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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 7970 | . . . 4 ⊢ 1 ∈ V | |
2 | 1 | snid 3638 | . . 3 ⊢ 1 ∈ {1} |
3 | 1ap0 8565 | . . 3 ⊢ 1 # 0 | |
4 | ax-1cn 7922 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | snssi 3751 | . . . . 5 ⊢ (1 ∈ ℂ → {1} ⊆ ℂ) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ {1} ⊆ ℂ |
7 | elsni 3625 | . . . . . 6 ⊢ (𝑥 ∈ {1} → 𝑥 = 1) | |
8 | elsni 3625 | . . . . . 6 ⊢ (𝑦 ∈ {1} → 𝑦 = 1) | |
9 | oveq12 5900 | . . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (1 · 1)) | |
10 | 1t1e1 9089 | . . . . . . 7 ⊢ (1 · 1) = 1 | |
11 | 9, 10 | eqtrdi 2238 | . . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = 1) |
12 | 7, 8, 11 | syl2an 289 | . . . . 5 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) = 1) |
13 | eleq1 2252 | . . . . . . . 8 ⊢ ((𝑥 · 𝑦) = 1 → ((𝑥 · 𝑦) ∈ V ↔ 1 ∈ V)) | |
14 | 1, 13 | mpbiri 168 | . . . . . . 7 ⊢ ((𝑥 · 𝑦) = 1 → (𝑥 · 𝑦) ∈ V) |
15 | elsng 3622 | . . . . . . 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 177 | . . . . 5 ⊢ ((𝑥 · 𝑦) = 1 → (𝑥 · 𝑦) ∈ {1}) |
18 | 12, 17 | syl 14 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) ∈ {1}) |
19 | 7 | oveq2d 5907 | . . . . . . 7 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = (1 / 1)) |
20 | 1div1e1 8679 | . . . . . . 7 ⊢ (1 / 1) = 1 | |
21 | 19, 20 | eqtrdi 2238 | . . . . . 6 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = 1) |
22 | eleq1 2252 | . . . . . . . . 9 ⊢ ((1 / 𝑥) = 1 → ((1 / 𝑥) ∈ V ↔ 1 ∈ V)) | |
23 | 1, 22 | mpbiri 168 | . . . . . . . 8 ⊢ ((1 / 𝑥) = 1 → (1 / 𝑥) ∈ V) |
24 | elsng 3622 | . . . . . . . 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 177 | . . . . . 6 ⊢ ((1 / 𝑥) = 1 → (1 / 𝑥) ∈ {1}) |
27 | 21, 26 | syl 14 | . . . . 5 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) ∈ {1}) |
28 | 27 | adantr 276 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑥 # 0) → (1 / 𝑥) ∈ {1}) |
29 | 6, 18, 2, 28 | expcl2lemap 10550 | . . 3 ⊢ ((1 ∈ {1} ∧ 1 # 0 ∧ 𝑁 ∈ ℤ) → (1↑𝑁) ∈ {1}) |
30 | 2, 3, 29 | mp3an12 1338 | . 2 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) ∈ {1}) |
31 | elsni 3625 | . 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 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 {csn 3607 class class class wbr 4018 (class class class)co 5891 ℂcc 7827 0cc0 7829 1c1 7830 · cmul 7834 # cap 8556 / cdiv 8647 ℤcz 9271 ↑cexp 10537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-inn 8938 df-n0 9195 df-z 9272 df-uz 9547 df-seqfrec 10464 df-exp 10538 |
This theorem is referenced by: exprecap 10579 sq1 10632 iexpcyc 10643 binom1p 11511 binom11 11512 esum 11688 ege2le3 11697 eirraplem 11802 odzdvds 12263 ef2kpi 14624 lgseisenlem1 14847 m1lgs 14849 |
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