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Mirrors > Home > MPE Home > Th. List > zgcdsq | Structured version Visualization version GIF version |
Description: nn0gcdsq 16693 extended to integers by symmetry. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
Ref | Expression |
---|---|
zgcdsq | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdabs 16475 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((abs‘𝐴) gcd (abs‘𝐵)) = (𝐴 gcd 𝐵)) | |
2 | 1 | eqcomd 2730 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = ((abs‘𝐴) gcd (abs‘𝐵))) |
3 | 2 | oveq1d 7417 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵)↑2) = (((abs‘𝐴) gcd (abs‘𝐵))↑2)) |
4 | nn0abscl 15261 | . . 3 ⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ0) | |
5 | nn0abscl 15261 | . . 3 ⊢ (𝐵 ∈ ℤ → (abs‘𝐵) ∈ ℕ0) | |
6 | nn0gcdsq 16693 | . . 3 ⊢ (((abs‘𝐴) ∈ ℕ0 ∧ (abs‘𝐵) ∈ ℕ0) → (((abs‘𝐴) gcd (abs‘𝐵))↑2) = (((abs‘𝐴)↑2) gcd ((abs‘𝐵)↑2))) | |
7 | 4, 5, 6 | syl2an 595 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((abs‘𝐴) gcd (abs‘𝐵))↑2) = (((abs‘𝐴)↑2) gcd ((abs‘𝐵)↑2))) |
8 | zre 12561 | . . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℝ) |
10 | absresq 15251 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2)) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((abs‘𝐴)↑2) = (𝐴↑2)) |
12 | zre 12561 | . . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
13 | 12 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℝ) |
14 | absresq 15251 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((abs‘𝐵)↑2) = (𝐵↑2)) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((abs‘𝐵)↑2) = (𝐵↑2)) |
16 | 11, 15 | oveq12d 7420 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((abs‘𝐴)↑2) gcd ((abs‘𝐵)↑2)) = ((𝐴↑2) gcd (𝐵↑2))) |
17 | 3, 7, 16 | 3eqtrd 2768 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6534 (class class class)co 7402 ℝcr 11106 2c2 12266 ℕ0cn0 12471 ℤcz 12557 ↑cexp 14028 abscabs 15183 gcd cgcd 16438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-fl 13758 df-mod 13836 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-dvds 16201 df-gcd 16439 |
This theorem is referenced by: numdensq 16695 |
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