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Theorem log2ublem3 22456
Description: Lemma for log2ub 22457. 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 10509 . . . . . . 7  |-  0  <_  0
2 0re 9484 . . . . . . . . . . . . 13  |-  0  e.  RR
3 ltm1 10267 . . . . . . . . . . . . 13  |-  ( 0  e.  RR  ->  (
0  -  1 )  <  0 )
42, 3ax-mp 5 . . . . . . . . . . . 12  |-  ( 0  -  1 )  <  0
5 0z 10755 . . . . . . . . . . . . 13  |-  0  e.  ZZ
6 peano2zm 10786 . . . . . . . . . . . . . 14  |-  ( 0  e.  ZZ  ->  (
0  -  1 )  e.  ZZ )
75, 6ax-mp 5 . . . . . . . . . . . . 13  |-  ( 0  -  1 )  e.  ZZ
8 fzn 11564 . . . . . . . . . . . . 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 13274 . . . . . . . . . 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 13297 . . . . . . . . . 10  |-  sum_ n  e.  (/)  ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  =  0
1311, 12eqtri 2479 . . . . . . . . 9  |-  sum_ n  e.  ( 0 ... (
0  -  1 ) ) ( 2  / 
( ( 3  x.  ( ( 2  x.  n )  +  1 ) )  x.  (
9 ^ n ) ) )  =  0
1413oveq2i 6198 . . . . . . . 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 10494 . . . . . . . . . . 11  |-  3  e.  CC
16 7nn0 10699 . . . . . . . . . . 11  |-  7  e.  NN0
17 expcl 11981 . . . . . . . . . . 11  |-  ( ( 3  e.  CC  /\  7  e.  NN0 )  -> 
( 3 ^ 7 )  e.  CC )
1815, 16, 17mp2an 672 . . . . . . . . . 10  |-  ( 3 ^ 7 )  e.  CC
19 5cn 10499 . . . . . . . . . . 11  |-  5  e.  CC
20 7cn 10503 . . . . . . . . . . 11  |-  7  e.  CC
2119, 20mulcli 9489 . . . . . . . . . 10  |-  ( 5  x.  7 )  e.  CC
2218, 21mulcli 9489 . . . . . . . . 9  |-  ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  e.  CC
2322mul01i 9657 . . . . . . . 8  |-  ( ( ( 3 ^ 7 )  x.  ( 5  x.  7 ) )  x.  0 )  =  0
2414, 23eqtri 2479 . . . . . . 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 10490 . . . . . . . 8  |-  2  e.  CC
2625mul01i 9657 . . . . . . 7  |-  ( 2  x.  0 )  =  0
271, 24, 263brtr4i 4415 . . . . . 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 10692 . . . . . 6  |-  0  e.  NN0
29 2nn0 10694 . . . . . . . . . 10  |-  2  e.  NN0
30 5nn0 10697 . . . . . . . . . 10  |-  5  e.  NN0
3129, 30deccl 10867 . . . . . . . . 9  |- ; 2 5  e.  NN0
3231, 30deccl 10867 . . . . . . . 8  |- ;; 2 5 5  e.  NN0
33 1nn0 10693 . . . . . . . 8  |-  1  e.  NN0
3432, 33deccl 10867 . . . . . . 7  |- ;;; 2 5 5 1  e.  NN0
3534, 30deccl 10867 . . . . . 6  |- ;;;; 2 5 5 1 5  e.  NN0
36 eqid 2451 . . . . . 6  |-  ( 0  -  1 )  =  ( 0  -  1 )
3735nn0cni 10689 . . . . . . 7  |- ;;;; 2 5 5 1 5  e.  CC
3837addid2i 9655 . . . . . 6  |-  ( 0  + ;;;; 2 5 5 1 5 )  = ;;;; 2 5 5 1 5
39 3nn0 10695 . . . . . 6  |-  3  e.  NN0
4015addid1i 9654 . . . . . 6  |-  ( 3  +  0 )  =  3
4137mulid2i 9487 . . . . . . 7  |-  ( 1  x. ;;;; 2 5 5 1 5 )  = ;;;; 2 5 5 1 5
4226oveq1i 6197 . . . . . . . . 9  |-  ( ( 2  x.  0 )  +  1 )  =  ( 0  +  1 )
43 0p1e1 10531 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
4442, 43eqtri 2479 . . . . . . . 8  |-  ( ( 2  x.  0 )  +  1 )  =  1
4544oveq1i 6197 . . . . . . 7  |-  ( ( ( 2  x.  0 )  +  1 )  x. ;;;; 2 5 5 1 5 )  =  ( 1  x. ;;;; 2 5 5 1 5 )
4630, 16nn0mulcli 10716 . . . . . . . 8  |-  ( 5  x.  7 )  e. 
