Theorem nn0gcdsq 15298

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 11171	. 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		elnn0 11171	. 2 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3		sqgcd 15116	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
4		nncn 10905	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5		abssq 13894	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵)↑2) = (abs‘(𝐵↑2)))
6	4, 5	syl 17	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((abs‘𝐵)↑2) = (abs‘(𝐵↑2)))
7		nnz 11276	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
8		gcd0id 15078	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
10	9	oveq1d 6564	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((0 gcd 𝐵)↑2) = ((abs‘𝐵)↑2))
11		sq0 12817	. . . . . . . . 9 ⊢ (0↑2) = 0
12	11	a1i 11	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → (0↑2) = 0)
13	12	oveq1d 6564	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (0 gcd (𝐵↑2)))
14		zsqcl 12796	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ)
15		gcd0id 15078	. . . . . . . 8 ⊢ ((𝐵↑2) ∈ ℤ → (0 gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
16	7, 14, 15	3syl 18	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → (0 gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
17	13, 16	eqtrd 2644	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
18	6, 10, 17	3eqtr4d 2654	. . . . 5 ⊢ (𝐵 ∈ ℕ → ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2)))
19	18	adantl 481	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2)))
20		oveq1 6556	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
21	20	oveq1d 6564	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 𝐵)↑2))
22		oveq1 6556	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴↑2) = (0↑2))
23	22	oveq1d 6564	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (𝐵↑2)))
24	21, 23	eqeq12d 2625	. . . . 5 ⊢ (𝐴 = 0 → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2))))
25	24	adantr 480	. . . 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 246	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
27		nncn 10905	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
28		abssq 13894	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2)))
29	27, 28	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2)))
30		nnz 11276	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
31		gcdid0 15079	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 gcd 0) = (abs‘𝐴))
32	30, 31	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐴 gcd 0) = (abs‘𝐴))
33	32	oveq1d 6564	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑2) = ((abs‘𝐴)↑2))
34	11	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (0↑2) = 0)
35	34	oveq2d 6565	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = ((𝐴↑2) gcd 0))
36		zsqcl 12796	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
37		gcdid0 15079	. . . . . . . 8 ⊢ ((𝐴↑2) ∈ ℤ → ((𝐴↑2) gcd 0) = (abs‘(𝐴↑2)))
38	30, 36, 37	3syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((𝐴↑2) gcd 0) = (abs‘(𝐴↑2)))
39	35, 38	eqtrd 2644	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = (abs‘(𝐴↑2)))
40	29, 33, 39	3eqtr4d 2654	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2)))
41	40	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2)))
42		oveq2 6557	. . . . . . 7 ⊢ (𝐵 = 0 → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
43	42	oveq1d 6564	. . . . . 6 ⊢ (𝐵 = 0 → ((𝐴 gcd 𝐵)↑2) = ((𝐴 gcd 0)↑2))
44		oveq1 6556	. . . . . . 7 ⊢ (𝐵 = 0 → (𝐵↑2) = (0↑2))
45	44	oveq2d 6565	. . . . . 6 ⊢ (𝐵 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((𝐴↑2) gcd (0↑2)))
46	43, 45	eqeq12d 2625	. . . . 5 ⊢ (𝐵 = 0 → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2))))
47	46	adantl 481	. . . 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 246	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
49		gcd0val 15057	. . . . . 6 ⊢ (0 gcd 0) = 0
50	49	oveq1i 6559	. . . . 5 ⊢ ((0 gcd 0)↑2) = (0↑2)
51	11, 11	oveq12i 6561	. . . . . 6 ⊢ ((0↑2) gcd (0↑2)) = (0 gcd 0)
52	51, 49	eqtri 2632	. . . . 5 ⊢ ((0↑2) gcd (0↑2)) = 0
53	11, 50, 52	3eqtr4i 2642	. . . 4 ⊢ ((0 gcd 0)↑2) = ((0↑2) gcd (0↑2))
54		oveq12 6558	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
55	54	oveq1d 6564	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 0)↑2))
56	22, 44	oveqan12d 6568	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (0↑2)))
57	53, 55, 56	3eqtr4a 2670	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
58	3, 26, 48, 57	ccase 984	. 2 ⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
59	1, 2, 58	syl2anb 495	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  ℕcn 10897  2c2 10947  ℕ₀cn0 11169  ℤcz 11254  ↑cexp 12722  abscabs 13822   gcd cgcd 15054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-gcd 15055

This theorem is referenced by:  zgcdsq  15299