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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 11704 | . . . . 5 ⊢ 2 ∈ ℂ | |
2 | 2nn0 11906 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
3 | decexp2.1 | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 | |
4 | 2, 3 | nn0expcli 13447 | . . . . . 6 ⊢ (2↑𝑀) ∈ ℕ0 |
5 | 4 | nn0cni 11901 | . . . . 5 ⊢ (2↑𝑀) ∈ ℂ |
6 | 1, 5 | mulcli 10640 | . . . 4 ⊢ (2 · (2↑𝑀)) ∈ ℂ |
7 | expp1 13428 | . . . . . . 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 10641 | . . . . . 6 ⊢ ((2↑𝑀) · 2) = (2 · (2↑𝑀)) |
10 | 8, 9 | eqtr2i 2843 | . . . . 5 ⊢ (2 · (2↑𝑀)) = (2↑(𝑀 + 1)) |
11 | 10 | oveq1i 7158 | . . . 4 ⊢ ((2 · (2↑𝑀)) · 2) = ((2↑(𝑀 + 1)) · 2) |
12 | 6, 1, 11 | mulcomli 10642 | . . 3 ⊢ (2 · (2 · (2↑𝑀))) = ((2↑(𝑀 + 1)) · 2) |
13 | 4 | decbin0 12230 | . . 3 ⊢ (4 · (2↑𝑀)) = (2 · (2 · (2↑𝑀))) |
14 | peano2nn0 11929 | . . . . 5 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0) | |
15 | 3, 14 | ax-mp 5 | . . . 4 ⊢ (𝑀 + 1) ∈ ℕ0 |
16 | expp1 13428 | . . . 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 2852 | . 2 ⊢ (4 · (2↑𝑀)) = (2↑((𝑀 + 1) + 1)) |
19 | 4nn0 11908 | . . . . 5 ⊢ 4 ∈ ℕ0 | |
20 | 19, 4 | nn0mulcli 11927 | . . . 4 ⊢ (4 · (2↑𝑀)) ∈ ℕ0 |
21 | 20 | nn0cni 11901 | . . 3 ⊢ (4 · (2↑𝑀)) ∈ ℂ |
22 | 21 | addid1i 10819 | . 2 ⊢ ((4 · (2↑𝑀)) + 0) = (4 · (2↑𝑀)) |
23 | 3 | nn0cni 11901 | . . . . 5 ⊢ 𝑀 ∈ ℂ |
24 | ax-1cn 10587 | . . . . 5 ⊢ 1 ∈ ℂ | |
25 | 23, 24, 24 | addassi 10643 | . . . 4 ⊢ ((𝑀 + 1) + 1) = (𝑀 + (1 + 1)) |
26 | df-2 11692 | . . . . 5 ⊢ 2 = (1 + 1) | |
27 | 26 | oveq2i 7159 | . . . 4 ⊢ (𝑀 + 2) = (𝑀 + (1 + 1)) |
28 | decexp2.2 | . . . 4 ⊢ (𝑀 + 2) = 𝑁 | |
29 | 25, 27, 28 | 3eqtr2ri 2849 | . . 3 ⊢ 𝑁 = ((𝑀 + 1) + 1) |
30 | 29 | oveq2i 7159 | . 2 ⊢ (2↑𝑁) = (2↑((𝑀 + 1) + 1)) |
31 | 18, 22, 30 | 3eqtr4i 2852 | 1 ⊢ ((4 · (2↑𝑀)) + 0) = (2↑𝑁) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1531 ∈ wcel 2108 (class class class)co 7148 ℂcc 10527 0cc0 10529 1c1 10530 + caddc 10532 · cmul 10534 2c2 11684 4c4 11686 ℕ0cn0 11889 ↑cexp 13421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-n0 11890 df-z 11974 df-uz 12236 df-seq 13362 df-exp 13422 |
This theorem is referenced by: (None) |
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