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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdsexpim | Structured version Visualization version GIF version |
Description: dvdssqim 16116 generalized to nonnegative exponents. (Contributed by Steven Nguyen, 2-Apr-2023.) |
Ref | Expression |
---|---|
dvdsexpim | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ 𝐵 → (𝐴↑𝑁) ∥ (𝐵↑𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divides 15817 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ 𝐵 ↔ ∃𝑘 ∈ ℤ (𝑘 · 𝐴) = 𝐵)) | |
2 | 1 | 3adant3 1134 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ 𝐵 ↔ ∃𝑘 ∈ ℤ (𝑘 · 𝐴) = 𝐵)) |
3 | zexpcl 13650 | . . . . . . . . 9 ⊢ ((𝑘 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑘↑𝑁) ∈ ℤ) | |
4 | 3 | ancoms 462 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → (𝑘↑𝑁) ∈ ℤ) |
5 | 4 | adantll 714 | . . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → (𝑘↑𝑁) ∈ ℤ) |
6 | zexpcl 13650 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ) | |
7 | 6 | adantr 484 | . . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → (𝐴↑𝑁) ∈ ℤ) |
8 | dvdsmul2 15840 | . . . . . . 7 ⊢ (((𝑘↑𝑁) ∈ ℤ ∧ (𝐴↑𝑁) ∈ ℤ) → (𝐴↑𝑁) ∥ ((𝑘↑𝑁) · (𝐴↑𝑁))) | |
9 | 5, 7, 8 | syl2anc 587 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → (𝐴↑𝑁) ∥ ((𝑘↑𝑁) · (𝐴↑𝑁))) |
10 | zcn 12181 | . . . . . . . 8 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℂ) | |
11 | 10 | adantl 485 | . . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℂ) |
12 | zcn 12181 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
13 | 12 | ad2antrr 726 | . . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → 𝐴 ∈ ℂ) |
14 | simplr 769 | . . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → 𝑁 ∈ ℕ0) | |
15 | 11, 13, 14 | mulexpd 13731 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → ((𝑘 · 𝐴)↑𝑁) = ((𝑘↑𝑁) · (𝐴↑𝑁))) |
16 | 9, 15 | breqtrrd 5081 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → (𝐴↑𝑁) ∥ ((𝑘 · 𝐴)↑𝑁)) |
17 | oveq1 7220 | . . . . . 6 ⊢ ((𝑘 · 𝐴) = 𝐵 → ((𝑘 · 𝐴)↑𝑁) = (𝐵↑𝑁)) | |
18 | 17 | breq2d 5065 | . . . . 5 ⊢ ((𝑘 · 𝐴) = 𝐵 → ((𝐴↑𝑁) ∥ ((𝑘 · 𝐴)↑𝑁) ↔ (𝐴↑𝑁) ∥ (𝐵↑𝑁))) |
19 | 16, 18 | syl5ibcom 248 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) ∧ 𝑘 ∈ ℤ) → ((𝑘 · 𝐴) = 𝐵 → (𝐴↑𝑁) ∥ (𝐵↑𝑁))) |
20 | 19 | rexlimdva 3203 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (∃𝑘 ∈ ℤ (𝑘 · 𝐴) = 𝐵 → (𝐴↑𝑁) ∥ (𝐵↑𝑁))) |
21 | 20 | 3adant2 1133 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (∃𝑘 ∈ ℤ (𝑘 · 𝐴) = 𝐵 → (𝐴↑𝑁) ∥ (𝐵↑𝑁))) |
22 | 2, 21 | sylbid 243 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ 𝐵 → (𝐴↑𝑁) ∥ (𝐵↑𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∃wrex 3062 class class class wbr 5053 (class class class)co 7213 ℂcc 10727 · cmul 10734 ℕ0cn0 12090 ℤcz 12176 ↑cexp 13635 ∥ cdvds 15815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-seq 13575 df-exp 13636 df-dvds 15816 |
This theorem is referenced by: dvdsexpad 40040 expgcd 40042 dvdsexpnn 40048 fltaccoprm 40180 |
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