NN0
4716, 29deccl 10867 . . . . . . . 8  |- ; 7 2  e.  NN0
48 9nn0 10701 . . . . . . . 8  |-  9  e.  NN0
49 2p1e3 10543 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
50 8nn0 10700 . . . . . . . . . 10  |-  8  e.  NN0
51 1p1e2 10533 . . . . . . . . . . 11  |-  ( 1  +  1 )  =  2
52 9cn 10507 . . . . . . . . . . . . . 14  |-  9  e.  CC
53 exp1 11969 . . . . . . . . . . . . . 14  |-  ( 9  e.  CC  ->  (
9 ^ 1 )  =  9 )
5452, 53ax-mp 5 . . . . . . . . . . . . 13  |-  ( 9 ^ 1 )  =  9
5554oveq1i 6197 . . . . . . . . . . . 12  |-  ( ( 9 ^ 1 )  x.  9 )  =  ( 9  x.  9 )
56 9t9e81 10955 . . . . . . . . . . . 12  |-  ( 9  x.  9 )  = ; 8
1
5755, 56eqtri 2479 . . . . . . . . . . 11  |-  ( ( 9 ^ 1 )  x.  9 )  = ; 8
1
5848, 33, 51, 57numexpp1 14206 . . . . . . . . . 10  |-  ( 9 ^ 2 )  = ; 8
1
59 8cn 10505 . . . . . . . . . . . . 13  |-  8  e.  CC
60 9t8e72 10954 . . . . . . . . . . . . 13  |-  ( 9  x.  8 )  = ; 7
2
6152, 59, 60mulcomli 9491 . . . . . . . . . . . 12  |-  ( 8  x.  9 )  = ; 7
2
6261oveq1i 6197 . . . . . . . . . . 11  |-  ( ( 8  x.  9 )  +  0 )  =  (; 7 2  +  0 )
6347nn0cni 10689 . . . . . . . . . . . 12  |- ; 7 2  e.  CC
6463addid1i 9654 . . . . . . . . . . 11  |-  (; 7 2  +  0 )  = ; 7 2
6562, 64eqtri 2479 . . . . . . . . . 10  |-  ( ( 8  x.  9 )  +  0 )  = ; 7
2
6652mulid2i 9487 . . . . . . . . . . 11  |-  ( 1  x.  9 )  =  9
6748dec0h 10869 . . . . . . . . . . 11  |-  9  = ; 0 9
6866, 67eqtri 2479 . . . . . . . . . 10  |-  ( 1  x.  9 )  = ; 0
9
6948, 50, 33, 58, 48, 28, 65, 68decmul1c 10900 . . . . . . . . 9  |-  ( ( 9 ^ 2 )  x.  9 )  = ;; 7 2 9
7048, 29, 49, 69numexpp1 14206 . . . . . . . 8  |-  ( 9 ^ 3 )  = ;; 7 2 9
7139, 33deccl 10867 . . . . . . . 8  |- ; 3 1  e.  NN0
72 eqid 2451 . . . . . . . . 9  |- ; 7 2  = ; 7 2
73 eqid 2451 . . . . . . . . 9  |- ; 3 1  = ; 3 1
74 7t5e35 10938 . . . . . . . . . . 11  |-  ( 7  x.  5 )  = ; 3
5
7520, 19, 74mulcomli 9491 . . . . . . . . . 10  |-  ( 5  x.  7 )  = ; 3
5
76 7p3e10 10565 . . . . . . . . . . . 12  |-  ( 7  +  3 )  =  10
7720, 15, 76addcomli 9659 . . . . . . . . . . 11  |-  ( 3  +  7 )  =  10
78 dec10 10883 . . . . . . . . . . 11  |-  10  = ; 1 0
7977, 78eqtri 2479 . . . . . . . . . 10  |-  ( 3  +  7 )  = ; 1
0
80 ax-1cn 9438 . . . . . . . . . . . . 13  |-  1  e.  CC
