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Mirrors > Home > MPE Home > Th. List > numexp2x | Structured version Visualization version GIF version |
Description: Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.) |
Ref | Expression |
---|---|
numexp.1 | ⊢ 𝐴 ∈ ℕ0 |
numexpp1.2 | ⊢ 𝑀 ∈ ℕ0 |
numexp2x.3 | ⊢ (2 · 𝑀) = 𝑁 |
numexp2x.4 | ⊢ (𝐴↑𝑀) = 𝐷 |
numexp2x.5 | ⊢ (𝐷 · 𝐷) = 𝐶 |
Ref | Expression |
---|---|
numexp2x | ⊢ (𝐴↑𝑁) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numexp2x.3 | . . . . 5 ⊢ (2 · 𝑀) = 𝑁 | |
2 | numexpp1.2 | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 | |
3 | 2 | nn0cni 11910 | . . . . . 6 ⊢ 𝑀 ∈ ℂ |
4 | 3 | 2timesi 11776 | . . . . 5 ⊢ (2 · 𝑀) = (𝑀 + 𝑀) |
5 | 1, 4 | eqtr3i 2846 | . . . 4 ⊢ 𝑁 = (𝑀 + 𝑀) |
6 | 5 | oveq2i 7167 | . . 3 ⊢ (𝐴↑𝑁) = (𝐴↑(𝑀 + 𝑀)) |
7 | numexp.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
8 | 7 | nn0cni 11910 | . . . 4 ⊢ 𝐴 ∈ ℂ |
9 | expadd 13472 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑀)) = ((𝐴↑𝑀) · (𝐴↑𝑀))) | |
10 | 8, 2, 2, 9 | mp3an 1457 | . . 3 ⊢ (𝐴↑(𝑀 + 𝑀)) = ((𝐴↑𝑀) · (𝐴↑𝑀)) |
11 | 6, 10 | eqtri 2844 | . 2 ⊢ (𝐴↑𝑁) = ((𝐴↑𝑀) · (𝐴↑𝑀)) |
12 | numexp2x.4 | . . . 4 ⊢ (𝐴↑𝑀) = 𝐷 | |
13 | 12, 12 | oveq12i 7168 | . . 3 ⊢ ((𝐴↑𝑀) · (𝐴↑𝑀)) = (𝐷 · 𝐷) |
14 | numexp2x.5 | . . 3 ⊢ (𝐷 · 𝐷) = 𝐶 | |
15 | 13, 14 | eqtri 2844 | . 2 ⊢ ((𝐴↑𝑀) · (𝐴↑𝑀)) = 𝐶 |
16 | 11, 15 | eqtri 2844 | 1 ⊢ (𝐴↑𝑁) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 + caddc 10540 · cmul 10542 2c2 11693 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-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-exp 13431 |
This theorem is referenced by: 2exp4 16421 2exp6 16422 2exp8 16423 2exp16 16424 1259lem1 16464 log2ub 25527 wallispi2lem2 42377 |
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