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Mirrors > Home > MPE Home > Th. List > dvdsexp | Structured version Visualization version GIF version |
Description: A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Ref | Expression |
---|---|
dvdsexp | ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ∥ (𝐴↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1131 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℤ) | |
2 | uznn0sub 12269 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ0) | |
3 | 2 | 3ad2ant3 1130 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝑁 − 𝑀) ∈ ℕ0) |
4 | zexpcl 13436 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝑁 − 𝑀) ∈ ℕ0) → (𝐴↑(𝑁 − 𝑀)) ∈ ℤ) | |
5 | 1, 3, 4 | syl2anc 586 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑(𝑁 − 𝑀)) ∈ ℤ) |
6 | zexpcl 13436 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0) → (𝐴↑𝑀) ∈ ℤ) | |
7 | 6 | 3adant3 1127 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ∈ ℤ) |
8 | dvdsmul2 15624 | . . 3 ⊢ (((𝐴↑(𝑁 − 𝑀)) ∈ ℤ ∧ (𝐴↑𝑀) ∈ ℤ) → (𝐴↑𝑀) ∥ ((𝐴↑(𝑁 − 𝑀)) · (𝐴↑𝑀))) | |
9 | 5, 7, 8 | syl2anc 586 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ∥ ((𝐴↑(𝑁 − 𝑀)) · (𝐴↑𝑀))) |
10 | 1 | zcnd 12080 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℂ) |
11 | simp2 1132 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℕ0) | |
12 | 10, 11, 3 | expaddd 13504 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑((𝑁 − 𝑀) + 𝑀)) = ((𝐴↑(𝑁 − 𝑀)) · (𝐴↑𝑀))) |
13 | eluzelcn 12247 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℂ) | |
14 | 13 | 3ad2ant3 1130 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℂ) |
15 | 11 | nn0cnd 11949 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℂ) |
16 | 14, 15 | npcand 10993 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((𝑁 − 𝑀) + 𝑀) = 𝑁) |
17 | 16 | oveq2d 7164 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑((𝑁 − 𝑀) + 𝑀)) = (𝐴↑𝑁)) |
18 | 12, 17 | eqtr3d 2856 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((𝐴↑(𝑁 − 𝑀)) · (𝐴↑𝑀)) = (𝐴↑𝑁)) |
19 | 9, 18 | breqtrd 5083 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ∥ (𝐴↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 ∈ wcel 2108 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 + caddc 10532 · cmul 10534 − cmin 10862 ℕ0cn0 11889 ℤcz 11973 ℤ≥cuz 12235 ↑cexp 13421 ∥ cdvds 15599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-n0 11890 df-z 11974 df-uz 12236 df-seq 13362 df-exp 13422 df-dvds 15600 |
This theorem is referenced by: bitsmod 15777 pcpremul 16172 pcdvdsb 16197 lt6abl 19007 ablfac1eu 19187 dvdsppwf1o 25755 jm2.20nn 39585 odz2prm2pw 43716 |
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