81 3p1e4 10545 . . . . . . . . . . . . 13  |-  ( 3  +  1 )  =  4
8215, 80, 81addcomli 9659 . . . . . . . . . . . 12  |-  ( 1  +  3 )  =  4
8382oveq2i 6198 . . . . . . . . . . 11  |-  ( ( 3  x.  7 )  +  ( 1  +  3 ) )  =  ( ( 3  x.  7 )  +  4 )
84 4nn0 10696 . . . . . . . . . . . 12  |-  4  e.  NN0
85 7t3e21 10936 . . . . . . . . . . . . 13  |-  ( 7  x.  3 )  = ; 2
1
8620, 15, 85mulcomli 9491 . . . . . . . . . . . 12  |-  ( 3  x.  7 )  = ; 2
1
87 4cn 10497 . . . . . . . . . . . . 13  |-  4  e.  CC
88 4p1e5 10546 . . . . . . . . . . . . 13  |-  ( 4  +  1 )  =  5
8987, 80, 88addcomli 9659 . . . . . . . . . . . 12  |-  ( 1  +  4 )  =  5
9029, 33, 84, 86, 89decaddi 10897 . . . . . . . . . . 11  |-  ( ( 3  x.  7 )  +  4 )  = ; 2
5
9183, 90eqtri 2479 . . . . . . . . . 10  |-  ( ( 3  x.  7 )  +  ( 1  +  3 ) )  = ; 2
5
9275oveq1i 6197 . . . . . . . . . . 11  |-  ( ( 5  x.  7 )  +  0 )  =  (; 3 5  +  0 )
9339, 30deccl 10867 . . . . . . . . . . . . 13  |- ; 3 5  e.  NN0
9493nn0cni 10689 . . . . . . . . . . . 12  |- ; 3 5  e.  CC
9594addid1i 9654 . . . . . . . . . . 11  |-  (; 3 5  +  0 )  = ; 3 5
9692, 95eqtri 2479 . . . . . . . . . 10  |-  ( ( 5  x.  7 )  +  0 )  = ; 3
5
9739, 30, 33, 28, 75, 79, 16, 30, 39, 91, 96decmac 10892 . . . . . . . . 9  |-  ( ( ( 5  x.  7 )  x.  7 )  +  ( 3  +  7 ) )  = ;; 2 5 5
9833dec0h 10869 . . . . . . . . . 10  |-  1  = ; 0 1
99 3t2e6 10571 . . . . . . . . . . . 12  |-  ( 3  x.  2 )  =  6
10099, 43oveq12i 6199 . . . . . . . . . . 11  |-  ( ( 3  x.  2 )  +  ( 0  +  1 ) )  =  ( 6  +  1 )
101 6p1e7 10548 . . . . . . . . . . 11  |-  ( 6  +  1 )  =  7
102100, 101eqtri 2479 . . . . . . . . . 10  |-  ( ( 3  x.  2 )  +  ( 0  +  1 ) )  =  7
103 5t2e10 10574 . . . . . . . . . . . 12  |-  ( 5  x.  2 )  =  10
104103, 78eqtri 2479 . . . . . . . . . . 11  |-  ( 5  x.  2 )  = ; 1
0
10533, 28, 43, 104decsuc 10876 . . . . . . . . . 10  |-  ( ( 5  x.  2 )  +  1 )  = ; 1
1
10639, 30, 28, 33, 75, 98, 29, 33, 33, 102, 105decmac 10892 . . . . . . . . 9  |-  ( ( ( 5  x.  7 )  x.  2 )  +  1 )  = ; 7
1
10716, 29, 39, 33, 72, 73, 46, 33, 16, 97, 106decma2c 10893 . . . . . . . 8  |-  ( ( ( 5  x.  7 )  x. ; 7 2 )  + ; 3
1 )  = ;;; 2 5 5 1
108 9t3e27 10949 . . . . . . . . . . 11  |-  ( 9  x.  3 )  = ; 2
7
