Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 2exp11 | Structured version Visualization version GIF version |
Description: Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021.) |
Ref | Expression |
---|---|
2exp11 | ⊢ (2↑;11) = ;;;2048 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8p3e11 11488 | . . . . 5 ⊢ (8 + 3) = ;11 | |
2 | 1 | eqcomi 2619 | . . . 4 ⊢ ;11 = (8 + 3) |
3 | 2 | oveq2i 6560 | . . 3 ⊢ (2↑;11) = (2↑(8 + 3)) |
4 | 2cn 10968 | . . . 4 ⊢ 2 ∈ ℂ | |
5 | 8nn0 11192 | . . . 4 ⊢ 8 ∈ ℕ0 | |
6 | 3nn0 11187 | . . . 4 ⊢ 3 ∈ ℕ0 | |
7 | expadd 12764 | . . . 4 ⊢ ((2 ∈ ℂ ∧ 8 ∈ ℕ0 ∧ 3 ∈ ℕ0) → (2↑(8 + 3)) = ((2↑8) · (2↑3))) | |
8 | 4, 5, 6, 7 | mp3an 1416 | . . 3 ⊢ (2↑(8 + 3)) = ((2↑8) · (2↑3)) |
9 | 3, 8 | eqtri 2632 | . 2 ⊢ (2↑;11) = ((2↑8) · (2↑3)) |
10 | 2exp8 15634 | . . . 4 ⊢ (2↑8) = ;;256 | |
11 | cu2 12825 | . . . 4 ⊢ (2↑3) = 8 | |
12 | 10, 11 | oveq12i 6561 | . . 3 ⊢ ((2↑8) · (2↑3)) = (;;256 · 8) |
13 | 2nn0 11186 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
14 | 5nn0 11189 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
15 | 13, 14 | deccl 11388 | . . . 4 ⊢ ;25 ∈ ℕ0 |
16 | 6nn0 11190 | . . . 4 ⊢ 6 ∈ ℕ0 | |
17 | eqid 2610 | . . . 4 ⊢ ;;256 = ;;256 | |
18 | 4nn0 11188 | . . . 4 ⊢ 4 ∈ ℕ0 | |
19 | 0nn0 11184 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
20 | 13, 19 | deccl 11388 | . . . . 5 ⊢ ;20 ∈ ℕ0 |
21 | eqid 2610 | . . . . . 6 ⊢ ;25 = ;25 | |
22 | 1nn0 11185 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
23 | 8cn 10983 | . . . . . . . 8 ⊢ 8 ∈ ℂ | |
24 | 8t2e16 11530 | . . . . . . . 8 ⊢ (8 · 2) = ;16 | |
25 | 23, 4, 24 | mulcomli 9926 | . . . . . . 7 ⊢ (2 · 8) = ;16 |
26 | 1p1e2 11011 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
27 | 6p4e10 11474 | . . . . . . 7 ⊢ (6 + 4) = ;10 | |
28 | 22, 16, 18, 25, 26, 19, 27 | decaddci 11456 | . . . . . 6 ⊢ ((2 · 8) + 4) = ;20 |
29 | 5cn 10977 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
30 | 8t5e40 11533 | . . . . . . 7 ⊢ (8 · 5) = ;40 | |
31 | 23, 29, 30 | mulcomli 9926 | . . . . . 6 ⊢ (5 · 8) = ;40 |
32 | 5, 13, 14, 21, 19, 18, 28, 31 | decmul1c 11463 | . . . . 5 ⊢ (;25 · 8) = ;;200 |
33 | 4cn 10975 | . . . . . 6 ⊢ 4 ∈ ℂ | |
34 | 33 | addid2i 10103 | . . . . 5 ⊢ (0 + 4) = 4 |
35 | 20, 19, 18, 32, 34 | decaddi 11455 | . . . 4 ⊢ ((;25 · 8) + 4) = ;;204 |
36 | 6cn 10979 | . . . . 5 ⊢ 6 ∈ ℂ | |
37 | 8t6e48 11535 | . . . . 5 ⊢ (8 · 6) = ;48 | |
38 | 23, 36, 37 | mulcomli 9926 | . . . 4 ⊢ (6 · 8) = ;48 |
39 | 5, 15, 16, 17, 5, 18, 35, 38 | decmul1c 11463 | . . 3 ⊢ (;;256 · 8) = ;;;2048 |
40 | 12, 39 | eqtri 2632 | . 2 ⊢ ((2↑8) · (2↑3)) = ;;;2048 |
41 | 9, 40 | eqtri 2632 | 1 ⊢ (2↑;11) = ;;;2048 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 2c2 10947 3c3 10948 4c4 10949 5c5 10950 6c6 10951 8c8 10953 ℕ0cn0 11169 ;cdc 11369 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-seq 12664 df-exp 12723 |
This theorem is referenced by: m11nprm 40056 |
Copyright terms: Public domain | W3C validator |