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 10035 | . . . 4 ⊢ 1 ∈ V | |
2 | 1 | snid 4208 | . . 3 ⊢ 1 ∈ {1} |
3 | ax-1ne0 10005 | . . 3 ⊢ 1 ≠ 0 | |
4 | ax-1cn 9994 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | snssi 4339 | . . . . 5 ⊢ (1 ∈ ℂ → {1} ⊆ ℂ) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ {1} ⊆ ℂ |
7 | elsni 4194 | . . . . . 6 ⊢ (𝑥 ∈ {1} → 𝑥 = 1) | |
8 | elsni 4194 | . . . . . 6 ⊢ (𝑦 ∈ {1} → 𝑦 = 1) | |
9 | oveq12 6659 | . . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (1 · 1)) | |
10 | 1t1e1 11175 | . . . . . . 7 ⊢ (1 · 1) = 1 | |
11 | 9, 10 | syl6eq 2672 | . . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = 1) |
12 | 7, 8, 11 | syl2an 494 | . . . . 5 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) = 1) |
13 | ovex 6678 | . . . . . 6 ⊢ (𝑥 · 𝑦) ∈ V | |
14 | 13 | elsn 4192 | . . . . 5 ⊢ ((𝑥 · 𝑦) ∈ {1} ↔ (𝑥 · 𝑦) = 1) |
15 | 12, 14 | sylibr 224 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) ∈ {1}) |
16 | 7 | oveq2d 6666 | . . . . . . 7 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = (1 / 1)) |
17 | 1div1e1 10717 | . . . . . . 7 ⊢ (1 / 1) = 1 | |
18 | 16, 17 | syl6eq 2672 | . . . . . 6 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = 1) |
19 | ovex 6678 | . . . . . . 7 ⊢ (1 / 𝑥) ∈ V | |
20 | 19 | elsn 4192 | . . . . . 6 ⊢ ((1 / 𝑥) ∈ {1} ↔ (1 / 𝑥) = 1) |
21 | 18, 20 | sylibr 224 | . . . . 5 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) ∈ {1}) |
22 | 21 | adantr 481 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ {1}) |
23 | 6, 15, 2, 22 | expcl2lem 12872 | . . 3 ⊢ ((1 ∈ {1} ∧ 1 ≠ 0 ∧ 𝑁 ∈ ℤ) → (1↑𝑁) ∈ {1}) |
24 | 2, 3, 23 | mp3an12 1414 | . 2 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) ∈ {1}) |
25 | elsni 4194 | . 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 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ⊆ wss 3574 {csn 4177 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 · cmul 9941 / cdiv 10684 ℤcz 11377 ↑cexp 12860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861 |
This theorem is referenced by: exprec 12901 sq1 12958 iexpcyc 12969 faclbnd4lem1 13080 iseraltlem2 14413 iseraltlem3 14414 binom1p 14563 binom11 14564 pwm1geoser 14600 esum 14811 ege2le3 14820 eirrlem 14932 odzdvds 15500 iblabsr 23596 iblmulc2 23597 abelthlem1 24185 abelthlem3 24187 abelthlem8 24193 abelthlem9 24194 ef2kpi 24230 root1cj 24497 cxpeq 24498 quart 24588 leibpi 24669 log2cnv 24671 mule1 24874 lgseisenlem1 25100 lgseisenlem4 25103 lgseisen 25104 lgsquadlem1 25105 lgsquad2lem1 25109 m1lgs 25113 dchrisum0flblem1 25197 subfaclim 31170 iblmulc2nc 33475 expdioph 37590 lhe4.4ex1a 38528 fprodexp 39826 stoweidlem7 40224 stirlinglem5 40295 stirlinglem7 40297 stirlinglem10 40300 pwm1geoserALT 41502 2pwp1prm 41503 m1expevenALTV 41560 altgsumbc 42130 |
Copyright terms: Public domain | W3C validator |