10952, 15, 108mulcomli 9491 . . . . . . . . . 10  |-  ( 3  x.  9 )  = ; 2
7
110 7p4e11 10905 . . . . . . . . . 10  |-  ( 7  +  4 )  = ; 1
1
11129, 16, 84, 109, 49, 33, 110decaddci 10898 . . . . . . . . 9  |-  ( ( 3  x.  9 )  +  4 )  = ; 3
1
112 9t5e45 10951 . . . . . . . . . 10  |-  ( 9  x.  5 )  = ; 4
5
11352, 19, 112mulcomli 9491 . . . . . . . . 9  |-  ( 5  x.  9 )  = ; 4
5
11448, 39, 30, 75, 30, 84, 111, 113decmul1c 10900 . . . . . . . 8  |-  ( ( 5  x.  7 )  x.  9 )  = ;; 3 1 5
11546, 47, 48, 70, 30, 71, 107, 114decmul2c 10901 . . . . . . 7  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 3 ) )  = ;;;; 2 5 5 1 5
11641, 45, 1153eqtr4ri 2490 . . . . . 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 22455 . . . . 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 10867 . . . . . 6  |- ; 9 4  e.  NN0
119118, 30deccl 10867 . . . . 5  |- ;; 9 4 5  e.  NN0
120 1m1e0 10488 . . . . 5  |-  ( 1  -  1 )  =  0
121 eqid 2451 . . . . . 6  |- ;;;; 2 5 5 1 5  = ;;;; 2 5 5 1 5
122 eqid 2451 . . . . . 6  |- ;; 9 4 5  = ;; 9 4 5
123 6nn0 10698 . . . . . . . . 9  |-  6  e.  NN0
12429, 123deccl 10867 . . . . . . . 8  |- ; 2 6  e.  NN0
125124, 84deccl 10867 . . . . . . 7  |- ;; 2 6 4  e.  NN0
126 5p1e6 10547 . . . . . . 7  |-  ( 5  +  1 )  =  6
127 eqid 2451 . . . . . . . 8  |- ;;; 2 5 5 1  = ;;; 2 5 5 1
128 eqid 2451 . . . . . . . 8  |- ; 9 4  = ; 9 4
129 eqid 2451 . . . . . . . . 9  |- ;; 2 5 5  = ;; 2 5 5
130 eqid 2451 . . . . . . . . . 10  |- ; 2 5  = ; 2 5
13129, 30, 126, 130decsuc 10876 . . . . . . . . 9  |-  (; 2 5  +  1 )  = ; 2 6
132 9p5e14 10918 . . . . . . . . . 10  |-  ( 9  +  5 )  = ; 1
4
13352, 19, 132addcomli 9659 . . . . . . . . 9  |-  ( 5  +  9 )  = ; 1
4
13431, 30, 48, 129, 131, 84, 133decaddci 10898 . . . . . . . 8  |-  (;; 2 5 5  +  9 )  = ;; 2 6 4
13532, 33, 48, 84, 127, 128, 134, 89decadd 10894 . . . . . . 7  |-  (;;; 2 5 5 1  + ; 9 4 )  = ;;; 2 6 4 5
136125, 30, 126, 135decsuc 10876 . . . . . 6  |-  ( (;;; 2 5 5 1  + ; 9
4 )  +  1 )  = ;;; 2 6 4 6
137 5p5e10 10560 . . . . . 6  |-  ( 5  +  5 )  =  10
13834, 30, 118, 30, 121, 122, 136, 137decaddc2 10896 . . . . 5  |-  (;;;; 2 5 5 1 5  + ;; 9 4 5 )  = ;;;; 2 6 4 6 0
13952sqvali 12043 . . . . . . . . 9  |-  ( 9 ^ 2 )  =  ( 9  x.  9 )
140 3t3e9 10572 . . . . . . . . . 10  |-  ( 3  x.  3 )  =  9
