . . . . . . . . 10  |-  ( 9  x.  5 )  = ; 4
5
11352, 19, 112mulcomli 9601 . . . . . . . . 9  |-  ( 5  x.  9 )  = ; 4
5
11448, 39, 30, 75, 30, 84, 111, 113decmul1c 11026 . . . . . . . 8  |-  ( ( 5  x.  7 )  x.  9 )  = ;; 3 1 5
11546, 47, 48, 70, 30, 71, 107, 114decmul2c 11027 . . . . . . 7  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 3 ) )  = ;;;; 2 5 5 1 5
11641, 45, 1153eqtr4ri 2481 . . . . . 6  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 3 ) )  =  ( ( ( 2  x.  0 )  +  1 )  x. ;;;; 2 5 5 1 5 )
11727, 28, 35, 28, 36, 38, 39, 40, 116log2ublem2 23143 . . . . 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 )
11848, 84deccl 10993 . . . . . 6  |- ; 9 4  e.  NN0
119118, 30deccl 10993 . . . . 5  |- ;; 9 4 5  e.  NN0
120 1m1e0 10605 . . . . 5  |-  ( 1  -  1 )  =  0
121 eqid 2441 . . . . . 6  |- ;;;; 2 5 5 1 5  = ;;;; 2 5 5 1 5
122 eqid 2441 . . . . . 6  |- ;; 9 4 5  = ;; 9 4 5
123 6nn0 10817 . . . . . . . . 9  |-  6  e.  NN0
12429, 123deccl 10993 . . . . . . . 8  |- ; 2 6  e.  NN0
125124, 84deccl 10993 . . . . . . 7  |- ;; 2 6 4  e.  NN0
126 5p1e6 10664 . . . . . . 7  |-  ( 5  +  1 )  =  6
127 eqid 2441 . . . . . . . 8  |- ;;; 2 5 5 1  = ;;; 2 5 5 1
128 eqid 2441 . . . . . . . 8  |- ; 9 4  = ; 9 4
129 eqid 2441 . . . . . . . . 9  |- ;; 2 5 5  = ;; 2 5 5
130 eqid 2441 . . . . . . . . . 10  |- ; 2 5  = ; 2 5
13129, 30, 126, 130decsuc 11002 . . . . . . . . 9  |-  (; 2 5  +  1 )  = ; 2 6
132 9p5e14 11044 . . . . . . . . . 10  |-  ( 9  +  5 )  = ; 1
4
13352, 19, 132addcomli 9770 . . . . . . . . 9  |-  ( 5  +  9 )  = ; 1
4
13431, 30, 48, 129, 131, 84, 133decaddci 11024 . . . . . . . 8  |-  (;; 2 5 5  +  9 )  = ;; 2 6 4
13532, 33, 48, 84, 127, 128, 134, 89decadd 11020 . . . . . . 7  |-  (;;; 2 5 5 1  + ; 9 4 )  = ;;; 2 6 4 5
136125, 30, 126, 135decsuc 11002 . . . . . 6  |-  ( (;;; 2 5 5 1  + ; 9
4 )  +  1 )  = ;;; 2 6 4 6
137 5p5e10 10677 . . . . . 6  |-  ( 5  +  5 )  =  10
13834, 30, 118, 30, 121, 122, 136, 137decaddc2 11022 . . . . 5  |-  (;;;; 2 5 5 1 5  + ;; 9 4 5 )  = ;;;; 2 6 4 6 0
13952sqvali 12221 . . . . . . . . 9  |-  ( 9 ^ 2 )  =  ( 9  x.  9 )
140 3t3e9 10689 . . . . . . . . . 10  |-  ( 3  x.  3 )  =  9
141140oveq1i 6287 . . . . . . . . 9  |-  ( ( 3  x.  3 )  x.  9 )  =  ( 9  x.  9 )
142139, 141eqtr4i 2473 . . . . . . . 8  |-  ( 9 ^ 2 )  =  ( ( 3  x.  3 )  x.  9 )
14315, 15, 52mulassi 9603 . . . . . . . 8  |-  ( ( 3  x.  3 )  x.  9 )  =  ( 3  x.  (
3  x.  9 ) )
144142, 143eqtri 2470 . . . . . . 7  |-  ( 9 ^ 2 )  =  ( 3  x.  (
3  x.  9 ) )
145144oveq2i 6288 . . . . . 6  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 2 ) )  =  ( ( 5  x.  7 )  x.  (
3  x.  ( 3  x.  9 ) ) )
14615, 52mulcli 9599 . . . . . . . 8  |-  ( 3  x.  9 )  e.  CC
