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Theorem 2exp16 13119
Description: Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
Assertion
Ref Expression
2exp16  |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6

Proof of Theorem 2exp16
StepHypRef Expression
1 2nn0 9998 . 2  |-  2  e.  NN0
2 8nn0 10004 . 2  |-  8  e.  NN0
32nn0cni 9993 . . 3  |-  8  e.  CC
4 2cn 9832 . . 3  |-  2  e.  CC
5 8t2e16 10228 . . 3  |-  ( 8  x.  2 )  = ; 1
6
63, 4, 5mulcomli 8860 . 2  |-  ( 2  x.  8 )  = ; 1
6
7 2exp8 13118 . 2  |-  ( 2 ^ 8 )  = ;; 2 5 6
8 5nn0 10001 . . . . 5  |-  5  e.  NN0
91, 8deccl 10154 . . . 4  |- ; 2 5  e.  NN0
10 6nn0 10002 . . . 4  |-  6  e.  NN0
119, 10deccl 10154 . . 3  |- ;; 2 5 6  e.  NN0
12 eqid 2296 . . 3  |- ;; 2 5 6  = ;; 2 5 6
13 1nn0 9997 . . . . 5  |-  1  e.  NN0
1413, 8deccl 10154 . . . 4  |- ; 1 5  e.  NN0
15 3nn0 9999 . . . 4  |-  3  e.  NN0
1614, 15deccl 10154 . . 3  |- ;; 1 5 3  e.  NN0
17 eqid 2296 . . . 4  |- ; 2 5  = ; 2 5
18 eqid 2296 . . . 4  |- ;; 1 5 3  = ;; 1 5 3
1913, 1deccl 10154 . . . . 5  |- ; 1 2  e.  NN0
2019, 2deccl 10154 . . . 4  |- ;; 1 2 8  e.  NN0
21 4nn0 10000 . . . . . 6  |-  4  e.  NN0
2213, 21deccl 10154 . . . . 5  |- ; 1 4  e.  NN0
23 eqid 2296 . . . . . 6  |- ; 1 5  = ; 1 5
24 eqid 2296 . . . . . 6  |- ;; 1 2 8  = ;; 1 2 8
25 0nn0 9996 . . . . . . . 8  |-  0  e.  NN0
2613dec0h 10156 . . . . . . . 8  |-  1  = ; 0 1
27 eqid 2296 . . . . . . . 8  |- ; 1 2  = ; 1 2
28 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
2928addid2i 9016 . . . . . . . 8  |-  ( 0  +  1 )  =  1
30 2p1e3 9863 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
314, 28, 30addcomli 9020 . . . . . . . 8  |-  ( 1  +  2 )  =  3
3225, 13, 13, 1, 26, 27, 29, 31decadd 10181 . . . . . . 7  |-  ( 1  + ; 1 2 )  = ; 1
3
33 3p1e4 9864 . . . . . . 7  |-  ( 3  +  1 )  =  4
3413, 15, 13, 32, 33decaddi 10184 . . . . . 6  |-  ( ( 1  + ; 1 2 )  +  1 )  = ; 1 4
35 5nn 9896 . . . . . . . 8  |-  5  e.  NN
3635nncni 9772 . . . . . . 7  |-  5  e.  CC
37 8p5e13 10198 . . . . . . 7  |-  ( 8  +  5 )  = ; 1
3
383, 36, 37addcomli 9020 . . . . . 6  |-  ( 5  +  8 )  = ; 1
3
3913, 8, 19, 2, 23, 24, 34, 15, 38decaddc 10182 . . . . 5  |-  (; 1 5  + ;; 1 2 8 )  = ;; 1 4 3
40 eqid 2296 . . . . . . 7  |- ; 1 4  = ; 1 4
41 4p1e5 9865 . . . . . . 7  |-  ( 4  +  1 )  =  5
4213, 21, 13, 40, 41decaddi 10184 . . . . . 6  |-  (; 1 4  +  1 )  = ; 1 5
43 2t2e4 9887 . . . . . . . 8  |-  ( 2  x.  2 )  =  4
44 1p1e2 9856 . . . . . . . 8  |-  ( 1  +  1 )  =  2
4543, 44oveq12i 5886 . . . . . . 7  |-  ( ( 2  x.  2 )  +  ( 1  +  1 ) )  =  ( 4  +  2 )
46 4p2e6 9873 . . . . . . 7  |-  ( 4  +  2 )  =  6
4745, 46eqtri 2316 . . . . . 6  |-  ( ( 2  x.  2 )  +  ( 1  +  1 ) )  =  6
48 5t2e10 9891 . . . . . . . 8  |-  ( 5  x.  2 )  =  10
49 dec10 10170 . . . . . . . 8  |-  10  = ; 1 0
5048, 49eqtri 2316 . . . . . . 7  |-  ( 5  x.  2 )  = ; 1
0
5136addid2i 9016 . . . . . . 7  |-  ( 0  +  5 )  =  5
5213, 25, 8, 50, 51decaddi 10184 . . . . . 6  |-  ( ( 5  x.  2 )  +  5 )  = ; 1
5
531, 8, 13, 8, 17, 42, 1, 8, 13, 47, 52decmac 10179 . . . . 5  |-  ( (; 2
5  x.  2 )  +  (; 1 4  +  1 ) )  = ; 6 5
