Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 1exp | Structured version Visualization version 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 10625 | . . . 4 ⊢ 1 ∈ V | |
2 | 1 | snid 4591 | . . 3 ⊢ 1 ∈ {1} |
3 | ax-1ne0 10594 | . . 3 ⊢ 1 ≠ 0 | |
4 | ax-1cn 10583 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | snssi 4733 | . . . . 5 ⊢ (1 ∈ ℂ → {1} ⊆ ℂ) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ {1} ⊆ ℂ |
7 | elsni 4574 | . . . . . 6 ⊢ (𝑥 ∈ {1} → 𝑥 = 1) | |
8 | elsni 4574 | . . . . . 6 ⊢ (𝑦 ∈ {1} → 𝑦 = 1) | |
9 | oveq12 7154 | . . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (1 · 1)) | |
10 | 1t1e1 11787 | . . . . . . 7 ⊢ (1 · 1) = 1 | |
11 | 9, 10 | syl6eq 2869 | . . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = 1) |
12 | 7, 8, 11 | syl2an 595 | . . . . 5 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) = 1) |
13 | ovex 7178 | . . . . . 6 ⊢ (𝑥 · 𝑦) ∈ V | |
14 | 13 | elsn 4572 | . . . . 5 ⊢ ((𝑥 · 𝑦) ∈ {1} ↔ (𝑥 · 𝑦) = 1) |
15 | 12, 14 | sylibr 235 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) ∈ {1}) |
16 | 7 | oveq2d 7161 | . . . . . . 7 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = (1 / 1)) |
17 | 1div1e1 11318 | . . . . . . 7 ⊢ (1 / 1) = 1 | |
18 | 16, 17 | syl6eq 2869 | . . . . . 6 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = 1) |
19 | ovex 7178 | . . . . . . 7 ⊢ (1 / 𝑥) ∈ V | |
20 | 19 | elsn 4572 | . . . . . 6 ⊢ ((1 / 𝑥) ∈ {1} ↔ (1 / 𝑥) = 1) |
21 | 18, 20 | sylibr 235 | . . . . 5 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) ∈ {1}) |
22 | 21 | adantr 481 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ {1}) |
23 | 6, 15, 2, 22 | expcl2lem 13429 | . . 3 ⊢ ((1 ∈ {1} ∧ 1 ≠ 0 ∧ 𝑁 ∈ ℤ) → (1↑𝑁) ∈ {1}) |
24 | 2, 3, 23 | mp3an12 1442 | . 2 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) ∈ {1}) |
25 | elsni 4574 | . 2 ⊢ ((1↑𝑁) ∈ {1} → (1↑𝑁) = 1) | |
26 | 24, 25 | syl 17 | 1 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ⊆ wss 3933 {csn 4557 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 · cmul 10530 / cdiv 11285 ℤcz 11969 ↑cexp 13417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13358 df-exp 13418 |
This theorem is referenced by: exprec 13458 sq1 13546 iexpcyc 13557 faclbnd4lem1 13641 iseraltlem2 15027 iseraltlem3 15028 binom1p 15174 binom11 15175 pwm1geoser 15212 pwm1geoserOLD 15213 esum 15422 ege2le3 15431 eirrlem 15545 odzdvds 16120 iblabsr 24357 iblmulc2 24358 abelthlem1 24946 abelthlem3 24948 abelthlem8 24954 abelthlem9 24955 ef2kpi 24991 root1cj 25264 cxpeq 25265 quart 25366 leibpi 25447 log2cnv 25449 mule1 25652 lgseisenlem1 25878 lgseisenlem4 25881 lgseisen 25882 lgsquadlem1 25883 lgsquad2lem1 25887 m1lgs 25891 dchrisum0flblem1 26011 subfaclim 32332 iblmulc2nc 34838 nn0rppwr 39060 numdenexp 39064 zrtelqelz 39070 expdioph 39498 lhe4.4ex1a 40538 fprodexp 41751 stoweidlem7 42169 stirlinglem5 42240 stirlinglem7 42242 stirlinglem10 42245 2pwp1prm 43628 m1expevenALTV 43689 4fppr1 43777 efmnd1hash 43989 altgsumbc 44328 |
Copyright terms: Public domain | W3C validator |