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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 13528 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ) |
4 | 3 | recnd 10669 | . . 3 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
5 | 4 | negidd 10987 | . 2 ⊢ (𝜑 → ((𝐴↑𝑁) + -(𝐴↑𝑁)) = 0) |
6 | 1 | recnd 10669 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
7 | 6 | mulm1d 11092 | . . . . . . 7 ⊢ (𝜑 → (-1 · 𝐴) = -𝐴) |
8 | 7 | eqcomd 2827 | . . . . . 6 ⊢ (𝜑 → -𝐴 = (-1 · 𝐴)) |
9 | 8 | oveq1d 7171 | . . . . 5 ⊢ (𝜑 → (-𝐴↑𝑁) = ((-1 · 𝐴)↑𝑁)) |
10 | nn0z 12006 | . . . . . . . . . . . 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 529 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∈ ℕ0 → (𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁))) |
14 | 2, 13 | mpd 15 | . . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁)) |
15 | m1expo 15726 | . . . . . . . . . 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 7171 | . . . . . . 7 ⊢ (𝜑 → ((-1↑𝑁) · (𝐴↑𝑁)) = (-1 · (𝐴↑𝑁))) |
19 | 4 | mulm1d 11092 | . . . . . . 7 ⊢ (𝜑 → (-1 · (𝐴↑𝑁)) = -(𝐴↑𝑁)) |
20 | 18, 19 | eqtr2d 2857 | . . . . . 6 ⊢ (𝜑 → -(𝐴↑𝑁) = ((-1↑𝑁) · (𝐴↑𝑁))) |
21 | neg1cn 11752 | . . . . . . . 8 ⊢ -1 ∈ ℂ | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → -1 ∈ ℂ) |
23 | 22, 6, 2 | mulexpd 13526 | . . . . . 6 ⊢ (𝜑 → ((-1 · 𝐴)↑𝑁) = ((-1↑𝑁) · (𝐴↑𝑁))) |
24 | 20, 23 | eqtr4d 2859 | . . . . 5 ⊢ (𝜑 → -(𝐴↑𝑁) = ((-1 · 𝐴)↑𝑁)) |
25 | 9, 24 | eqtr4d 2859 | . . . 4 ⊢ (𝜑 → (-𝐴↑𝑁) = -(𝐴↑𝑁)) |
26 | 25 | oveq2d 7172 | . . 3 ⊢ (𝜑 → ((𝐴↑𝑁) + (-𝐴↑𝑁)) = ((𝐴↑𝑁) + -(𝐴↑𝑁))) |
27 | 26 | eqeq1d 2823 | . 2 ⊢ (𝜑 → (((𝐴↑𝑁) + (-𝐴↑𝑁)) = 0 ↔ ((𝐴↑𝑁) + -(𝐴↑𝑁)) = 0)) |
28 | 5, 27 | mpbird 259 | 1 ⊢ (𝜑 → ((𝐴↑𝑁) + (-𝐴↑𝑁)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 -cneg 10871 2c2 11693 ℕ0cn0 11898 ℤcz 11982 ↑cexp 13430 ∥ cdvds 15607 |
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-rmo 3146 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-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-exp 13431 df-dvds 15608 |
This theorem is referenced by: 3cubeslem3r 39333 |
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