54 6t2e12 10217 . . . . . 6  |-  ( 6  x.  2 )  = ; 1
2
55 3cn 9834 . . . . . . 7  |-  3  e.  CC
56 3p2e5 9871 . . . . . . 7  |-  ( 3  +  2 )  =  5
5755, 4, 56addcomli 9020 . . . . . 6  |-  ( 2  +  3 )  =  5
5813, 1, 15, 54, 57decaddi 10184 . . . . 5  |-  ( ( 6  x.  2 )  +  3 )  = ; 1
5
599, 10, 22, 15, 12, 39, 1, 8, 13, 53, 58decmac 10179 . . . 4  |-  ( (;; 2 5 6  x.  2 )  +  (; 1
5  + ;; 1 2 8 ) )  = ;; 6 5 5
6015dec0h 10156 . . . . 5  |-  3  = ; 0 3
6155addid2i 9016 . . . . . . 7  |-  ( 0  +  3 )  =  3
6261, 60eqtri 2316 . . . . . 6  |-  ( 0  +  3 )  = ; 0
3
634addid2i 9016 . . . . . . . 8  |-  ( 0  +  2 )  =  2
6463oveq2i 5885 . . . . . . 7  |-  ( ( 2  x.  5 )  +  ( 0  +  2 ) )  =  ( ( 2  x.  5 )  +  2 )
6536, 4, 48mulcomli 8860 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
6665, 49eqtri 2316 . . . . . . . 8  |-  ( 2  x.  5 )  = ; 1
0
6713, 25, 1, 66, 63decaddi 10184 . . . . . . 7  |-  ( ( 2  x.  5 )  +  2 )  = ; 1
2
6864, 67eqtri 2316 . . . . . 6  |-  ( ( 2  x.  5 )  +  ( 0  +  2 ) )  = ; 1
2
69 5t5e25 10216 . . . . . . 7  |-  ( 5  x.  5 )  = ; 2
5
70 5p3e8 9877 . . . . . . 7  |-  ( 5  +  3 )  =  8
711, 8, 15, 69, 70decaddi 10184 . . . . . 6  |-  ( ( 5  x.  5 )  +  3 )  = ; 2
8
721, 8, 25, 15, 17, 62, 8, 2, 1, 68, 71decmac 10179 . . . . 5  |-  ( (; 2
5  x.  5 )  +  ( 0  +  3 ) )  = ;; 1 2 8
73 6t5e30 10220 . . . . . 6  |-  ( 6  x.  5 )  = ; 3
0
7415, 25, 15, 73, 61decaddi 10184 . . . . 5  |-  ( ( 6  x.  5 )  +  3 )  = ; 3
3
759, 10, 25, 15, 12, 60, 8, 15, 15, 72, 74decmac 10179 . . . 4  |-  ( (;; 2 5 6  x.  5 )  +  3 )  = ;;; 1 2 8 3
761, 8, 14, 15, 17, 18, 11, 15, 20, 59, 75decma2c 10180 . . 3  |-  ( (;; 2 5 6  x. ; 2
5 )  + ;; 1 5 3 )  = ;;; 6 5 5 3
7761oveq2i 5885 . . . . . 6  |-  ( ( 2  x.  6 )  +  ( 0  +  3 ) )  =  ( ( 2  x.  6 )  +  3 )
78 6nn 9897 . . . . . . . . 9  |-  6  e.  NN
7978nncni 9772 . . . . . . . 8  |-  6  e.  CC
8079, 4, 54mulcomli 8860 . . . . . . 7  |-  ( 2  x.  6 )  = ; 1
2
8113, 1, 15, 80, 57decaddi 10184 . . . . . 6  |-  ( ( 2  x.  6 )  +  3 )  = ; 1
5
8277, 81eqtri 2316 . . . . 5  |-  ( ( 2  x.  6 )  +  ( 0  +  3 ) )  = ; 1
5
8379, 36, 73mulcomli 8860 . . . . . 6  |-  ( 5  x.  6 )  = ; 3
0
8415, 25, 15, 83, 61decaddi 10184 . . . . 5  |-  ( ( 5  x.  6 )  +  3 )  = ; 3
3
851, 8, 25, 15, 17, 60, 10, 15, 15, 82, 84decmac 10179 . . . 4  |-  ( (; 2
5  x.  6 )  +  3 )  = ;; 1 5 3
86 6t6e36 10221 . . . 4  |-  ( 6  x.  6 )  = ; 3
6
8710, 9, 10, 12, 10, 15, 85, 86decmul1c 10187 . . 3  |-  (;; 2 5 6  x.  6 )  = ;;; 1 5 3 6
8811, 9, 10, 12, 10, 16, 76, 87decmul2c 10188 . 2  |-  (;; 2 5 6  x. ;; 2 5 6 )  = ;;;; 6 5 5 3 6
891, 2, 6, 7, 88numexp2x 13110 1  |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
Colors of variables: wff set class
Syntax hints:    = wceq 1632  (class class class)co 5874   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   2c2 9811   3c3 9812   4c4 9813   5c5 9814   6c6 9815   8c8 9817   10c10 9819  ;cdc 10140   ^cexp 11120
This theorem is referenced by:  1259lem1  13145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-seq 11063  df-exp 11121
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