141140oveq1i 6197 . . . . . . . . 9  |-  ( ( 3  x.  3 )  x.  9 )  =  ( 9  x.  9 )
142139, 141eqtr4i 2482 . . . . . . . 8  |-  ( 9 ^ 2 )  =  ( ( 3  x.  3 )  x.  9 )
14315, 15, 52mulassi 9493 . . . . . . . 8  |-  ( ( 3  x.  3 )  x.  9 )  =  ( 3  x.  (
3  x.  9 ) )
144142, 143eqtri 2479 . . . . . . 7  |-  ( 9 ^ 2 )  =  ( 3  x.  (
3  x.  9 ) )
145144oveq2i 6198 . . . . . 6  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 2 ) )  =  ( ( 5  x.  7 )  x.  (
3  x.  ( 3  x.  9 ) ) )
14615, 52mulcli 9489 . . . . . . . 8  |-  ( 3  x.  9 )  e.  CC
14721, 15, 146mul12i 9662 . . . . . . 7  |-  ( ( 5  x.  7 )  x.  ( 3  x.  ( 3  x.  9 ) ) )  =  ( 3  x.  (
( 5  x.  7 )  x.  ( 3  x.  9 ) ) )
14829, 84deccl 10867 . . . . . . . . 9  |- ; 2 4  e.  NN0
149 eqid 2451 . . . . . . . . . 10  |- ; 2 4  = ; 2 4
15099, 49oveq12i 6199 . . . . . . . . . . 11  |-  ( ( 3  x.  2 )  +  ( 2  +  1 ) )  =  ( 6  +  3 )
151 6p3e9 10562 . . . . . . . . . . 11  |-  ( 6  +  3 )  =  9
152150, 151eqtri 2479 . . . . . . . . . 10  |-  ( ( 3  x.  2 )  +  ( 2  +  1 ) )  =  9
15387addid2i 9655 . . . . . . . . . . 11  |-  ( 0  +  4 )  =  4
15433, 28, 84, 104, 153decaddi 10897 . . . . . . . . . 10  |-  ( ( 5  x.  2 )  +  4 )  = ; 1
4
15539, 30, 29, 84, 75, 149, 29, 84, 33, 152, 154decmac 10892 . . . . . . . . 9  |-  ( ( ( 5  x.  7 )  x.  2 )  + ; 2 4 )  = ; 9
4
15629, 33, 39, 86, 82decaddi 10897 . . . . . . . . . 10  |-  ( ( 3  x.  7 )  +  3 )  = ; 2
4
15716, 39, 30, 75, 30, 39, 156, 75decmul1c 10900 . . . . . . . . 9  |-  ( ( 5  x.  7 )  x.  7 )  = ;; 2 4 5
15846, 29, 16, 109, 30, 148, 155, 157decmul2c 10901 . . . . . . . 8  |-  ( ( 5  x.  7 )  x.  ( 3  x.  9 ) )  = ;; 9 4 5
159158oveq2i 6198 . . . . . . 7  |-  ( 3  x.  ( ( 5  x.  7 )  x.  ( 3  x.  9 ) ) )  =  ( 3  x. ;; 9 4 5 )
160147, 159eqtri 2479 . . . . . 6  |-  ( ( 5  x.  7 )  x.  ( 3  x.  ( 3  x.  9 ) ) )  =  ( 3  x. ;; 9 4 5 )
161 df-3 10479 . . . . . . . 8  |-  3  =  ( 2  +  1 )
16225mulid1i 9486 . . . . . . . . 9  |-  ( 2  x.  1 )  =  2
163162oveq1i 6197 . . . . . . . 8  |-  ( ( 2  x.  1 )  +  1 )  =  ( 2  +  1 )
164161, 163eqtr4i 2482 . . . . . . 7  |-  3  =  ( ( 2  x.  1 )  +  1 )
165164oveq1i 6197 . . . . . 6  |-  ( 3  x. ;; 9 4 5 )  =  ( ( ( 2  x.  1 )  +  1 )  x. ;; 9 4 5 )
