Theorem nn0gcdsq 16890

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 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))

Proof of Theorem nn0gcdsq

Step	Hyp	Ref	Expression
1		elnn0 12577	. 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		elnn0 12577	. 2 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3		sqgcd 16699	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
4		nncn 12312	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5		abssq 15440	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵)↑2) = (abs‘(𝐵↑2)))
6	4, 5	syl 18	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((abs‘𝐵)↑2) = (abs‘(𝐵↑2)))
7		nnz 12683	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
8		gcd0id 16656	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
9	7, 8	syl 18	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
10	9	oveq1d 7423	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((0 gcd 𝐵)↑2) = ((abs‘𝐵)↑2))
11		sq0 14303	. . . . . . . . 9 ⊢ (0↑2) = 0
12	11	a1i 11	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → (0↑2) = 0)
13	12	oveq1d 7423	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (0 gcd (𝐵↑2)))
14		zsqcl 14240	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ)
15		gcd0id 16656	. . . . . . . 8 ⊢ ((𝐵↑2) ∈ ℤ → (0 gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
16	7, 14, 15	3syl 19	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → (0 gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
17	13, 16	eqtrd 2795	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
18	6, 10, 17	3eqtr4d 2805	. . . . 5 ⊢ (𝐵 ∈ ℕ → ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2)))
19	18	adantl 487	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2)))
20		oveq1 7415	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
21	20	oveq1d 7423	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 𝐵)↑2))
22		oveq1 7415	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴↑2) = (0↑2))
23	22	oveq1d 7423	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (𝐵↑2)))
24	21, 23	eqeq12d 2776	. . . . 5 ⊢ (𝐴 = 0 → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2))))
25	24	adantr 486	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2))))
26	19, 25	mpbird 260	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
27		nncn 12312	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
28		abssq 15440	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2)))
29	27, 28	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2)))
30		nnz 12683	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
31		gcdid0 16657	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 gcd 0) = (abs‘𝐴))
32	30, 31	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐴 gcd 0) = (abs‘𝐴))
33	32	oveq1d 7423	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑2) = ((abs‘𝐴)↑2))
34	11	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (0↑2) = 0)
35	34	oveq2d 7424	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = ((𝐴↑2) gcd 0))
36		zsqcl 14240	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
37		gcdid0 16657	. . . . . . . 8 ⊢ ((𝐴↑2) ∈ ℤ → ((𝐴↑2) gcd 0) = (abs‘(𝐴↑2)))
38	30, 36, 37	3syl 19	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((𝐴↑2) gcd 0) = (abs‘(𝐴↑2)))
39	35, 38	eqtrd 2795	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = (abs‘(𝐴↑2)))
40	29, 33, 39	3eqtr4d 2805	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2)))
41	40	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2)))
42		oveq2 7416	. . . . . . 7 ⊢ (𝐵 = 0 → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
43	42	oveq1d 7423	. . . . . 6 ⊢ (𝐵 = 0 → ((𝐴 gcd 𝐵)↑2) = ((𝐴 gcd 0)↑2))
44		oveq1 7415	. . . . . . 7 ⊢ (𝐵 = 0 → (𝐵↑2) = (0↑2))
45	44	oveq2d 7424	. . . . . 6 ⊢ (𝐵 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((𝐴↑2) gcd (0↑2)))
46	43, 45	eqeq12d 2776	. . . . 5 ⊢ (𝐵 = 0 → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2))))
47	46	adantl 487	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2))))
48	41, 47	mpbird 260	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
49		gcd0val 16634	. . . . . 6 ⊢ (0 gcd 0) = 0
50	49	oveq1i 7418	. . . . 5 ⊢ ((0 gcd 0)↑2) = (0↑2)
51	11, 11	oveq12i 7420	. . . . . 6 ⊢ ((0↑2) gcd (0↑2)) = (0 gcd 0)
52	51, 49	eqtri 2783	. . . . 5 ⊢ ((0↑2) gcd (0↑2)) = 0
53	11, 50, 52	3eqtr4i 2793	. . . 4 ⊢ ((0 gcd 0)↑2) = ((0↑2) gcd (0↑2))
54		oveq12 7417	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
55	54	oveq1d 7423	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 0)↑2))
56	22, 44	oveqan12d 7427	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (0↑2)))
57	53, 55, 56	3eqtr4a 2821	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
58	3, 26, 48, 57	ccase 1053	. 2 ⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
59	1, 2, 58	syl2anb 610	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ‘cfv 6527  (class class class)co 7408  ℂcc 11169  0cc0 11171  ℕcn 12304  2c2 12366  ℕ₀cn0 12575  ℤcz 12662  ↑cexp 14172  abscabs 15368   gcd cgcd 16631

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248  ax-pre-sup 11249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-sup 9412  df-inf 9413  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-div 11943  df-nn 12305  df-2 12374  df-3 12375  df-n0 12576  df-z 12663  df-uz 12935  df-rp 13090  df-fl 13900  df-mod 13978  df-seq 14113  df-exp 14173  df-cj 15233  df-re 15234  df-im 15235  df-sqrt 15369  df-abs 15370  df-dvds 16390  df-gcd 16632

This theorem is used by:  zgcdsq  16891