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Mirrors > Home > ILE Home > Th. List > 1exp | GIF version |
Description: Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Ref | Expression |
---|---|
1exp | ⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ex 7679 | . . . 4 ⊢ 1 ∈ V | |
2 | 1 | snid 3520 | . . 3 ⊢ 1 ∈ {1} |
3 | 1ap0 8264 | . . 3 ⊢ 1 # 0 | |
4 | ax-1cn 7632 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | snssi 3628 | . . . . 5 ⊢ (1 ∈ ℂ → {1} ⊆ ℂ) | |
6 | 4, 5 | ax-mp 7 | . . . 4 ⊢ {1} ⊆ ℂ |
7 | elsni 3509 | . . . . . 6 ⊢ (𝑥 ∈ {1} → 𝑥 = 1) | |
8 | elsni 3509 | . . . . . 6 ⊢ (𝑦 ∈ {1} → 𝑦 = 1) | |
9 | oveq12 5735 | . . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (1 · 1)) | |
10 | 1t1e1 8770 | . . . . . . 7 ⊢ (1 · 1) = 1 | |
11 | 9, 10 | syl6eq 2161 | . . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = 1) |
12 | 7, 8, 11 | syl2an 285 | . . . . 5 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) = 1) |
13 | eleq1 2175 | . . . . . . . 8 ⊢ ((𝑥 · 𝑦) = 1 → ((𝑥 · 𝑦) ∈ V ↔ 1 ∈ V)) | |
14 | 1, 13 | mpbiri 167 | . . . . . . 7 ⊢ ((𝑥 · 𝑦) = 1 → (𝑥 · 𝑦) ∈ V) |
15 | elsng 3506 | . . . . . . 7 ⊢ ((𝑥 · 𝑦) ∈ V → ((𝑥 · 𝑦) ∈ {1} ↔ (𝑥 · 𝑦) = 1)) | |
16 | 14, 15 | syl 14 | . . . . . 6 ⊢ ((𝑥 · 𝑦) = 1 → ((𝑥 · 𝑦) ∈ {1} ↔ (𝑥 · 𝑦) = 1)) |
17 | 16 | ibir 176 | . . . . 5 ⊢ ((𝑥 · 𝑦) = 1 → (𝑥 · 𝑦) ∈ {1}) |
18 | 12, 17 | syl 14 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) ∈ {1}) |
19 | 7 | oveq2d 5742 | . . . . . . 7 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = (1 / 1)) |
20 | 1div1e1 8371 | . . . . . . 7 ⊢ (1 / 1) = 1 | |
21 | 19, 20 | syl6eq 2161 | . . . . . 6 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = 1) |
22 | eleq1 2175 | . . . . . . . . 9 ⊢ ((1 / 𝑥) = 1 → ((1 / 𝑥) ∈ V ↔ 1 ∈ V)) | |
23 | 1, 22 | mpbiri 167 | . . . . . . . 8 ⊢ ((1 / 𝑥) = 1 → (1 / 𝑥) ∈ V) |
24 | elsng 3506 | . . . . . . . 8 ⊢ ((1 / 𝑥) ∈ V → ((1 / 𝑥) ∈ {1} ↔ (1 / 𝑥) = 1)) | |
25 | 23, 24 | syl 14 | . . . . . . 7 ⊢ ((1 / 𝑥) = 1 → ((1 / 𝑥) ∈ {1} ↔ (1 / 𝑥) = 1)) |
26 | 25 | ibir 176 | . . . . . 6 ⊢ ((1 / 𝑥) = 1 → (1 / 𝑥) ∈ {1}) |
27 | 21, 26 | syl 14 | . . . . 5 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) ∈ {1}) |
28 | 27 | adantr 272 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑥 # 0) → (1 / 𝑥) ∈ {1}) |
29 | 6, 18, 2, 28 | expcl2lemap 10192 | . . 3 ⊢ ((1 ∈ {1} ∧ 1 # 0 ∧ 𝑁 ∈ ℤ) → (1↑𝑁) ∈ {1}) |
30 | 2, 3, 29 | mp3an12 1286 | . 2 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) ∈ {1}) |
31 | elsni 3509 | . 2 ⊢ ((1↑𝑁) ∈ {1} → (1↑𝑁) = 1) | |
32 | 30, 31 | syl 14 | 1 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1312 ∈ wcel 1461 Vcvv 2655 ⊆ wss 3035 {csn 3491 class class class wbr 3893 (class class class)co 5726 ℂcc 7539 0cc0 7541 1c1 7542 · cmul 7546 # cap 8255 / cdiv 8339 ℤcz 8952 ↑cexp 10179 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-mulrcl 7638 ax-addcom 7639 ax-mulcom 7640 ax-addass 7641 ax-mulass 7642 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-1rid 7646 ax-0id 7647 ax-rnegex 7648 ax-precex 7649 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-apti 7654 ax-pre-ltadd 7655 ax-pre-mulgt0 7656 ax-pre-mulext 7657 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rmo 2396 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-if 3439 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-po 4176 df-iso 4177 df-iord 4246 df-on 4248 df-ilim 4249 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-recs 6154 df-frec 6240 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-reap 8249 df-ap 8256 df-div 8340 df-inn 8625 df-n0 8876 df-z 8953 df-uz 9223 df-seqfrec 10106 df-exp 10180 |
This theorem is referenced by: exprecap 10221 sq1 10273 iexpcyc 10284 binom1p 11140 binom11 11141 esum 11213 ege2le3 11222 eirraplem 11325 |
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