166145, 160, 1653eqtri 2483 . . . . 5  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 2 ) )  =  ( ( ( 2  x.  1 )  +  1 )  x. ;; 9 4 5 )
167117, 35, 119, 33, 120, 138, 29, 49, 166log2ublem2 22455 . . . 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 10867 . . . . 5  |- ;;; 2 6 4 6  e.  NN0
169168, 28deccl 10867 . . . 4  |- ;;;; 2 6 4 6 0  e.  NN0
170123, 39deccl 10867 . . . 4  |- ; 6 3  e.  NN0
171 2m1e1 10534 . . . 4  |-  ( 2  -  1 )  =  1
172 eqid 2451 . . . . 5  |- ;;;; 2 6 4 6 0  = ;;;; 2 6 4 6 0
173 eqid 2451 . . . . 5  |- ; 6 3  = ; 6 3
174 eqid 2451 . . . . . 6  |- ;;; 2 6 4 6  = ;;; 2 6 4 6
175 eqid 2451 . . . . . . 7  |- ;; 2 6 4  = ;; 2 6 4
176124, 84, 88, 175decsuc 10876 . . . . . 6  |-  (;; 2 6 4  +  1 )  = ;; 2 6 5
177 6p6e12 10904 . . . . . 6  |-  ( 6  +  6 )  = ; 1
2
178125, 123, 123, 174, 176, 29, 177decaddci 10898 . . . . 5  |-  (;;; 2 6 4 6  +  6 )  = ;;; 2 6 5 2
17915addid2i 9655 . . . . 5  |-  ( 0  +  3 )  =  3
180168, 28, 123, 39, 172, 173, 178, 179decadd 10894 . . . 4  |-  (;;;; 2 6 4 6 0  + ; 6 3 )  = ;;;; 2 6 5 2 3
181 1p2e3 10544 . . . 4  |-  ( 1  +  2 )  =  3
18254oveq2i 6198 . . . . 5  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 1 ) )  =  ( ( 5  x.  7 )  x.  9 )
18319, 20, 52mulassi 9493 . . . . . 6  |-  ( ( 5  x.  7 )  x.  9 )  =  ( 5  x.  (
7  x.  9 ) )
184 9t7e63 10953 . . . . . . . 8  |-  ( 9  x.  7 )  = ; 6
3
18552, 20, 184mulcomli 9491 . . . . . . 7  |-  ( 7  x.  9 )  = ; 6
3
186185oveq2i 6198 . . . . . 6  |-  ( 5  x.  ( 7  x.  9 ) )  =  ( 5  x. ; 6 3 )
187183, 186eqtri 2479 . . . . 5  |-  ( ( 5  x.  7 )  x.  9 )  =  ( 5  x. ; 6 3 )
188 df-5 10481 . . . . . . 7  |-  5  =  ( 4  +  1 )
189 2t2e4 10569 . . . . . . . 8  |-  ( 2  x.  2 )  =  4
190189oveq1i 6197 . . . . . . 7  |-  ( ( 2  x.  2 )  +  1 )  =  ( 4  +  1 )
191188, 190eqtr4i 2482 . . . . . 6  |-  5  =  ( ( 2  x.  2 )  +  1 )
192191oveq1i 6197 . . . . 5  |-  ( 5  x. ; 6 3 )  =  ( ( ( 2  x.  2 )  +  1 )  x. ; 6 3 )
193182, 187, 1923eqtri 2483 . . . 4  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 1 ) )  =  ( ( ( 2  x.  2 )  +  1 )  x. ; 6 3 )
194167, 169, 170, 29, 171, 180, 33, 181, 193log2ublem2 22455 . . 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 10867 . . . . 5  |- ;; 2 6 5  e.  NN0
