2sqcoprm - Metamath Proof Explorer

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Theorem 2sqcoprm 27346

Description: If the sum of two squares is prime, the two original numbers are coprime. (Contributed by Thierry Arnoux, 2-Feb-2020.)

**Hypotheses**
Ref	Expression
2sqcoprm.1	⊢ (𝜑 → 𝑃 ∈ ℙ)
2sqcoprm.2	⊢ (𝜑 → 𝐴 ∈ ℤ)
2sqcoprm.3	⊢ (𝜑 → 𝐵 ∈ ℤ)
2sqcoprm.4	⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃)

**Assertion**
Ref	Expression
2sqcoprm	⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)

**Proof of Theorem 2sqcoprm**
Step	Hyp	Ref	Expression
1		2sqcoprm.1	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
2		2sqcoprm.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℤ)
3		2sqcoprm.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℤ)
4		2sqcoprm.4	. . 3 ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = 𝑃)
5	1, 2, 3, 4	2sqn0 27345	. 2 ⊢ (𝜑 → 𝐴 ≠ 0)
6	2, 3	gcdcld 16478	. . . 4 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ₀)
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴 gcd 𝐵) ∈ ℕ₀)
8	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℤ)
9	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℤ)
10		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
11	10	neneqd 2930	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ¬ 𝐴 = 0)
12	11	intnanrd 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
13		gcdn0cl 16472	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
14	8, 9, 12, 13	syl21anc 837	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴 gcd 𝐵) ∈ ℕ)
15	14	nnsqcld 14209	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝐴 gcd 𝐵)↑2) ∈ ℕ)
16	6	nn0zd 12555	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ)
17		sqnprm 16672	. . . . . . . . . . 11 ⊢ ((𝐴 gcd 𝐵) ∈ ℤ → ¬ ((𝐴 gcd 𝐵)↑2) ∈ ℙ)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ¬ ((𝐴 gcd 𝐵)↑2) ∈ ℙ)
19		zsqcl 14094	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 gcd 𝐵) ∈ ℤ → ((𝐴 gcd 𝐵)↑2) ∈ ℤ)
20	16, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∈ ℤ)
21		zsqcl 14094	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
22	2, 21	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴↑2) ∈ ℤ)
23		zsqcl 14094	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ)
24	3, 23	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵↑2) ∈ ℤ)
25		gcddvds 16473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
26	2, 3, 25	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
27	26	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐴)
28		dvdssqim 16524	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2)))
29	28	imp 406	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝐴 gcd 𝐵) ∥ 𝐴) → ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2))
30	16, 2, 27, 29	syl21anc 837	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2))
31	26	simprd 495	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐵)
32		dvdssqim 16524	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐵 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
33	32	imp 406	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 gcd 𝐵) ∥ 𝐵) → ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2))
34	16, 3, 31, 33	syl21anc 837	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2))
35	20, 22, 24, 30, 34	dvds2addd 16262	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2)))
36	35, 4	breqtrd 5133	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ 𝑃)
37	36	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → ((𝐴 gcd 𝐵)↑2) ∥ 𝑃)
38		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2))
39	1	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → 𝑃 ∈ ℙ)
40		dvdsprm 16673	. . . . . . . . . . . . 13 ⊢ ((((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (((𝐴 gcd 𝐵)↑2) ∥ 𝑃 ↔ ((𝐴 gcd 𝐵)↑2) = 𝑃))
41	38, 39, 40	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → (((𝐴 gcd 𝐵)↑2) ∥ 𝑃 ↔ ((𝐴 gcd 𝐵)↑2) = 𝑃))
42	37, 41	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → ((𝐴 gcd 𝐵)↑2) = 𝑃)
43	42, 39	eqeltrd 2828	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → ((𝐴 gcd 𝐵)↑2) ∈ ℙ)
44	18, 43	mtand 815	. . . . . . . . 9 ⊢ (𝜑 → ¬ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2))
45		eluz2b3 12881	. . . . . . . . 9 ⊢ (((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2) ↔ (((𝐴 gcd 𝐵)↑2) ∈ ℕ ∧ ((𝐴 gcd 𝐵)↑2) ≠ 1))
46	44, 45	sylnib 328	. . . . . . . 8 ⊢ (𝜑 → ¬ (((𝐴 gcd 𝐵)↑2) ∈ ℕ ∧ ((𝐴 gcd 𝐵)↑2) ≠ 1))
47		imnan 399	. . . . . . . 8 ⊢ ((((𝐴 gcd 𝐵)↑2) ∈ ℕ → ¬ ((𝐴 gcd 𝐵)↑2) ≠ 1) ↔ ¬ (((𝐴 gcd 𝐵)↑2) ∈ ℕ ∧ ((𝐴 gcd 𝐵)↑2) ≠ 1))
48	46, 47	sylibr 234	. . . . . . 7 ⊢ (𝜑 → (((𝐴 gcd 𝐵)↑2) ∈ ℕ → ¬ ((𝐴 gcd 𝐵)↑2) ≠ 1))
49	48	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (((𝐴 gcd 𝐵)↑2) ∈ ℕ → ¬ ((𝐴 gcd 𝐵)↑2) ≠ 1))
50	15, 49	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ¬ ((𝐴 gcd 𝐵)↑2) ≠ 1)
51		df-ne 2926	. . . . 5 ⊢ (((𝐴 gcd 𝐵)↑2) ≠ 1 ↔ ¬ ((𝐴 gcd 𝐵)↑2) = 1)
52	50, 51	sylnib 328	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ¬ ¬ ((𝐴 gcd 𝐵)↑2) = 1)
53	52	notnotrd 133	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝐴 gcd 𝐵)↑2) = 1)
54		nn0sqeq1 15242	. . 3 ⊢ (((𝐴 gcd 𝐵) ∈ ℕ₀ ∧ ((𝐴 gcd 𝐵)↑2) = 1) → (𝐴 gcd 𝐵) = 1)
55	7, 53, 54	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴 gcd 𝐵) = 1)
56	5, 55	mpdan 687	1 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 0cc0 11068 1c1 11069 + caddc 11071 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ↑cexp 14026 ∥ cdvds 16222 gcd cgcd 16464 ℙcprime 16641

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-gcd 16465 df-prm 16642

This theorem is referenced by: 2sqmod 27347

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