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| Mirrors > Home > MPE Home > Th. List > 1exp | Structured version Visualization version GIF version | ||
| Description: Value of 1 raised to an 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 11257 | . . . 4 ⊢ 1 ∈ V | |
| 2 | 1 | snid 4662 | . . 3 ⊢ 1 ∈ {1} |
| 3 | ax-1ne0 11224 | . . 3 ⊢ 1 ≠ 0 | |
| 4 | ax-1cn 11213 | . . . . 5 ⊢ 1 ∈ ℂ | |
| 5 | snssi 4808 | . . . . 5 ⊢ (1 ∈ ℂ → {1} ⊆ ℂ) | |
| 6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ {1} ⊆ ℂ |
| 7 | elsni 4643 | . . . . . 6 ⊢ (𝑥 ∈ {1} → 𝑥 = 1) | |
| 8 | elsni 4643 | . . . . . 6 ⊢ (𝑦 ∈ {1} → 𝑦 = 1) | |
| 9 | oveq12 7440 | . . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (1 · 1)) | |
| 10 | 1t1e1 12428 | . . . . . . 7 ⊢ (1 · 1) = 1 | |
| 11 | 9, 10 | eqtrdi 2793 | . . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = 1) |
| 12 | 7, 8, 11 | syl2an 596 | . . . . 5 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) = 1) |
| 13 | ovex 7464 | . . . . . 6 ⊢ (𝑥 · 𝑦) ∈ V | |
| 14 | 13 | elsn 4641 | . . . . 5 ⊢ ((𝑥 · 𝑦) ∈ {1} ↔ (𝑥 · 𝑦) = 1) |
| 15 | 12, 14 | sylibr 234 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑦 ∈ {1}) → (𝑥 · 𝑦) ∈ {1}) |
| 16 | 7 | oveq2d 7447 | . . . . . . 7 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = (1 / 1)) |
| 17 | 1div1e1 11958 | . . . . . . 7 ⊢ (1 / 1) = 1 | |
| 18 | 16, 17 | eqtrdi 2793 | . . . . . 6 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) = 1) |
| 19 | ovex 7464 | . . . . . . 7 ⊢ (1 / 𝑥) ∈ V | |
| 20 | 19 | elsn 4641 | . . . . . 6 ⊢ ((1 / 𝑥) ∈ {1} ↔ (1 / 𝑥) = 1) |
| 21 | 18, 20 | sylibr 234 | . . . . 5 ⊢ (𝑥 ∈ {1} → (1 / 𝑥) ∈ {1}) |
| 22 | 21 | adantr 480 | . . . 4 ⊢ ((𝑥 ∈ {1} ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ {1}) |
| 23 | 6, 15, 2, 22 | expcl2lem 14114 | . . 3 ⊢ ((1 ∈ {1} ∧ 1 ≠ 0 ∧ 𝑁 ∈ ℤ) → (1↑𝑁) ∈ {1}) |
| 24 | 2, 3, 23 | mp3an12 1453 | . 2 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) ∈ {1}) |
| 25 | elsni 4643 | . 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 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 {csn 4626 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 / cdiv 11920 ℤcz 12613 ↑cexp 14102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: exprec 14144 sq1 14234 iexpcyc 14246 faclbnd4lem1 14332 iseraltlem2 15719 iseraltlem3 15720 binom1p 15867 binom11 15868 pwm1geoser 15905 esum 16116 ege2le3 16126 eirrlem 16240 nn0rppwr 16598 numdenexp 16797 odzdvds 16833 efmnd1hash 18905 iblabsr 25865 iblmulc2 25866 abelthlem1 26475 abelthlem3 26477 abelthlem8 26483 abelthlem9 26484 ef2kpi 26520 root1cj 26799 cxpeq 26800 zrtelqelz 26801 quart 26904 leibpi 26985 log2cnv 26987 mule1 27191 lgseisenlem1 27419 lgseisenlem4 27422 lgseisen 27423 lgsquadlem1 27424 lgsquad2lem1 27428 m1lgs 27432 dchrisum0flblem1 27552 subfaclim 35193 iblmulc2nc 37692 lcmineqlem1 42030 lcmineqlem3 42032 lcmineqlem12 42041 aks4d1p1p2 42071 explt1d 42358 expeq1d 42359 expeqidd 42360 expdioph 43035 lhe4.4ex1a 44348 fprodexp 45609 stoweidlem7 46022 stirlinglem5 46093 stirlinglem7 46095 stirlinglem10 46098 2pwp1prm 47576 m1expevenALTV 47634 4fppr1 47722 altgsumbc 48268 |
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