14721, 15, 146mul12i 9773 . . . . . . 7  |-  ( ( 5  x.  7 )  x.  ( 3  x.  ( 3  x.  9 ) ) )  =  ( 3  x.  (
( 5  x.  7 )  x.  ( 3  x.  9 ) ) )
14829, 84deccl 10993 . . . . . . . . 9  |- ; 2 4  e.  NN0
149 eqid 2441 . . . . . . . . . 10  |- ; 2 4  = ; 2 4
15099, 49oveq12i 6289 . . . . . . . . . . 11  |-  ( ( 3  x.  2 )  +  ( 2  +  1 ) )  =  ( 6  +  3 )
151 6p3e9 10679 . . . . . . . . . . 11  |-  ( 6  +  3 )  =  9
152150, 151eqtri 2470 . . . . . . . . . 10  |-  ( ( 3  x.  2 )  +  ( 2  +  1 ) )  =  9
15387addid2i 9766 . . . . . . . . . . 11  |-  ( 0  +  4 )  =  4
15433, 28, 84, 104, 153decaddi 11023 . . . . . . . . . 10  |-  ( ( 5  x.  2 )  +  4 )  = ; 1
4
15539, 30, 29, 84, 75, 149, 29, 84, 33, 152, 154decmac 11018 . . . . . . . . 9  |-  ( ( ( 5  x.  7 )  x.  2 )  + ; 2 4 )  = ; 9
4
15629, 33, 39, 86, 82decaddi 11023 . . . . . . . . . 10  |-  ( ( 3  x.  7 )  +  3 )  = ; 2
4
15716, 39, 30, 75, 30, 39, 156, 75decmul1c 11026 . . . . . . . . 9  |-  ( ( 5  x.  7 )  x.  7 )  = ;; 2 4 5
15846, 29, 16, 109, 30, 148, 155, 157decmul2c 11027 . . . . . . . 8  |-  ( ( 5  x.  7 )  x.  ( 3  x.  9 ) )  = ;; 9 4 5
159158oveq2i 6288 . . . . . . 7  |-  ( 3  x.  ( ( 5  x.  7 )  x.  ( 3  x.  9 ) ) )  =  ( 3  x. ;; 9 4 5 )
160147, 159eqtri 2470 . . . . . 6  |-  ( ( 5  x.  7 )  x.  ( 3  x.  ( 3  x.  9 ) ) )  =  ( 3  x. ;; 9 4 5 )
161 df-3 10596 . . . . . . . 8  |-  3  =  ( 2  +  1 )
16225mulid1i 9596 . . . . . . . . 9  |-  ( 2  x.  1 )  =  2
163162oveq1i 6287 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
164161, 163eqtr4i 2473 . . . . . . 7  |-  3  =  ( ( 2  x.  1 )  +  1 )
165164oveq1i 6287 . . . . . 6  |-  ( 3  x. ;; 9 4 5 )  =  ( ( ( 2  x.  1 )  +  1 )  x. ;; 9 4 5 )
166145, 160, 1653eqtri 2474 . . . . 5  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 2 ) )  =  ( ( ( 2  x.  1 )  +  1 )  x. ;; 9 4 5 )
167117, 35, 119, 33, 120, 138, 29, 49, 166log2ublem2 23143 . . . 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 )
168125, 123deccl 10993 . . . . 5  |- ;;; 2 6 4 6  e.  NN0
169168, 28deccl 10993 . . . 4  |- ;;;; 2 6 4 6 0  e.  NN0
170123, 39deccl 10993 . . . 4  |- ; 6 3  e.  NN0
171 2m1e1 10651 . . . 4  |-  ( 2  -  1 )  =  1
172 eqid 2441 . . . . 5  |- ;;;; 2 6 4 6 0  = ;;;; 2 6 4 6 0
173 eqid 2441 . . . . 5  |- ; 6 3  = ; 6 3
174 eqid 2441 . . . . . 6  |- ;;; 2 6 4 6  = ;;; 2 6 4 6
175 eqid 2441 . . . . . . 7  |- ;; 2 6 4  = ;; 2 6 4
176124, 84, 88, 175decsuc 11002 . . . . . 6  |-  (;; 2 6 4  +  1 )  = ;; 2 6 5
177 6p6e12 11030 . . . . . 6  |-  ( 6  +  6 )  = ; 1
2
178125, 123, 123, 174, 176, 29, 177decaddci 11024 . . . . 5  |-  (;;; 2 6 4 6  +  6 )  = ;;; 2 6 5 2
17915addid2i 9766 . . . . 5  |-  ( 0  +  3 )  =  3
