Theorem nn0gcdsq 12790

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 9404	. 2 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		elnn0 9404	. 2 ⊢ (𝐵 ∈ ℕ0 ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3		sqgcd 12618	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
4		nncn 9151	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5		abssq 11659	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵)↑2) = (abs‘(𝐵↑2)))
6	4, 5	syl 14	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((abs‘𝐵)↑2) = (abs‘(𝐵↑2)))
7		nnz 9498	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
8		gcd0id 12568	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
9	7, 8	syl 14	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
10	9	oveq1d 6033	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((0 gcd 𝐵)↑2) = ((abs‘𝐵)↑2))
11		sq0 10893	. . . . . . . . 9 ⊢ (0↑2) = 0
12	11	a1i 9	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → (0↑2) = 0)
13	12	oveq1d 6033	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (0 gcd (𝐵↑2)))
14		zsqcl 10873	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ)
15		gcd0id 12568	. . . . . . . 8 ⊢ ((𝐵↑2) ∈ ℤ → (0 gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
16	7, 14, 15	3syl 17	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → (0 gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
17	13, 16	eqtrd 2264	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
18	6, 10, 17	3eqtr4d 2274	. . . . 5 ⊢ (𝐵 ∈ ℕ → ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2)))
19	18	adantl 277	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2)))
20		oveq1 6025	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
21	20	oveq1d 6033	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 𝐵)↑2))
22		oveq1 6025	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴↑2) = (0↑2))
23	22	oveq1d 6033	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (𝐵↑2)))
24	21, 23	eqeq12d 2246	. . . . 5 ⊢ (𝐴 = 0 → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((0 gcd 𝐵)↑2) = ((0↑2) gcd (𝐵↑2))))
25	24	adantr 276	. . . 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 167	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
27		nncn 9151	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
28		abssq 11659	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2)))
29	27, 28	syl 14	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((abs‘𝐴)↑2) = (abs‘(𝐴↑2)))
30		nnz 9498	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
31		gcdid0 12569	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 gcd 0) = (abs‘𝐴))
32	30, 31	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐴 gcd 0) = (abs‘𝐴))
33	32	oveq1d 6033	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑2) = ((abs‘𝐴)↑2))
34	11	a1i 9	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (0↑2) = 0)
35	34	oveq2d 6034	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = ((𝐴↑2) gcd 0))
36		zsqcl 10873	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
37		gcdid0 12569	. . . . . . . 8 ⊢ ((𝐴↑2) ∈ ℤ → ((𝐴↑2) gcd 0) = (abs‘(𝐴↑2)))
38	30, 36, 37	3syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((𝐴↑2) gcd 0) = (abs‘(𝐴↑2)))
39	35, 38	eqtrd 2264	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = (abs‘(𝐴↑2)))
40	29, 33, 39	3eqtr4d 2274	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2)))
41	40	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2)))
42		oveq2 6026	. . . . . . 7 ⊢ (𝐵 = 0 → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
43	42	oveq1d 6033	. . . . . 6 ⊢ (𝐵 = 0 → ((𝐴 gcd 𝐵)↑2) = ((𝐴 gcd 0)↑2))
44		oveq1 6025	. . . . . . 7 ⊢ (𝐵 = 0 → (𝐵↑2) = (0↑2))
45	44	oveq2d 6034	. . . . . 6 ⊢ (𝐵 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((𝐴↑2) gcd (0↑2)))
46	43, 45	eqeq12d 2246	. . . . 5 ⊢ (𝐵 = 0 → (((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)) ↔ ((𝐴 gcd 0)↑2) = ((𝐴↑2) gcd (0↑2))))
47	46	adantl 277	. . . 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 167	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
49		gcd0val 12549	. . . . . 6 ⊢ (0 gcd 0) = 0
50	49	oveq1i 6028	. . . . 5 ⊢ ((0 gcd 0)↑2) = (0↑2)
51	11, 11	oveq12i 6030	. . . . . 6 ⊢ ((0↑2) gcd (0↑2)) = (0 gcd 0)
52	51, 49	eqtri 2252	. . . . 5 ⊢ ((0↑2) gcd (0↑2)) = 0
53	11, 50, 52	3eqtr4i 2262	. . . 4 ⊢ ((0 gcd 0)↑2) = ((0↑2) gcd (0↑2))
54		oveq12 6027	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
55	54	oveq1d 6033	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 0)↑2))
56	22, 44	oveqan12d 6037	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (0↑2)))
57	53, 55, 56	3eqtr4a 2290	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
58	3, 26, 48, 57	ccase 972	. 2 ⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
59	1, 2, 58	syl2anb 291	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715   = wceq 1397   ∈ wcel 2202  ‘cfv 5326  (class class class)co 6018  ℂcc 8030  0cc0 8032  ℕcn 9143  2c2 9194  ℕ₀cn0 9402  ℤcz 9479  ↑cexp 10801  abscabs 11575   gcd cgcd 12542

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-sup 7183  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-dvds 12367  df-gcd 12543

This theorem is referenced by:  zgcdsq  12791