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Mirrors > Home > MPE Home > Th. List > Mathboxes > m1expevenALTV | Structured version Visualization version GIF version |
Description: Exponentiation of -1 by an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 6-Jul-2020.) |
Ref | Expression |
---|---|
m1expevenALTV | ⊢ (𝑁 ∈ Even → (-1↑𝑁) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2614 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 = (2 · 𝑖) ↔ 𝑁 = (2 · 𝑖))) | |
2 | 1 | rexbidv 3034 | . . 3 ⊢ (𝑛 = 𝑁 → (∃𝑖 ∈ ℤ 𝑛 = (2 · 𝑖) ↔ ∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖))) |
3 | dfeven4 40089 | . . 3 ⊢ Even = {𝑛 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑛 = (2 · 𝑖)} | |
4 | 2, 3 | elrab2 3333 | . 2 ⊢ (𝑁 ∈ Even ↔ (𝑁 ∈ ℤ ∧ ∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖))) |
5 | oveq2 6557 | . . . . . 6 ⊢ (𝑁 = (2 · 𝑖) → (-1↑𝑁) = (-1↑(2 · 𝑖))) | |
6 | neg1cn 11001 | . . . . . . . . . 10 ⊢ -1 ∈ ℂ | |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ (𝑖 ∈ ℤ → -1 ∈ ℂ) |
8 | neg1ne0 11003 | . . . . . . . . . 10 ⊢ -1 ≠ 0 | |
9 | 8 | a1i 11 | . . . . . . . . 9 ⊢ (𝑖 ∈ ℤ → -1 ≠ 0) |
10 | 2z 11286 | . . . . . . . . . 10 ⊢ 2 ∈ ℤ | |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝑖 ∈ ℤ → 2 ∈ ℤ) |
12 | id 22 | . . . . . . . . 9 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℤ) | |
13 | expmulz 12768 | . . . . . . . . 9 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (2 ∈ ℤ ∧ 𝑖 ∈ ℤ)) → (-1↑(2 · 𝑖)) = ((-1↑2)↑𝑖)) | |
14 | 7, 9, 11, 12, 13 | syl22anc 1319 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → (-1↑(2 · 𝑖)) = ((-1↑2)↑𝑖)) |
15 | neg1sqe1 12821 | . . . . . . . . . 10 ⊢ (-1↑2) = 1 | |
16 | 15 | oveq1i 6559 | . . . . . . . . 9 ⊢ ((-1↑2)↑𝑖) = (1↑𝑖) |
17 | 1exp 12751 | . . . . . . . . 9 ⊢ (𝑖 ∈ ℤ → (1↑𝑖) = 1) | |
18 | 16, 17 | syl5eq 2656 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → ((-1↑2)↑𝑖) = 1) |
19 | 14, 18 | eqtrd 2644 | . . . . . . 7 ⊢ (𝑖 ∈ ℤ → (-1↑(2 · 𝑖)) = 1) |
20 | 19 | adantl 481 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (-1↑(2 · 𝑖)) = 1) |
21 | 5, 20 | sylan9eqr 2666 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ 𝑁 = (2 · 𝑖)) → (-1↑𝑁) = 1) |
22 | 21 | ex 449 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑁 = (2 · 𝑖) → (-1↑𝑁) = 1)) |
23 | 22 | rexlimdva 3013 | . . 3 ⊢ (𝑁 ∈ ℤ → (∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖) → (-1↑𝑁) = 1)) |
24 | 23 | imp 444 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖)) → (-1↑𝑁) = 1) |
25 | 4, 24 | sylbi 206 | 1 ⊢ (𝑁 ∈ Even → (-1↑𝑁) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 -cneg 10146 2c2 10947 ℤcz 11254 ↑cexp 12722 Even ceven 40075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-exp 12723 df-even 40077 |
This theorem is referenced by: m1expoddALTV 40099 |
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