180168, 28, 123, 39, 172, 173, 178, 179decadd 11020 . . . 4  |-  (;;;; 2 6 4 6 0  + ; 6 3 )  = ;;;; 2 6 5 2 3
181 1p2e3 10661 . . . 4  |-  ( 1  +  2 )  =  3
18254oveq2i 6288 . . . . 5  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 1 ) )  =  ( ( 5  x.  7 )  x.  9 )
18319, 20, 52mulassi 9603 . . . . . 6  |-  ( ( 5  x.  7 )  x.  9 )  =  ( 5  x.  (
7  x.  9 ) )
184 9t7e63 11079 . . . . . . . 8  |-  ( 9  x.  7 )  = ; 6
3
18552, 20, 184mulcomli 9601 . . . . . . 7  |-  ( 7  x.  9 )  = ; 6
3
186185oveq2i 6288 . . . . . 6  |-  ( 5  x.  ( 7  x.  9 ) )  =  ( 5  x. ; 6 3 )
187183, 186eqtri 2470 . . . . 5  |-  ( ( 5  x.  7 )  x.  9 )  =  ( 5  x. ; 6 3 )
188 df-5 10598 . . . . . . 7  |-  5  =  ( 4  +  1 )
189 2t2e4 10686 . . . . . . . 8  |-  ( 2  x.  2 )  =  4
190189oveq1i 6287 . . . . . . 7  |-  ( ( 2  x.  2 )  +  1 )  =  ( 4  +  1 )
191188, 190eqtr4i 2473 . . . . . 6  |-  5  =  ( ( 2  x.  2 )  +  1 )
192191oveq1i 6287 . . . . 5  |-  ( 5  x. ; 6 3 )  =  ( ( ( 2  x.  2 )  +  1 )  x. ; 6 3 )
193182, 187, 1923eqtri 2474 . . . 4  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 1 ) )  =  ( ( ( 2  x.  2 )  +  1 )  x. ; 6 3 )
194167, 169, 170, 29, 171, 180, 33, 181, 193log2ublem2 23143 . . 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 )
195124, 30deccl 10993 . . . . 5  |- ;; 2 6 5  e.  NN0
196195, 29deccl 10993 . . . 4  |- ;;; 2 6 5 2  e.  NN0
197196, 39deccl 10993 . . 3  |- ;;;; 2 6 5 2 3  e.  NN0
198 3m1e2 10653 . . 3  |-  ( 3  -  1 )  =  2
199 eqid 2441 . . . 4  |- ;;;; 2 6 5 2 3  = ;;;; 2 6 5 2 3
200 5p3e8 10675 . . . . 5  |-  ( 5  +  3 )  =  8
20119, 15, 200addcomli 9770 . . . 4  |-  ( 3  +  5 )  =  8
202196, 39, 30, 199, 201decaddi 11023 . . 3  |-  (;;;; 2 6 5 2 3  +  5 )  = ;;;; 2 6 5 2 8
20320, 19mulcli 9599 . . . . 5  |-  ( 7  x.  5 )  e.  CC
204203mulid1i 9596 . . . 4  |-  ( ( 7  x.  5 )  x.  1 )  =  ( 7  x.  5 )
20519, 20mulcomi 9600 . . . . 5  |-  ( 5  x.  7 )  =  ( 7  x.  5 )
206 exp0 12144 . . . . . 6  |-  ( 9  e.  CC  ->  (
9 ^ 0 )  =  1 )
20752, 206ax-mp 5 . . . . 5  |-  ( 9 ^ 0 )  =  1
208205, 207oveq12i 6289 . . . 4  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 0 ) )  =  ( ( 7  x.  5 )  x.  1 )
20915, 25, 99mulcomli 9601 . . . . . . 7  |-  ( 2  x.  3 )  =  6
210209oveq1i 6287 . . . . . 6  |-  ( ( 2  x.  3 )  +  1 )  =  ( 6  +  1 )
211 df-7 10600 . . . . . 6  |-  7  =  ( 6  +  1 )
212210, 211eqtr4i 2473 . . . . 5  |-  ( ( 2  x.  3 )  +  1 )  =  7
213212oveq1i 6287 . . . 4  |-  ( ( ( 2  x.  3 )  +  1 )  x.  5 )  =  ( 7  x.  5 )
214204, 208, 2133eqtr4i 2480 . . 3  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 0 ) )  =  ( ( ( 2  x.  3 )  +  1 )  x.  5 )
