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Mirrors > Home > MPE Home > Th. List > Mathboxes > zexpgcd | Structured version Visualization version GIF version |
Description: Exponentiation distributes over GCD. zgcdsq 16081 extended to nonnegative exponents. nn0expgcd 39062 extended to integer bases by symmetry. (Contributed by Steven Nguyen, 5-Apr-2023.) |
Ref | Expression |
---|---|
zexpgcd | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdabs 15865 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((abs‘𝐴) gcd (abs‘𝐵)) = (𝐴 gcd 𝐵)) | |
2 | 1 | 3adant3 1124 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((abs‘𝐴) gcd (abs‘𝐵)) = (𝐴 gcd 𝐵)) |
3 | 2 | eqcomd 2824 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴 gcd 𝐵) = ((abs‘𝐴) gcd (abs‘𝐵))) |
4 | 3 | oveq1d 7160 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = (((abs‘𝐴) gcd (abs‘𝐵))↑𝑁)) |
5 | nn0abscl 14660 | . . 3 ⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ0) | |
6 | nn0abscl 14660 | . . 3 ⊢ (𝐵 ∈ ℤ → (abs‘𝐵) ∈ ℕ0) | |
7 | id 22 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
8 | nn0expgcd 39062 | . . 3 ⊢ (((abs‘𝐴) ∈ ℕ0 ∧ (abs‘𝐵) ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (((abs‘𝐴) gcd (abs‘𝐵))↑𝑁) = (((abs‘𝐴)↑𝑁) gcd ((abs‘𝐵)↑𝑁))) | |
9 | 5, 6, 7, 8 | syl3an 1152 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (((abs‘𝐴) gcd (abs‘𝐵))↑𝑁) = (((abs‘𝐴)↑𝑁) gcd ((abs‘𝐵)↑𝑁))) |
10 | zcn 11974 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
11 | 10 | 3ad2ant1 1125 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 𝐴 ∈ ℂ) |
12 | simp3 1130 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
13 | 11, 12 | absexpd 14800 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁)) |
14 | 13 | eqcomd 2824 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((abs‘𝐴)↑𝑁) = (abs‘(𝐴↑𝑁))) |
15 | zcn 11974 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
16 | 15 | 3ad2ant2 1126 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 𝐵 ∈ ℂ) |
17 | 16, 12 | absexpd 14800 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (abs‘(𝐵↑𝑁)) = ((abs‘𝐵)↑𝑁)) |
18 | 17 | eqcomd 2824 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((abs‘𝐵)↑𝑁) = (abs‘(𝐵↑𝑁))) |
19 | 14, 18 | oveq12d 7163 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (((abs‘𝐴)↑𝑁) gcd ((abs‘𝐵)↑𝑁)) = ((abs‘(𝐴↑𝑁)) gcd (abs‘(𝐵↑𝑁)))) |
20 | zexpcl 13432 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ) | |
21 | 20 | 3adant2 1123 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ) |
22 | zexpcl 13432 | . . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐵↑𝑁) ∈ ℤ) | |
23 | 22 | 3adant1 1122 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐵↑𝑁) ∈ ℤ) |
24 | gcdabs 15865 | . . . 4 ⊢ (((𝐴↑𝑁) ∈ ℤ ∧ (𝐵↑𝑁) ∈ ℤ) → ((abs‘(𝐴↑𝑁)) gcd (abs‘(𝐵↑𝑁))) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))) | |
25 | 21, 23, 24 | syl2anc 584 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((abs‘(𝐴↑𝑁)) gcd (abs‘(𝐵↑𝑁))) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))) |
26 | 19, 25 | eqtrd 2853 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (((abs‘𝐴)↑𝑁) gcd ((abs‘𝐵)↑𝑁)) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))) |
27 | 4, 9, 26 | 3eqtrd 2857 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℕ0cn0 11885 ℤcz 11969 ↑cexp 13417 abscabs 14581 gcd cgcd 15831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-gcd 15832 |
This theorem is referenced by: numdenexp 39064 |
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