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Theorem nn0gcdsq 12167
Description: Squaring commutes with GCD, in particular two coprime numbers have coprime squares. (Contributed by Stefan O'Rear, 15-Sep-2014.)
Assertion
Ref Expression
nn0gcdsq ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))

Proof of Theorem nn0gcdsq
StepHypRef Expression
1 elnn0 9151 . 2 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 elnn0 9151 . 2 (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3 sqgcd 11997 . . 3 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
4 nncn 8900 . . . . . . 7 (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5 abssq 11058 . . . . . . 7 (𝐵 ∈ ℂ → ((abs‘𝐵)↑2) = (abs‘(𝐵↑2)))
64, 5syl 14 . . . . . 6 (𝐵 ∈ ℕ → ((abs‘𝐵)↑2) = (abs‘(𝐵↑2)))
7 nnz 9245 . . . . . . . 8 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
8 gcd0id 11947 . . . . . . . 8 (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
97, 8syl 14 . . . . . . 7 (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
109oveq1d 5880 . . . . . 6 (𝐵 ∈ ℕ → ((0 gcd 𝐵)↑2) = ((abs‘𝐵)↑2))
11 sq0 10580 . . . . . . . . 9 (0↑2) = 0
1211a1i 9 . . . . . . . 8 (𝐵 ∈ ℕ → (0↑2) = 0)
1312oveq1d 5880 . . . . . . 7 (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (0 gcd (𝐵↑2)))
14 zsqcl 10560 . . . . . . . 8 (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ)
15 gcd0id 11947 . . . . . . . 8 ((𝐵↑2) ∈ ℤ → (0 gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
167, 14, 153syl 17 . . . . . . 7 (𝐵 ∈ ℕ → (0 gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
1713, 16eqtrd 2208 . . . . . 6 (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
186, 10, 173eqtr4d 2218 . . . . 5 (𝐵 ∈ ℕ → ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2)))
1918adantl 277 . . . 4 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2)))
20 oveq1 5872 . . . . . . 7 (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
2120oveq1d 5880 . . . . . 6 (𝐴 = 0 → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 𝐵)↑2))
22 oveq1 5872 . . . . . . 7 (𝐴 = 0 → (𝐴↑2) = (0↑2))
2322oveq1d 5880 . . . . . 6 (𝐴 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (𝐵↑2)))
2421, 23eqeq12d 2190 . . . . 5 (𝐴 = 0 → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2))))
2524adantr 276 . . . 4 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2))))
2619, 25mpbird 167 . . 3 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
27 nncn 8900 . . . . . . 7 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
28 abssq 11058 . . . . . . 7 (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2)))
2927, 28syl 14 . . . . . 6 (𝐴 ∈ ℕ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2)))
30 nnz 9245 . . . . . . . 8 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
31 gcdid0 11948 . . . . . . . 8 (𝐴 ∈ ℤ → (𝐴 gcd 0) = (abs‘𝐴))
3230, 31syl 14 . . . . . . 7 (𝐴 ∈ ℕ → (𝐴 gcd 0) = (abs‘𝐴))
3332oveq1d 5880 . . . . . 6 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑2) = ((abs‘𝐴)↑2))
3411a1i 9 . . . . . . . 8 (𝐴 ∈ ℕ → (0↑2) = 0)
3534oveq2d 5881 . . . . . . 7 (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = ((𝐴↑2) gcd 0))
36 zsqcl 10560 . . . . . . . 8 (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
37 gcdid0 11948 . . . . . . . 8 ((𝐴↑2) ∈ ℤ → ((𝐴↑2) gcd 0) = (abs‘(𝐴↑2)))
3830, 36, 373syl 17 . . . . . . 7 (𝐴 ∈ ℕ → ((𝐴↑2) gcd 0) = (abs‘(𝐴↑2)))
3935, 38eqtrd 2208 . . . . . 6 (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = (abs‘(𝐴↑2)))
4029, 33, 393eqtr4d 2218 . . . . 5 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2)))
4140adantr 276 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2)))
42 oveq2 5873 . . . . . . 7 (𝐵 = 0 → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
4342oveq1d 5880 . . . . . 6 (𝐵 = 0 → ((𝐴 gcd 𝐵)↑2) = ((𝐴 gcd 0)↑2))
44 oveq1 5872 . . . . . . 7 (𝐵 = 0 → (𝐵↑2) = (0↑2))
4544oveq2d 5881 . . . . . 6 (𝐵 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((𝐴↑2) gcd (0↑2)))
4643, 45eqeq12d 2190 . . . . 5 (𝐵 = 0 → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2))))
4746adantl 277 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2))))
4841, 47mpbird 167 . . 3 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
49 gcd0val 11928 . . . . . 6 (0 gcd 0) = 0
5049oveq1i 5875 . . . . 5 ((0 gcd 0)↑2) = (0↑2)
5111, 11oveq12i 5877 . . . . . 6 ((0↑2) gcd (0↑2)) = (0 gcd 0)
5251, 49eqtri 2196 . . . . 5 ((0↑2) gcd (0↑2)) = 0
5311, 50, 523eqtr4i 2206 . . . 4 ((0 gcd 0)↑2) = ((0↑2) gcd (0↑2))
54 oveq12 5874 . . . . 5 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
5554oveq1d 5880 . . . 4 ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 0)↑2))
5622, 44oveqan12d 5884 . . . 4 ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (0↑2)))
5753, 55, 563eqtr4a 2234 . . 3 ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
583, 26, 48, 57ccase 964 . 2 (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
591, 2, 58syl2anb 291 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wcel 2146  cfv 5208  (class class class)co 5865  cc 7784  0cc0 7786  cn 8892  2c2 8943  0cn0 9149  cz 9226  cexp 10489  abscabs 10974   gcd cgcd 11910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-3 8952  df-4 8953  df-n0 9150  df-z 9227  df-uz 9502  df-q 9593  df-rp 9625  df-fz 9980  df-fzo 10113  df-fl 10240  df-mod 10293  df-seqfrec 10416  df-exp 10490  df-cj 10819  df-re 10820  df-im 10821  df-rsqrt 10975  df-abs 10976  df-dvds 11763  df-gcd 11911
This theorem is referenced by:  zgcdsq  12168
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