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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 7228 | . . . 4 ⊢ 1 ∈ V | |
2 | 1 | snid 3443 | . . 3 ⊢ 1 ∈ {1} |
3 | 1ap0 7809 | . . 3 ⊢ 1 # 0 | |
4 | ax-1cn 7183 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | snssi 3549 | . . . . 5 ⊢ (1 ∈ ℂ → {1} ⊆ ℂ) | |
6 | 4, 5 | ax-mp 7 | . . . 4 ⊢ {1} ⊆ ℂ |
7 | elsni 3434 | . . . . . 6 ⊢ (𝑥 ∈ {1} → 𝑥 = 1) | |
8 | elsni 3434 | . . . . . 6 ⊢ (𝑦 ∈ {1} → 𝑦 = 1) | |
9 | oveq12 5572 | . . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (1 · 1)) | |
10 | 1t1e1 8303 | . . . . . . 7 ⊢ (1 · 1) = 1 | |
11 | 9, 10 | syl6eq 2131 | . . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = 1) |
12 | 7, 8, 11 | syl2an 283 | . . . . 5 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) = 1) |
13 | eleq1 2145 | . . . . . . . 8 ⊢ ((𝑥 · 𝑦) = 1 → ((𝑥 · 𝑦) ∈ V ↔ 1 ∈ V)) | |
14 | 1, 13 | mpbiri 166 | . . . . . . 7 ⊢ ((𝑥 · 𝑦) = 1 → (𝑥 · 𝑦) ∈ V) |
15 | elsng 3431 | . . . . . . 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 175 | . . . . 5 ⊢ ((𝑥 · 𝑦) = 1 → (𝑥 · 𝑦) ∈ {1}) |
18 | 12, 17 | syl 14 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) ∈ {1}) |
19 | 7 | oveq2d 5579 | . . . . . . 7 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = (1 / 1)) |
20 | 1div1e1 7911 | . . . . . . 7 ⊢ (1 / 1) = 1 | |
21 | 19, 20 | syl6eq 2131 | . . . . . 6 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = 1) |
22 | eleq1 2145 | . . . . . . . . 9 ⊢ ((1 / 𝑥) = 1 → ((1 / 𝑥) ∈ V ↔ 1 ∈ V)) | |
23 | 1, 22 | mpbiri 166 | . . . . . . . 8 ⊢ ((1 / 𝑥) = 1 → (1 / 𝑥) ∈ V) |
24 | elsng 3431 | . . . . . . . 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 175 | . . . . . 6 ⊢ ((1 / 𝑥) = 1 → (1 / 𝑥) ∈ {1}) |
27 | 21, 26 | syl 14 | . . . . 5 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) ∈ {1}) |
28 | 27 | adantr 270 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑥 # 0) → (1 / 𝑥) ∈ {1}) |
29 | 6, 18, 2, 28 | expcl2lemap 9637 | . . 3 ⊢ ((1 ∈ {1} ∧ 1 # 0 ∧ 𝑁 ∈ ℤ) → (1↑𝑁) ∈ {1}) |
30 | 2, 3, 29 | mp3an12 1259 | . 2 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) ∈ {1}) |
31 | elsni 3434 | . 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 102 ↔ wb 103 = wceq 1285 ∈ wcel 1434 Vcvv 2610 ⊆ wss 2982 {csn 3416 class class class wbr 3805 (class class class)co 5563 ℂcc 7093 0cc0 7095 1c1 7096 · cmul 7100 # cap 7800 / cdiv 7879 ℤcz 8484 ↑cexp 9624 |
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 3913 ax-sep 3916 ax-nul 3924 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-iinf 4357 ax-cnex 7181 ax-resscn 7182 ax-1cn 7183 ax-1re 7184 ax-icn 7185 ax-addcl 7186 ax-addrcl 7187 ax-mulcl 7188 ax-mulrcl 7189 ax-addcom 7190 ax-mulcom 7191 ax-addass 7192 ax-mulass 7193 ax-distr 7194 ax-i2m1 7195 ax-0lt1 7196 ax-1rid 7197 ax-0id 7198 ax-rnegex 7199 ax-precex 7200 ax-cnre 7201 ax-pre-ltirr 7202 ax-pre-ltwlin 7203 ax-pre-lttrn 7204 ax-pre-apti 7205 ax-pre-ltadd 7206 ax-pre-mulgt0 7207 ax-pre-mulext 7208 |
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 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-nul 3268 df-if 3369 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-tr 3896 df-id 4076 df-po 4079 df-iso 4080 df-iord 4149 df-on 4151 df-ilim 4152 df-suc 4154 df-iom 4360 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-f1 4957 df-fo 4958 df-f1o 4959 df-fv 4960 df-riota 5519 df-ov 5566 df-oprab 5567 df-mpt2 5568 df-1st 5818 df-2nd 5819 df-recs 5974 df-frec 6060 df-pnf 7269 df-mnf 7270 df-xr 7271 df-ltxr 7272 df-le 7273 df-sub 7400 df-neg 7401 df-reap 7794 df-ap 7801 df-div 7880 df-inn 8159 df-n0 8408 df-z 8485 df-uz 8753 df-iseq 9574 df-iexp 9625 |
This theorem is referenced by: exprecap 9666 sq1 9718 iexpcyc 9728 |
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