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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 12182 | . . . . 5 ⊢ (8 + 3) = ;11 | |
2 | 1 | eqcomi 2832 | . . . 4 ⊢ ;11 = (8 + 3) |
3 | 2 | oveq2i 7169 | . . 3 ⊢ (2↑;11) = (2↑(8 + 3)) |
4 | 2cn 11715 | . . . 4 ⊢ 2 ∈ ℂ | |
5 | 8nn0 11923 | . . . 4 ⊢ 8 ∈ ℕ0 | |
6 | 3nn0 11918 | . . . 4 ⊢ 3 ∈ ℕ0 | |
7 | expadd 13474 | . . . 4 ⊢ ((2 ∈ ℂ ∧ 8 ∈ ℕ0 ∧ 3 ∈ ℕ0) → (2↑(8 + 3)) = ((2↑8) · (2↑3))) | |
8 | 4, 5, 6, 7 | mp3an 1457 | . . 3 ⊢ (2↑(8 + 3)) = ((2↑8) · (2↑3)) |
9 | 3, 8 | eqtri 2846 | . 2 ⊢ (2↑;11) = ((2↑8) · (2↑3)) |
10 | 2exp8 16425 | . . . 4 ⊢ (2↑8) = ;;256 | |
11 | cu2 13566 | . . . 4 ⊢ (2↑3) = 8 | |
12 | 10, 11 | oveq12i 7170 | . . 3 ⊢ ((2↑8) · (2↑3)) = (;;256 · 8) |
13 | 2nn0 11917 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
14 | 5nn0 11920 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
15 | 13, 14 | deccl 12116 | . . . 4 ⊢ ;25 ∈ ℕ0 |
16 | 6nn0 11921 | . . . 4 ⊢ 6 ∈ ℕ0 | |
17 | eqid 2823 | . . . 4 ⊢ ;;256 = ;;256 | |
18 | 4nn0 11919 | . . . 4 ⊢ 4 ∈ ℕ0 | |
19 | 0nn0 11915 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
20 | 13, 19 | deccl 12116 | . . . . 5 ⊢ ;20 ∈ ℕ0 |
21 | eqid 2823 | . . . . . 6 ⊢ ;25 = ;25 | |
22 | 1nn0 11916 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
23 | 8cn 11737 | . . . . . . . 8 ⊢ 8 ∈ ℂ | |
24 | 8t2e16 12216 | . . . . . . . 8 ⊢ (8 · 2) = ;16 | |
25 | 23, 4, 24 | mulcomli 10652 | . . . . . . 7 ⊢ (2 · 8) = ;16 |
26 | 1p1e2 11765 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
27 | 6p4e10 12173 | . . . . . . 7 ⊢ (6 + 4) = ;10 | |
28 | 22, 16, 18, 25, 26, 19, 27 | decaddci 12162 | . . . . . 6 ⊢ ((2 · 8) + 4) = ;20 |
29 | 5cn 11728 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
30 | 8t5e40 12219 | . . . . . . 7 ⊢ (8 · 5) = ;40 | |
31 | 23, 29, 30 | mulcomli 10652 | . . . . . 6 ⊢ (5 · 8) = ;40 |
32 | 5, 13, 14, 21, 19, 18, 28, 31 | decmul1c 12166 | . . . . 5 ⊢ (;25 · 8) = ;;200 |
33 | 4cn 11725 | . . . . . 6 ⊢ 4 ∈ ℂ | |
34 | 33 | addid2i 10830 | . . . . 5 ⊢ (0 + 4) = 4 |
35 | 20, 19, 18, 32, 34 | decaddi 12161 | . . . 4 ⊢ ((;25 · 8) + 4) = ;;204 |
36 | 6cn 11731 | . . . . 5 ⊢ 6 ∈ ℂ | |
37 | 8t6e48 12220 | . . . . 5 ⊢ (8 · 6) = ;48 | |
38 | 23, 36, 37 | mulcomli 10652 | . . . 4 ⊢ (6 · 8) = ;48 |
39 | 5, 15, 16, 17, 5, 18, 35, 38 | decmul1c 12166 | . . 3 ⊢ (;;256 · 8) = ;;;2048 |
40 | 12, 39 | eqtri 2846 | . 2 ⊢ ((2↑8) · (2↑3)) = ;;;2048 |
41 | 9, 40 | eqtri 2846 | 1 ⊢ (2↑;11) = ;;;2048 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 2c2 11695 3c3 11696 4c4 11697 5c5 11698 6c6 11699 8c8 11701 ℕ0cn0 11900 ;cdc 12101 ↑cexp 13432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-seq 13373 df-exp 13433 |
This theorem is referenced by: m11nprm 43773 |
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