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Mirrors > Home > MPE Home > Th. List > iddvdsexp | Structured version Visualization version GIF version |
Description: An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
Ref | Expression |
---|---|
iddvdsexp | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑀 ∥ (𝑀↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnm1nn0 12272 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
2 | zexpcl 13795 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℕ0) → (𝑀↑(𝑁 − 1)) ∈ ℤ) | |
3 | 1, 2 | sylan2 593 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀↑(𝑁 − 1)) ∈ ℤ) |
4 | simpl 483 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℤ) | |
5 | dvdsmul2 15986 | . . 3 ⊢ (((𝑀↑(𝑁 − 1)) ∈ ℤ ∧ 𝑀 ∈ ℤ) → 𝑀 ∥ ((𝑀↑(𝑁 − 1)) · 𝑀)) | |
6 | 3, 4, 5 | syl2anc 584 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑀 ∥ ((𝑀↑(𝑁 − 1)) · 𝑀)) |
7 | zcn 12322 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
8 | expm1t 13809 | . . 3 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝑀↑𝑁) = ((𝑀↑(𝑁 − 1)) · 𝑀)) | |
9 | 7, 8 | sylan 580 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀↑𝑁) = ((𝑀↑(𝑁 − 1)) · 𝑀)) |
10 | 6, 9 | breqtrrd 5104 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑀 ∥ (𝑀↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5076 (class class class)co 7277 ℂcc 10867 1c1 10870 · cmul 10874 − cmin 11203 ℕcn 11971 ℕ0cn0 12231 ℤcz 12317 ↑cexp 13780 ∥ cdvds 15961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-n0 12232 df-z 12318 df-uz 12581 df-seq 13720 df-exp 13781 df-dvds 15962 |
This theorem is referenced by: dvdsexp2im 16034 prmexpb 16423 rpexp 16425 difsqpwdvds 16586 pockthlem 16604 ablfac1eu 19674 vmappw 26263 vmasum 26362 perfectlem1 26375 oddpwdc 32318 zrtdvds 40343 lighneallem1 45024 lighneallem3 45026 lighneallem4 45029 proththdlem 45032 nnpw2evenALTV 45121 |
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