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Theorem nn0gcdsq 16090
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 11896 . 2 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 elnn0 11896 . 2 (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3 sqgcd 15907 . . 3 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
4 nncn 11642 . . . . . . 7 (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5 abssq 14666 . . . . . . 7 (𝐵 ∈ ℂ → ((abs‘𝐵)↑2) = (abs‘(𝐵↑2)))
64, 5syl 17 . . . . . 6 (𝐵 ∈ ℕ → ((abs‘𝐵)↑2) = (abs‘(𝐵↑2)))
7 nnz 12001 . . . . . . . 8 (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
8 gcd0id 15865 . . . . . . . 8 (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
97, 8syl 17 . . . . . . 7 (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
109oveq1d 7164 . . . . . 6 (𝐵 ∈ ℕ → ((0 gcd 𝐵)↑2) = ((abs‘𝐵)↑2))
11 sq0 13560 . . . . . . . . 9 (0↑2) = 0
1211a1i 11 . . . . . . . 8 (𝐵 ∈ ℕ → (0↑2) = 0)
1312oveq1d 7164 . . . . . . 7 (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (0 gcd (𝐵↑2)))
14 zsqcl 13499 . . . . . . . 8 (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ)
15 gcd0id 15865 . . . . . . . 8 ((𝐵↑2) ∈ ℤ → (0 gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
167, 14, 153syl 18 . . . . . . 7 (𝐵 ∈ ℕ → (0 gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
1713, 16eqtrd 2859 . . . . . 6 (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
186, 10, 173eqtr4d 2869 . . . . 5 (𝐵 ∈ ℕ → ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2)))
1918adantl 485 . . . 4 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2)))
20 oveq1 7156 . . . . . . 7 (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
2120oveq1d 7164 . . . . . 6 (𝐴 = 0 → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 𝐵)↑2))
22 oveq1 7156 . . . . . . 7 (𝐴 = 0 → (𝐴↑2) = (0↑2))
2322oveq1d 7164 . . . . . 6 (𝐴 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (𝐵↑2)))
2421, 23eqeq12d 2840 . . . . 5 (𝐴 = 0 → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2))))
2524adantr 484 . . . 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 260 . . 3 ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
27 nncn 11642 . . . . . . 7 (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
28 abssq 14666 . . . . . . 7 (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2)))
2927, 28syl 17 . . . . . 6 (𝐴 ∈ ℕ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2)))
30 nnz 12001 . . . . . . . 8 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
31 gcdid0 15866 . . . . . . . 8 (𝐴 ∈ ℤ → (𝐴 gcd 0) = (abs‘𝐴))
3230, 31syl 17 . . . . . . 7 (𝐴 ∈ ℕ → (𝐴 gcd 0) = (abs‘𝐴))
3332oveq1d 7164 . . . . . 6 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑2) = ((abs‘𝐴)↑2))
3411a1i 11 . . . . . . . 8 (𝐴 ∈ ℕ → (0↑2) = 0)
3534oveq2d 7165 . . . . . . 7 (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = ((𝐴↑2) gcd 0))
36 zsqcl 13499 . . . . . . . 8 (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
37 gcdid0 15866 . . . . . . . 8 ((𝐴↑2) ∈ ℤ → ((𝐴↑2) gcd 0) = (abs‘(𝐴↑2)))
3830, 36, 373syl 18 . . . . . . 7 (𝐴 ∈ ℕ → ((𝐴↑2) gcd 0) = (abs‘(𝐴↑2)))
3935, 38eqtrd 2859 . . . . . 6 (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = (abs‘(𝐴↑2)))
4029, 33, 393eqtr4d 2869 . . . . 5 (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2)))
4140adantr 484 . . . 4 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2)))
42 oveq2 7157 . . . . . . 7 (𝐵 = 0 → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
4342oveq1d 7164 . . . . . 6 (𝐵 = 0 → ((𝐴 gcd 𝐵)↑2) = ((𝐴 gcd 0)↑2))
44 oveq1 7156 . . . . . . 7 (𝐵 = 0 → (𝐵↑2) = (0↑2))
4544oveq2d 7165 . . . . . 6 (𝐵 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((𝐴↑2) gcd (0↑2)))
4643, 45eqeq12d 2840 . . . . 5 (𝐵 = 0 → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2))))
4746adantl 485 . . . 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 260 . . 3 ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
49 gcd0val 15844 . . . . . 6 (0 gcd 0) = 0
5049oveq1i 7159 . . . . 5 ((0 gcd 0)↑2) = (0↑2)
5111, 11oveq12i 7161 . . . . . 6 ((0↑2) gcd (0↑2)) = (0 gcd 0)
5251, 49eqtri 2847 . . . . 5 ((0↑2) gcd (0↑2)) = 0
5311, 50, 523eqtr4i 2857 . . . 4 ((0 gcd 0)↑2) = ((0↑2) gcd (0↑2))
54 oveq12 7158 . . . . 5 ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
5554oveq1d 7164 . . . 4 ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 0)↑2))
5622, 44oveqan12d 7168 . . . 4 ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (0↑2)))
5753, 55, 563eqtr4a 2885 . . 3 ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
583, 26, 48, 57ccase 1033 . 2 (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
591, 2, 58syl2anb 600 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  cfv 6343  (class class class)co 7149  cc 10533  0cc0 10535  cn 11634  2c2 11689  0cn0 11894  cz 11978  cexp 13434  abscabs 14593   gcd cgcd 15841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-sup 8903  df-inf 8904  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-fl 13166  df-mod 13242  df-seq 13374  df-exp 13435  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-dvds 15608  df-gcd 15842
This theorem is referenced by:  zgcdsq  16091
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