196195, 29deccl 10867 . . . 4  |- ;;; 2 6 5 2  e.  NN0
197196, 39deccl 10867 . . 3  |- ;;;; 2 6 5 2 3  e.  NN0
198 3m1e2 10536 . . 3  |-  ( 3  -  1 )  =  2
199 eqid 2451 . . . 4  |- ;;;; 2 6 5 2 3  = ;;;; 2 6 5 2 3
200 5p3e8 10558 . . . . 5  |-  ( 5  +  3 )  =  8
20119, 15, 200addcomli 9659 . . . 4  |-  ( 3  +  5 )  =  8
202196, 39, 30, 199, 201decaddi 10897 . . 3  |-  (;;;; 2 6 5 2 3  +  5 )  = ;;;; 2 6 5 2 8
20320, 19mulcli 9489 . . . . 5  |-  ( 7  x.  5 )  e.  CC
204203mulid1i 9486 . . . 4  |-  ( ( 7  x.  5 )  x.  1 )  =  ( 7  x.  5 )
20519, 20mulcomi 9490 . . . . 5  |-  ( 5  x.  7 )  =  ( 7  x.  5 )
206 exp0 11967 . . . . . 6  |-  ( 9  e.  CC  ->  (
9 ^ 0 )  =  1 )
20752, 206ax-mp 5 . . . . 5  |-  ( 9 ^ 0 )  =  1
208205, 207oveq12i 6199 . . . 4  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 0 ) )  =  ( ( 7  x.  5 )  x.  1 )
20915, 25, 99mulcomli 9491 . . . . . . 7  |-  ( 2  x.  3 )  =  6
210209oveq1i 6197 . . . . . 6  |-  ( ( 2  x.  3 )  +  1 )  =  ( 6  +  1 )
211 df-7 10483 . . . . . 6  |-  7  =  ( 6  +  1 )
212210, 211eqtr4i 2482 . . . . 5  |-  ( ( 2  x.  3 )  +  1 )  =  7
213212oveq1i 6197 . . . 4  |-  ( ( ( 2  x.  3 )  +  1 )  x.  5 )  =  ( 7  x.  5 )
214204, 208, 2133eqtr4i 2489 . . 3  |-  ( ( 5  x.  7 )  x.  ( 9 ^ 0 ) )  =  ( ( ( 2  x.  3 )  +  1 )  x.  5 )
215194, 197, 30, 39, 198, 202, 28, 179, 214log2ublem2 22455 . 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 2451 . . 3  |- ;;;; 2 6 5 2 8  = ;;;; 2 6 5 2 8
217 eqid 2451 . . . 4  |- ;;; 2 6 5 2  = ;;; 2 6 5 2
218 eqid 2451 . . . . 5  |- ;; 2 6 5  = ;; 2 6 5
219 00id 9642 . . . . . 6  |-  ( 0  +  0 )  =  0
22028dec0h 10869 . . . . . 6  |-  0  = ; 0 0
221219, 220eqtri 2479 . . . . 5  |-  ( 0  +  0 )  = ; 0
0
222 eqid 2451 . . . . . 6  |- ; 2 6  = ; 2 6
22343, 98eqtri 2479 . . . . . 6  |-  ( 0  +  1 )  = ; 0
1
224189, 43oveq12i 6199 . . . . . . 7  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  ( 4  +  1 )
225224, 88eqtri 2479 . . . . . 6  |-  ( ( 2  x.  2 )  +  ( 0  +  1 ) )  =  5
226 6cn 10501 . . . . . . . 8  |-  6  e.  CC
227 6t2e12 10930 . . . . . . . 8  |-  ( 6  x.  2 )  = ; 1
2
228226, 25, 227mulcomli 9491 . . . . . . 7  |-  ( 2  x.  6 )  = ; 1
2
22933, 29, 49, 228decsuc 10876 . . . . . 6  |-  ( ( 2  x.  6 )  +  1 )  = ; 1
3
