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Theorem 2exp16 13103
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 9982 . 2  |-  2  e.  NN0
2 8nn0 9988 . 2  |-  8  e.  NN0
32nn0cni 9977 . . 3  |-  8  e.  CC
4 2cn 9816 . . 3  |-  2  e.  CC
5 8t2e16 10212 . . 3  |-  ( 8  x.  2 )  = ; 1
6
63, 4, 5mulcomli 8844 . 2  |-  ( 2  x.  8 )  = ; 1
6
7 2exp8 13102 . 2  |-  ( 2 ^ 8 )  = ;; 2 5 6
8 5nn0 9985 . . . . 5  |-  5  e.  NN0
91, 8deccl 10138 . . . 4  |- ; 2 5  e.  NN0
10 6nn0 9986 . . . 4  |-  6  e.  NN0
119, 10deccl 10138 . . 3  |- ;; 2 5 6  e.  NN0
12 eqid 2283 . . 3  |- ;; 2 5 6  = ;; 2 5 6
13 1nn0 9981 . . . . 5  |-  1  e.  NN0
1413, 8deccl 10138 . . . 4  |- ; 1 5  e.  NN0
15 3nn0 9983 . . . 4  |-  3  e.  NN0
1614, 15deccl 10138 . . 3  |- ;; 1 5 3  e.  NN0
17 eqid 2283 . . . 4  |- ; 2 5  = ; 2 5
18 eqid 2283 . . . 4  |- ;; 1 5 3  = ;; 1 5 3
1913, 1deccl 10138 . . . . 5  |- ; 1 2  e.  NN0
2019, 2deccl 10138 . . . 4  |- ;; 1 2 8  e.  NN0
21 4nn0 9984 . . . . . 6  |-  4  e.  NN0
2213, 21deccl 10138 . . . . 5  |- ; 1 4  e.  NN0
23 eqid 2283 . . . . . 6  |- ; 1 5  = ; 1 5
24 eqid 2283 . . . . . 6  |- ;; 1 2 8  = ;; 1 2 8
25 0nn0 9980 . . . . . . . 8  |-  0  e.  NN0
2613dec0h 10140 . . . . . . . 8  |-  1  = ; 0 1
27 eqid 2283 . . . . . . . 8  |- ; 1 2  = ; 1 2
28 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
2928addid2i 9000 . . . . . . . 8  |-  ( 0  +  1 )  =  1
30 2p1e3 9847 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
314, 28, 30addcomli 9004 . . . . . . . 8  |-  ( 1  +  2 )  =  3
3225, 13, 13, 1, 26, 27, 29, 31decadd 10165 . . . . . . 7  |-  ( 1  + ; 1 2 )  = ; 1
3
33 3p1e4 9848 . . . . . . 7  |-  ( 3  +  1 )  =  4
3413, 15, 13, 32, 33decaddi 10168 . . . . . 6  |-  ( ( 1  + ; 1 2 )  +  1 )  = ; 1 4
35 5nn 9880 . . . . . . . 8  |-  5  e.  NN
3635nncni 9756 . . . . . . 7  |-  5  e.  CC
37 8p5e13 10182 . . . . . . 7  |-  ( 8  +  5 )  = ; 1
3
383, 36, 37addcomli 9004 . . . . . 6  |-  ( 5  +  8 )  = ; 1
3
3913, 8, 19, 2, 23, 24, 34, 15, 38decaddc 10166 . . . . 5  |-  (; 1 5  + ;; 1 2 8 )  = ;; 1 4 3
40 eqid 2283 . . . . . . 7  |- ; 1 4  = ; 1 4
41 4p1e5 9849 . . . . . . 7  |-  ( 4  +  1 )  =  5
4213, 21, 13, 40, 41decaddi 10168 . . . . . 6  |-  (; 1 4  +  1 )  = ; 1 5
43 2t2e4 9871 . . . . . . . 8  |-  ( 2  x.  2 )  =  4
44 1p1e2 9840 . . . . . . . 8  |-  ( 1  +  1 )  =  2
4543, 44oveq12i 5870 . . . . . . 7  |-  ( ( 2  x.  2 )  +  ( 1  +  1 ) )  =  ( 4  +  2 )
46 4p2e6 9857 . . . . . . 7  |-  ( 4  +  2 )  =  6
4745, 46eqtri 2303 . . . . . 6  |-  ( ( 2  x.  2 )  +  ( 1  +  1 ) )  =  6
48 5t2e10 9875 . . . . . . . 8  |-  ( 5  x.  2 )  =  10
49 dec10 10154 . . . . . . . 8  |-  10  = ; 1 0
5048, 49eqtri 2303 . . . . . . 7  |-  ( 5  x.  2 )  = ; 1
0
5136addid2i 9000 . . . . . . 7  |-  ( 0  +  5 )  =  5
5213, 25, 8, 50, 51decaddi 10168 . . . . . 6  |-  ( ( 5  x.  2 )  +  5 )  = ; 1
5
531, 8, 13, 8, 17, 42, 1, 8, 13, 47, 52decmac 10163 . . . . 5  |-  ( (; 2
5  x.  2 )  +  (; 1 4  +  1 ) )  = ; 6 5
54 6t2e12 10201 . . . . . 6  |-  ( 6  x.  2 )  = ; 1
2
55 3cn 9818 . . . . . . 7  |-  3  e.  CC
56 3p2e5 9855 . . . . . . 7  |-  ( 3  +  2 )  =  5
5755, 4, 56addcomli 9004 . . . . . 6  |-  ( 2  +  3 )  =  5
