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Mirrors > Home > MPE Home > Th. List > expnegz | Structured version Visualization version GIF version |
Description: Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.) |
Ref | Expression |
---|---|
expnegz | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elznn0 11806 | . . 3 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) | |
2 | expneg 13250 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁))) | |
3 | 2 | ex 405 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑁 ∈ ℕ0 → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))) |
4 | 3 | ad2antrr 714 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℝ) → (𝑁 ∈ ℕ0 → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))) |
5 | simpll 755 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → 𝐴 ∈ ℂ) | |
6 | simprl 759 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → 𝑁 ∈ ℝ) | |
7 | 6 | recnd 10466 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → 𝑁 ∈ ℂ) |
8 | simprr 761 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → -𝑁 ∈ ℕ0) | |
9 | expneg2 13251 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁))) | |
10 | 5, 7, 8, 9 | syl3anc 1352 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁))) |
11 | 10 | oveq2d 6990 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → (1 / (𝐴↑𝑁)) = (1 / (1 / (𝐴↑-𝑁)))) |
12 | expcl 13260 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -𝑁 ∈ ℕ0) → (𝐴↑-𝑁) ∈ ℂ) | |
13 | 12 | ad2ant2rl 737 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → (𝐴↑-𝑁) ∈ ℂ) |
14 | simplr 757 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → 𝐴 ≠ 0) | |
15 | 8 | nn0zd 11896 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → -𝑁 ∈ ℤ) |
16 | expne0i 13274 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ -𝑁 ∈ ℤ) → (𝐴↑-𝑁) ≠ 0) | |
17 | 5, 14, 15, 16 | syl3anc 1352 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → (𝐴↑-𝑁) ≠ 0) |
18 | 13, 17 | recrecd 11212 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → (1 / (1 / (𝐴↑-𝑁))) = (𝐴↑-𝑁)) |
19 | 11, 18 | eqtr2d 2808 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ0)) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁))) |
20 | 19 | expr 449 | . . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℝ) → (-𝑁 ∈ ℕ0 → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))) |
21 | 4, 20 | jaod 846 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℝ) → ((𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))) |
22 | 21 | expimpd 446 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0)) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))) |
23 | 1, 22 | syl5bi 234 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑁 ∈ ℤ → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))) |
24 | 23 | 3impia 1098 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∨ wo 834 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 (class class class)co 6974 ℂcc 10331 ℝcr 10332 0cc0 10333 1c1 10334 -cneg 10669 / cdiv 11096 ℕ0cn0 11705 ℤcz 11791 ↑cexp 13242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-n0 11706 df-z 11792 df-uz 12057 df-seq 13183 df-exp 13243 |
This theorem is referenced by: expsub 13290 expnegd 13330 |
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