nn0gcdsq - Metamath Proof Explorer

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Theorem nn0gcdsq 16717

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 12498	. 2 ⊢ (𝐴 ∈ ℕ₀ ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		elnn0 12498	. 2 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℕ ∨ 𝐵 = 0))
3		sqgcd 16529	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
4		nncn 12244	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
5		abssq 15279	. . . . . . 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 12603	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
8		gcd0id 16487	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (0 gcd 𝐵) = (abs‘𝐵))
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → (0 gcd 𝐵) = (abs‘𝐵))
10	9	oveq1d 7429	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((0 gcd 𝐵)↑2) = ((abs‘𝐵)↑2))
11		sq0 14181	. . . . . . . . 9 ⊢ (0↑2) = 0
12	11	a1i 11	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ → (0↑2) = 0)
13	12	oveq1d 7429	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (0 gcd (𝐵↑2)))
14		zsqcl 14119	. . . . . . . 8 ⊢ (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ)
15		gcd0id 16487	. . . . . . . 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 2768	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((0↑2) gcd (𝐵↑2)) = (abs‘(𝐵↑2)))
18	6, 10, 17	3eqtr4d 2778	. . . . 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 7421	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴 gcd 𝐵) = (0 gcd 𝐵))
21	20	oveq1d 7429	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 𝐵)↑2))
22		oveq1 7421	. . . . . . 7 ⊢ (𝐴 = 0 → (𝐴↑2) = (0↑2))
23	22	oveq1d 7429	. . . . . 6 ⊢ (𝐴 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (𝐵↑2)))
24	21, 23	eqeq12d 2744	. . . . 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 257	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
27		nncn 12244	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
28		abssq 15279	. . . . . . 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 12603	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
31		gcdid0 16488	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴 gcd 0) = (abs‘𝐴))
32	30, 31	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐴 gcd 0) = (abs‘𝐴))
33	32	oveq1d 7429	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐴 gcd 0)↑2) = ((abs‘𝐴)↑2))
34	11	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → (0↑2) = 0)
35	34	oveq2d 7430	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = ((𝐴↑2) gcd 0))
36		zsqcl 14119	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
37		gcdid0 16488	. . . . . . . 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 2768	. . . . . 6 ⊢ (𝐴 ∈ ℕ → ((𝐴↑2) gcd (0↑2)) = (abs‘(𝐴↑2)))
40	29, 33, 39	3eqtr4d 2778	. . . . 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 7422	. . . . . . 7 ⊢ (𝐵 = 0 → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
43	42	oveq1d 7429	. . . . . 6 ⊢ (𝐵 = 0 → ((𝐴 gcd 𝐵)↑2) = ((𝐴 gcd 0)↑2))
44		oveq1 7421	. . . . . . 7 ⊢ (𝐵 = 0 → (𝐵↑2) = (0↑2))
45	44	oveq2d 7430	. . . . . 6 ⊢ (𝐵 = 0 → ((𝐴↑2) gcd (𝐵↑2)) = ((𝐴↑2) gcd (0↑2)))
46	43, 45	eqeq12d 2744	. . . . 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 257	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
49		gcd0val 16465	. . . . . 6 ⊢ (0 gcd 0) = 0
50	49	oveq1i 7424	. . . . 5 ⊢ ((0 gcd 0)↑2) = (0↑2)
51	11, 11	oveq12i 7426	. . . . . 6 ⊢ ((0↑2) gcd (0↑2)) = (0 gcd 0)
52	51, 49	eqtri 2756	. . . . 5 ⊢ ((0↑2) gcd (0↑2)) = 0
53	11, 50, 52	3eqtr4i 2766	. . . 4 ⊢ ((0 gcd 0)↑2) = ((0↑2) gcd (0↑2))
54		oveq12 7423	. . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → (𝐴 gcd 𝐵) = (0 gcd 0))
55	54	oveq1d 7429	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((0 gcd 0)↑2))
56	22, 44	oveqan12d 7433	. . . 4 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴↑2) gcd (𝐵↑2)) = ((0↑2) gcd (0↑2)))
57	53, 55, 56	3eqtr4a 2794	. . 3 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
58	3, 26, 48, 57	ccase 1036	. 2 ⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ ∨ 𝐵 = 0)) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))
59	1, 2, 58	syl2anb 597	1 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ₀) → ((𝐴 gcd 𝐵)↑2) = ((𝐴↑2) gcd (𝐵↑2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 ℂcc 11130 0cc0 11132 ℕcn 12236 2c2 12291 ℕ₀cn0 12496 ℤcz 12582 ↑cexp 14052 abscabs 15207 gcd cgcd 16462

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-fl 13783 df-mod 13861 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16225 df-gcd 16463

This theorem is referenced by: zgcdsq 16718

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