5813, 1, 15, 54, 57decaddi 10168 . . . . 5  |-  ( ( 6  x.  2 )  +  3 )  = ; 1
5
599, 10, 22, 15, 12, 39, 1, 8, 13, 53, 58decmac 10163 . . . 4  |-  ( (;; 2 5 6  x.  2 )  +  (; 1
5  + ;; 1 2 8 ) )  = ;; 6 5 5
6015dec0h 10140 . . . . 5  |-  3  = ; 0 3
6155addid2i 9000 . . . . . . 7  |-  ( 0  +  3 )  =  3
6261, 60eqtri 2303 . . . . . 6  |-  ( 0  +  3 )  = ; 0
3
634addid2i 9000 . . . . . . . 8  |-  ( 0  +  2 )  =  2
6463oveq2i 5869 . . . . . . 7  |-  ( ( 2  x.  5 )  +  ( 0  +  2 ) )  =  ( ( 2  x.  5 )  +  2 )
6536, 4, 48mulcomli 8844 . . . . . . . . 9  |-  ( 2  x.  5 )  =  10
6665, 49eqtri 2303 . . . . . . . 8  |-  ( 2  x.  5 )  = ; 1
0
6713, 25, 1, 66, 63decaddi 10168 . . . . . . 7  |-  ( ( 2  x.  5 )  +  2 )  = ; 1
2
6864, 67eqtri 2303 . . . . . 6  |-  ( ( 2  x.  5 )  +  ( 0  +  2 ) )  = ; 1
2
69 5t5e25 10200 . . . . . . 7  |-  ( 5  x.  5 )  = ; 2
5
70 5p3e8 9861 . . . . . . 7  |-  ( 5  +  3 )  =  8
711, 8, 15, 69, 70decaddi 10168 . . . . . 6  |-  ( ( 5  x.  5 )  +  3 )  = ; 2
8
721, 8, 25, 15, 17, 62, 8, 2, 1, 68, 71decmac 10163 . . . . 5  |-  ( (; 2
5  x.  5 )  +  ( 0  +  3 ) )  = ;; 1 2 8
73 6t5e30 10204 . . . . . 6  |-  ( 6  x.  5 )  = ; 3
0
7415, 25, 15, 73, 61decaddi 10168 . . . . 5  |-  ( ( 6  x.  5 )  +  3 )  = ; 3
3
759, 10, 25, 15, 12, 60, 8, 15, 15, 72, 74decmac 10163 . . . 4  |-  ( (;; 2 5 6  x.  5 )  +  3 )  = ;;; 1 2 8 3
761, 8, 14, 15, 17, 18, 11, 15, 20, 59, 75decma2c 10164 . . 3  |-  ( (;; 2 5 6  x. ; 2
5 )  + ;; 1 5 3 )  = ;;; 6 5 5 3
7761oveq2i 5869 . . . . . 6  |-  ( ( 2  x.  6 )  +  ( 0  +  3 ) )  =  ( ( 2  x.  6 )  +  3 )
78 6nn 9881 . . . . . . . . 9  |-  6  e.  NN
7978nncni 9756 . . . . . . . 8  |-  6  e.  CC
8079, 4, 54mulcomli 8844 . . . . . . 7  |-  ( 2  x.  6 )  = ; 1
2
8113, 1, 15, 80, 57decaddi 10168 . . . . . 6  |-  ( ( 2  x.  6 )  +  3 )  = ; 1
5
8277, 81eqtri 2303 . . . . 5  |-  ( ( 2  x.  6 )  +  ( 0  +  3 ) )  = ; 1
5
8379, 36, 73mulcomli 8844 . . . . . 6  |-  ( 5  x.  6 )  = ; 3
0
8415, 25, 15, 83, 61decaddi 10168 . . . . 5  |-  ( ( 5  x.  6 )  +  3 )  = ; 3
3
851, 8, 25, 15, 17, 60, 10, 15, 15, 82, 84decmac 10163 . . . 4  |-  ( (; 2
5  x.  6 )  +  3 )  = ;; 1 5 3
86 6t6e36 10205 . . . 4  |-  ( 6  x.  6 )  = ; 3
6
8710, 9, 10, 12, 10, 15, 85, 86decmul1c 10171 . . 3  |-  (;; 2 5 6  x.  6 )  = ;;; 1 5 3 6
8811, 9, 10, 12, 10, 16, 76, 87decmul2c 10172 . 2  |-  (;; 2 5 6  x. ;; 2 5 6 )  = ;;;; 6 5 5 3 6
891, 2, 6, 7, 88numexp2x 13094 1  |-  ( 2 ^; 1 6 )  = ;;;; 6 5 5 3 6
Colors of variables: wff set class
Syntax hints:    = wceq 1623  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   2c2 9795   3c3 9796   4c4 9797   5c5 9798   6c6 9799   8c8 9801   10c10 9803  ;cdc 10124   ^cexp 11104
This theorem is referenced by:  1259lem1  13129
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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
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-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-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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  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-seq 11047  df-exp 11105
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