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Mirrors > Home > ILE Home > Th. List > expp1zap | GIF version |
Description: Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.) |
Ref | Expression |
---|---|
expp1zap | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 8535 | . . . 4 ⊢ 1 ∈ ℤ | |
2 | expaddzap 9694 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑁 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · (𝐴↑1))) | |
3 | 1, 2 | mpanr2 429 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · (𝐴↑1))) |
4 | 3 | 3impa 1134 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · (𝐴↑1))) |
5 | exp1 9656 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
6 | 5 | 3ad2ant1 960 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑1) = 𝐴) |
7 | 6 | oveq2d 5581 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → ((𝐴↑𝑁) · (𝐴↑1)) = ((𝐴↑𝑁) · 𝐴)) |
8 | 4, 7 | eqtrd 2115 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 class class class wbr 3806 (class class class)co 5565 ℂcc 7118 0cc0 7120 1c1 7121 + caddc 7123 · cmul 7125 # cap 7825 ℤcz 8509 ↑cexp 9649 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3914 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-iinf 4358 ax-cnex 7206 ax-resscn 7207 ax-1cn 7208 ax-1re 7209 ax-icn 7210 ax-addcl 7211 ax-addrcl 7212 ax-mulcl 7213 ax-mulrcl 7214 ax-addcom 7215 ax-mulcom 7216 ax-addass 7217 ax-mulass 7218 ax-distr 7219 ax-i2m1 7220 ax-0lt1 7221 ax-1rid 7222 ax-0id 7223 ax-rnegex 7224 ax-precex 7225 ax-cnre 7226 ax-pre-ltirr 7227 ax-pre-ltwlin 7228 ax-pre-lttrn 7229 ax-pre-apti 7230 ax-pre-ltadd 7231 ax-pre-mulgt0 7232 ax-pre-mulext 7233 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2613 df-sbc 2826 df-csb 2919 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-if 3370 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-int 3658 df-iun 3701 df-br 3807 df-opab 3861 df-mpt 3862 df-tr 3897 df-id 4077 df-po 4080 df-iso 4081 df-iord 4150 df-on 4152 df-ilim 4153 df-suc 4155 df-iom 4361 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-f1 4958 df-fo 4959 df-f1o 4960 df-fv 4961 df-riota 5521 df-ov 5568 df-oprab 5569 df-mpt2 5570 df-1st 5820 df-2nd 5821 df-recs 5976 df-frec 6062 df-pnf 7294 df-mnf 7295 df-xr 7296 df-ltxr 7297 df-le 7298 df-sub 7425 df-neg 7426 df-reap 7819 df-ap 7826 df-div 7905 df-inn 8184 df-n0 8433 df-z 8510 df-uz 8778 df-iseq 9599 df-iexp 9650 |
This theorem is referenced by: expp1zapd 9788 |
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