23029, 123, 28, 33, 222, 223, 29, 39, 33, 225, 229decma2c 10893 . . . . 5  |-  ( ( 2  x. ; 2 6 )  +  ( 0  +  1 ) )  = ; 5 3
23119, 25, 103mulcomli 9491 . . . . . . 7  |-  ( 2  x.  5 )  =  10
232231oveq1i 6197 . . . . . 6  |-  ( ( 2  x.  5 )  +  0 )  =  ( 10  +  0 )
233 dec10p 10882 . . . . . 6  |-  ( 10  +  0 )  = ; 1
0
234232, 233eqtri 2479 . . . . 5  |-  ( ( 2  x.  5 )  +  0 )  = ; 1
0
235124, 30, 28, 28, 218, 221, 29, 28, 33, 230, 234decma2c 10893 . . . 4  |-  ( ( 2  x. ;; 2 6 5 )  +  ( 0  +  0 ) )  = ;; 5 3 0
23630dec0h 10869 . . . . 5  |-  5  = ; 0 5
237190, 88, 2363eqtri 2483 . . . 4  |-  ( ( 2  x.  2 )  +  1 )  = ; 0
5
238195, 29, 28, 33, 217, 98, 29, 30, 28, 235, 237decma2c 10893 . . 3  |-  ( ( 2  x. ;;; 2 6 5 2 )  +  1 )  = ;;; 5 3 0 5
239 8t2e16 10941 . . . 4  |-  ( 8  x.  2 )  = ; 1
6
24059, 25, 239mulcomli 9491 . . 3  |-  ( 2  x.  8 )  = ; 1
6
24129, 196, 50, 216, 123, 33, 238, 240decmul2c 10901 . 2  |-  ( 2  x. ;;;; 2 6 5 2 8 )  = ;;;; 5 3 0 5 6
242215, 241breqtri 4410 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 1370    e. wcel 1758   (/)c0 3732   class class class wbr 4387  (class class class)co 6187   CCcc 9378   RRcr 9379   0cc0 9380   1c1 9381    + caddc 9383    x. cmul 9385    < clt 9516    <_ cle 9517    - cmin 9693    / cdiv 10091   2c2 10469   3c3 10470   4c4 10471   5c5 10472   6c6 10473   7c7 10474   8c8 10475   9c9 10476   10c10 10477   NN0cn0 10677   ZZcz 10744  ;cdc 10853   ...cfz 11535   ^cexp 11963   sum_csu 13262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-inf2 7945  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457  ax-pre-sup 9458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-se 4775  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-sup 7789  df-oi 7822  df-card 8207  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092  df-nn 10421  df-2 10478  df-3 10479  df-4 10480  df-5 10481  df-6 10482  df-7 10483  df-8 10484  df-9 10485  df-10 10486  df-n0 10678  df-z 10745  df-dec 10854  df-uz 10960  df-rp 11090  df-fz 11536  df-fzo 11647  df-seq 11905  df-exp 11964  df-hash 12202  df-cj 12687  df-re 12688  df-im 12689  df-sqr 12823  df-abs 12824  df-clim 13065  df-sum 13263
This theorem is referenced by:  log2ub  22457
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