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Mirrors > Home > MPE Home > Th. List > Mathboxes > negexpidd | Structured version Visualization version GIF version |
Description: The sum of a real number to the power of N and the negative of the number to the power of N equals zero if N is a nonnegative odd integer. (Contributed by Igor Ieskov, 21-Jan-2024.) |
Ref | Expression |
---|---|
negexpidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
negexpidd.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
negexpidd.3 | ⊢ (𝜑 → ¬ 2 ∥ 𝑁) |
Ref | Expression |
---|---|
negexpidd | ⊢ (𝜑 → ((𝐴↑𝑁) + (-𝐴↑𝑁)) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negexpidd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | negexpidd.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
3 | 1, 2 | reexpcld 13931 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ) |
4 | 3 | recnd 11053 | . . 3 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
5 | 4 | negidd 11372 | . 2 ⊢ (𝜑 → ((𝐴↑𝑁) + -(𝐴↑𝑁)) = 0) |
6 | 1 | recnd 11053 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
7 | 6 | mulm1d 11477 | . . . . . . 7 ⊢ (𝜑 → (-1 · 𝐴) = -𝐴) |
8 | 7 | eqcomd 2742 | . . . . . 6 ⊢ (𝜑 → -𝐴 = (-1 · 𝐴)) |
9 | 8 | oveq1d 7322 | . . . . 5 ⊢ (𝜑 → (-𝐴↑𝑁) = ((-1 · 𝐴)↑𝑁)) |
10 | nn0z 12393 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
11 | 10 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)) |
12 | negexpidd.3 | . . . . . . . . . . 11 ⊢ (𝜑 → ¬ 2 ∥ 𝑁) | |
13 | 11, 12 | jctird 528 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∈ ℕ0 → (𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁))) |
14 | 2, 13 | mpd 15 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁)) |
15 | m1expo 16133 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (-1↑𝑁) = -1) | |
16 | 15 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → (-1↑𝑁) = -1)) |
17 | 14, 16 | mpd 15 | . . . . . . . 8 ⊢ (𝜑 → (-1↑𝑁) = -1) |
18 | 17 | oveq1d 7322 | . . . . . . 7 ⊢ (𝜑 → ((-1↑𝑁) · (𝐴↑𝑁)) = (-1 · (𝐴↑𝑁))) |
19 | 4 | mulm1d 11477 | . . . . . . 7 ⊢ (𝜑 → (-1 · (𝐴↑𝑁)) = -(𝐴↑𝑁)) |
20 | 18, 19 | eqtr2d 2777 | . . . . . 6 ⊢ (𝜑 → -(𝐴↑𝑁) = ((-1↑𝑁) · (𝐴↑𝑁))) |
21 | neg1cn 12137 | . . . . . . . 8 ⊢ -1 ∈ ℂ | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → -1 ∈ ℂ) |
23 | 22, 6, 2 | mulexpd 13929 | . . . . . 6 ⊢ (𝜑 → ((-1 · 𝐴)↑𝑁) = ((-1↑𝑁) · (𝐴↑𝑁))) |
24 | 20, 23 | eqtr4d 2779 | . . . . 5 ⊢ (𝜑 → -(𝐴↑𝑁) = ((-1 · 𝐴)↑𝑁)) |
25 | 9, 24 | eqtr4d 2779 | . . . 4 ⊢ (𝜑 → (-𝐴↑𝑁) = -(𝐴↑𝑁)) |
26 | 25 | oveq2d 7323 | . . 3 ⊢ (𝜑 → ((𝐴↑𝑁) + (-𝐴↑𝑁)) = ((𝐴↑𝑁) + -(𝐴↑𝑁))) |
27 | 26 | eqeq1d 2738 | . 2 ⊢ (𝜑 → (((𝐴↑𝑁) + (-𝐴↑𝑁)) = 0 ↔ ((𝐴↑𝑁) + -(𝐴↑𝑁)) = 0)) |
28 | 5, 27 | mpbird 257 | 1 ⊢ (𝜑 → ((𝐴↑𝑁) + (-𝐴↑𝑁)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1539 ∈ wcel 2104 class class class wbr 5081 (class class class)co 7307 ℂcc 10919 ℝcr 10920 0cc0 10921 1c1 10922 + caddc 10924 · cmul 10926 -cneg 11256 2c2 12078 ℕ0cn0 12283 ℤcz 12369 ↑cexp 13832 ∥ cdvds 16012 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10977 ax-resscn 10978 ax-1cn 10979 ax-icn 10980 ax-addcl 10981 ax-addrcl 10982 ax-mulcl 10983 ax-mulrcl 10984 ax-mulcom 10985 ax-addass 10986 ax-mulass 10987 ax-distr 10988 ax-i2m1 10989 ax-1ne0 10990 ax-1rid 10991 ax-rnegex 10992 ax-rrecex 10993 ax-cnre 10994 ax-pre-lttri 10995 ax-pre-lttrn 10996 ax-pre-ltadd 10997 ax-pre-mulgt0 10998 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3304 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-pnf 11061 df-mnf 11062 df-xr 11063 df-ltxr 11064 df-le 11065 df-sub 11257 df-neg 11258 df-div 11683 df-nn 12024 df-2 12086 df-n0 12284 df-z 12370 df-uz 12633 df-seq 13772 df-exp 13833 df-dvds 16013 |
This theorem is referenced by: 3cubeslem3r 40704 |
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