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Mirrors > Home > MPE Home > Th. List > decexp2 | Structured version Visualization version GIF version |
Description: Calculate a power of two. (Contributed by Mario Carneiro, 19-Feb-2014.) |
Ref | Expression |
---|---|
decexp2.1 | ⊢ 𝑀 ∈ ℕ0 |
decexp2.2 | ⊢ (𝑀 + 2) = 𝑁 |
Ref | Expression |
---|---|
decexp2 | ⊢ ((4 · (2↑𝑀)) + 0) = (2↑𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 11713 | . . . . 5 ⊢ 2 ∈ ℂ | |
2 | 2nn0 11915 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
3 | decexp2.1 | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 | |
4 | 2, 3 | nn0expcli 13456 | . . . . . 6 ⊢ (2↑𝑀) ∈ ℕ0 |
5 | 4 | nn0cni 11910 | . . . . 5 ⊢ (2↑𝑀) ∈ ℂ |
6 | 1, 5 | mulcli 10648 | . . . 4 ⊢ (2 · (2↑𝑀)) ∈ ℂ |
7 | expp1 13437 | . . . . . . 7 ⊢ ((2 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → (2↑(𝑀 + 1)) = ((2↑𝑀) · 2)) | |
8 | 1, 3, 7 | mp2an 690 | . . . . . 6 ⊢ (2↑(𝑀 + 1)) = ((2↑𝑀) · 2) |
9 | 5, 1 | mulcomi 10649 | . . . . . 6 ⊢ ((2↑𝑀) · 2) = (2 · (2↑𝑀)) |
10 | 8, 9 | eqtr2i 2845 | . . . . 5 ⊢ (2 · (2↑𝑀)) = (2↑(𝑀 + 1)) |
11 | 10 | oveq1i 7166 | . . . 4 ⊢ ((2 · (2↑𝑀)) · 2) = ((2↑(𝑀 + 1)) · 2) |
12 | 6, 1, 11 | mulcomli 10650 | . . 3 ⊢ (2 · (2 · (2↑𝑀))) = ((2↑(𝑀 + 1)) · 2) |
13 | 4 | decbin0 12239 | . . 3 ⊢ (4 · (2↑𝑀)) = (2 · (2 · (2↑𝑀))) |
14 | peano2nn0 11938 | . . . . 5 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0) | |
15 | 3, 14 | ax-mp 5 | . . . 4 ⊢ (𝑀 + 1) ∈ ℕ0 |
16 | expp1 13437 | . . . 4 ⊢ ((2 ∈ ℂ ∧ (𝑀 + 1) ∈ ℕ0) → (2↑((𝑀 + 1) + 1)) = ((2↑(𝑀 + 1)) · 2)) | |
17 | 1, 15, 16 | mp2an 690 | . . 3 ⊢ (2↑((𝑀 + 1) + 1)) = ((2↑(𝑀 + 1)) · 2) |
18 | 12, 13, 17 | 3eqtr4i 2854 | . 2 ⊢ (4 · (2↑𝑀)) = (2↑((𝑀 + 1) + 1)) |
19 | 4nn0 11917 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
20 | 19, 4 | nn0mulcli 11936 | . . . 4 ⊢ (4 · (2↑𝑀)) ∈ ℕ0 |
21 | 20 | nn0cni 11910 | . . 3 ⊢ (4 · (2↑𝑀)) ∈ ℂ |
22 | 21 | addid1i 10827 | . 2 ⊢ ((4 · (2↑𝑀)) + 0) = (4 · (2↑𝑀)) |
23 | 3 | nn0cni 11910 | . . . . 5 ⊢ 𝑀 ∈ ℂ |
24 | ax-1cn 10595 | . . . . 5 ⊢ 1 ∈ ℂ | |
25 | 23, 24, 24 | addassi 10651 | . . . 4 ⊢ ((𝑀 + 1) + 1) = (𝑀 + (1 + 1)) |
26 | df-2 11701 | . . . . 5 ⊢ 2 = (1 + 1) | |
27 | 26 | oveq2i 7167 | . . . 4 ⊢ (𝑀 + 2) = (𝑀 + (1 + 1)) |
28 | decexp2.2 | . . . 4 ⊢ (𝑀 + 2) = 𝑁 | |
29 | 25, 27, 28 | 3eqtr2ri 2851 | . . 3 ⊢ 𝑁 = ((𝑀 + 1) + 1) |
30 | 29 | oveq2i 7167 | . 2 ⊢ (2↑𝑁) = (2↑((𝑀 + 1) + 1)) |
31 | 18, 22, 30 | 3eqtr4i 2854 | 1 ⊢ ((4 · (2↑𝑀)) + 0) = (2↑𝑁) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 2c2 11693 4c4 11695 ℕ0cn0 11898 ↑cexp 13430 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-exp 13431 |
This theorem is referenced by: (None) |
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