215194, 197, 30, 39, 198, 202, 28, 179, 214log2ublem2 23143 . 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 )
216 eqid 2441 . . 3  |- ;;;; 2 6 5 2 8  = ;;;; 2 6 5 2 8
217 eqid 2441 . . . 4  |- ;;; 2 6 5 2  = ;;; 2 6 5 2
218 eqid 2441 . . . . 5  |- ;; 2 6 5  = ;; 2 6 5
219 00id 9753 . . . . . 6  |-  ( 0  +  0 )  =  0
22028dec0h 10995 . . . . . 6  |-  0  = ; 0 0
221219, 220eqtri 2470 . . . . 5  |-  ( 0  +  0 )  = ; 0
0
222 eqid 2441 . . . . . 6  |- ; 2 6  = ; 2 6
22343, 98eqtri 2470 . . . . . 6  |-  ( 0  +  1 )  = ; 0
1
224189, 43oveq12i 6289 . . . . . . 7  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
225224, 88eqtri 2470 . . . . . 6  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  5
226 6cn 10618 . . . . . . . 8  |-  6  e.  CC
227 6t2e12 11056 . . . . . . . 8  |-  ( 6  x.  2 )  = ; 1
2
228226, 25, 227mulcomli 9601 . . . . . . 7  |-  ( 2  x.  6 )  = ; 1
2
22933, 29, 49, 228decsuc 11002 . . . . . 6  |-  ( ( 2  x.  6 )  +  1 )  = ; 1
3
23029, 123, 28, 33, 222, 223, 29, 39, 33, 225, 229decma2c 11019 . . . . 5  |-  ( ( 2  x. ; 2 6 )  +  ( 0  +  1 ) )  = ; 5 3
23119, 25, 103mulcomli 9601 . . . . . . 7  |-  ( 2  x.  5 )  =  10
232231oveq1i 6287 . . . . . 6  |-  ( ( 2  x.  5 )  +  0 )  =  ( 10  +  0 )
233 dec10p 11008 . . . . . 6  |-  ( 10  +  0 )  = ; 1
0
234232, 233eqtri 2470 . . . . 5  |-  ( ( 2  x.  5 )  +  0 )  = ; 1
0
235124, 30, 28, 28, 218, 221, 29, 28, 33, 230, 234decma2c 11019 . . . 4  |-  ( ( 2  x. ;; 2 6 5 )  +  ( 0  +  0 ) )  = ;; 5 3 0
23630dec0h 10995 . . . . 5  |-  5  = ; 0 5
237190, 88, 2363eqtri 2474 . . . 4  |-  ( ( 2  x.  2 )  +  1 )  = ; 0
5
238195, 29, 28, 33, 217, 98, 29, 30, 28, 235, 237decma2c 11019 . . 3  |-  ( ( 2  x. ;;; 2 6 5 2 )  +  1 )  = ;;; 5 3 0 5
239 8t2e16 11067 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
24059, 25, 239mulcomli 9601 . . 3  |-  ( 2  x.  8 )  = ; 1
6
24129, 196, 50, 216, 123, 33, 238, 240decmul2c 11027 . 2  |-  ( 2  x. ;;;; 2 6 5 2 8 )  = ;;;; 5 3 0 5 6
242215, 241breqtri 4456 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 setvar class
Syntax hints:    <-> wb 184    = wceq 1381    e. wcel 1802   (/)c0 3767   class class class wbr 4433  (class class class)co 6277   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495    < clt 9626    <_ cle 9627    - cmin 9805    / cdiv 10207   2c2 10586   3c3 10587   4c4 10588   5c5 10589   6c6 10590   7c7 10591   8c8 10592   9c9 10593   10c10 10594   NN0cn0 10796   ZZcz 10865  ;cdc 10979   ...cfz 11676   ^cexp 12140   sum_csu 13482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-rp 11225  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-sum 13483
This theorem is referenced by:  log2ub  23145
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