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| Mirrors > Home > MPE Home > Th. List > 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 12814 | . . . . 5 ⊢ (8 + 3) = ;11 | |
| 2 | 1 | eqcomi 2746 | . . . 4 ⊢ ;11 = (8 + 3) |
| 3 | 2 | oveq2i 7442 | . . 3 ⊢ (2↑;11) = (2↑(8 + 3)) |
| 4 | 2cn 12341 | . . . 4 ⊢ 2 ∈ ℂ | |
| 5 | 8nn0 12549 | . . . 4 ⊢ 8 ∈ ℕ0 | |
| 6 | 3nn0 12544 | . . . 4 ⊢ 3 ∈ ℕ0 | |
| 7 | expadd 14145 | . . . 4 ⊢ ((2 ∈ ℂ ∧ 8 ∈ ℕ0 ∧ 3 ∈ ℕ0) → (2↑(8 + 3)) = ((2↑8) · (2↑3))) | |
| 8 | 4, 5, 6, 7 | mp3an 1463 | . . 3 ⊢ (2↑(8 + 3)) = ((2↑8) · (2↑3)) |
| 9 | 3, 8 | eqtri 2765 | . 2 ⊢ (2↑;11) = ((2↑8) · (2↑3)) |
| 10 | 2exp8 17126 | . . . 4 ⊢ (2↑8) = ;;256 | |
| 11 | cu2 14239 | . . . 4 ⊢ (2↑3) = 8 | |
| 12 | 10, 11 | oveq12i 7443 | . . 3 ⊢ ((2↑8) · (2↑3)) = (;;256 · 8) |
| 13 | 2nn0 12543 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
| 14 | 5nn0 12546 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
| 15 | 13, 14 | deccl 12748 | . . . 4 ⊢ ;25 ∈ ℕ0 |
| 16 | 6nn0 12547 | . . . 4 ⊢ 6 ∈ ℕ0 | |
| 17 | eqid 2737 | . . . 4 ⊢ ;;256 = ;;256 | |
| 18 | 4nn0 12545 | . . . 4 ⊢ 4 ∈ ℕ0 | |
| 19 | 0nn0 12541 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
| 20 | 13, 19 | deccl 12748 | . . . . 5 ⊢ ;20 ∈ ℕ0 |
| 21 | eqid 2737 | . . . . . 6 ⊢ ;25 = ;25 | |
| 22 | 1nn0 12542 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
| 23 | 8cn 12363 | . . . . . . . 8 ⊢ 8 ∈ ℂ | |
| 24 | 8t2e16 12848 | . . . . . . . 8 ⊢ (8 · 2) = ;16 | |
| 25 | 23, 4, 24 | mulcomli 11270 | . . . . . . 7 ⊢ (2 · 8) = ;16 |
| 26 | 1p1e2 12391 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
| 27 | 6p4e10 12805 | . . . . . . 7 ⊢ (6 + 4) = ;10 | |
| 28 | 22, 16, 18, 25, 26, 19, 27 | decaddci 12794 | . . . . . 6 ⊢ ((2 · 8) + 4) = ;20 |
| 29 | 5cn 12354 | . . . . . . 7 ⊢ 5 ∈ ℂ | |
| 30 | 8t5e40 12851 | . . . . . . 7 ⊢ (8 · 5) = ;40 | |
| 31 | 23, 29, 30 | mulcomli 11270 | . . . . . 6 ⊢ (5 · 8) = ;40 |
| 32 | 5, 13, 14, 21, 19, 18, 28, 31 | decmul1c 12798 | . . . . 5 ⊢ (;25 · 8) = ;;200 |
| 33 | 4cn 12351 | . . . . . 6 ⊢ 4 ∈ ℂ | |
| 34 | 33 | addlidi 11449 | . . . . 5 ⊢ (0 + 4) = 4 |
| 35 | 20, 19, 18, 32, 34 | decaddi 12793 | . . . 4 ⊢ ((;25 · 8) + 4) = ;;204 |
| 36 | 6cn 12357 | . . . . 5 ⊢ 6 ∈ ℂ | |
| 37 | 8t6e48 12852 | . . . . 5 ⊢ (8 · 6) = ;48 | |
| 38 | 23, 36, 37 | mulcomli 11270 | . . . 4 ⊢ (6 · 8) = ;48 |
| 39 | 5, 15, 16, 17, 5, 18, 35, 38 | decmul1c 12798 | . . 3 ⊢ (;;256 · 8) = ;;;2048 |
| 40 | 12, 39 | eqtri 2765 | . 2 ⊢ ((2↑8) · (2↑3)) = ;;;2048 |
| 41 | 9, 40 | eqtri 2765 | 1 ⊢ (2↑;11) = ;;;2048 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 2c2 12321 3c3 12322 4c4 12323 5c5 12324 6c6 12325 8c8 12327 ℕ0cn0 12526 ;cdc 12733 ↑cexp 14102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: 3lexlogpow5ineq2 42056 m11nprm 47588 |
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