2sqcoprm - Metamath Proof Explorer

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Theorem 2sqcoprm 26783

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 26782	. 2 ⊢ (𝜑 → 𝐴 ≠ 0)
6	2, 3	gcdcld 16388	. . . 4 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ₀)
7	6	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴 gcd 𝐵) ∈ ℕ₀)
8	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℤ)
9	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℤ)
10		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
11	10	neneqd 2948	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ¬ 𝐴 = 0)
12	11	intnanrd 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
13		gcdn0cl 16382	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
14	8, 9, 12, 13	syl21anc 836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴 gcd 𝐵) ∈ ℕ)
15	14	nnsqcld 14147	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝐴 gcd 𝐵)↑2) ∈ ℕ)
16	6	nn0zd 12525	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ)
17		sqnprm 16578	. . . . . . . . . . 11 ⊢ ((𝐴 gcd 𝐵) ∈ ℤ → ¬ ((𝐴 gcd 𝐵)↑2) ∈ ℙ)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ¬ ((𝐴 gcd 𝐵)↑2) ∈ ℙ)
19		zsqcl 14034	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 gcd 𝐵) ∈ ℤ → ((𝐴 gcd 𝐵)↑2) ∈ ℤ)
20	16, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∈ ℤ)
21		zsqcl 14034	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
22	2, 21	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴↑2) ∈ ℤ)
23		zsqcl 14034	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ)
24	3, 23	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵↑2) ∈ ℤ)
25		gcddvds 16383	. . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐴)
28		dvdssqim 16435	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2)))
29	28	imp 407	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝐴 gcd 𝐵) ∥ 𝐴) → ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2))
30	16, 2, 27, 29	syl21anc 836	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2))
31	26	simprd 496	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐵)
32		dvdssqim 16435	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐵 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
33	32	imp 407	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 gcd 𝐵) ∥ 𝐵) → ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2))
34	16, 3, 31, 33	syl21anc 836	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2))
35	20, 22, 24, 30, 34	dvds2addd 16174	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2)))
36	35, 4	breqtrd 5131	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ 𝑃)
37	36	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → ((𝐴 gcd 𝐵)↑2) ∥ 𝑃)
38		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2))
39	1	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → 𝑃 ∈ ℙ)
40		dvdsprm 16579	. . . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → ((𝐴 gcd 𝐵)↑2) = 𝑃)
43	42, 39	eqeltrd 2838	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → ((𝐴 gcd 𝐵)↑2) ∈ ℙ)
44	18, 43	mtand 814	. . . . . . . . 9 ⊢ (𝜑 → ¬ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2))
45		eluz2b3 12847	. . . . . . . . 9 ⊢ (((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2) ↔ (((𝐴 gcd 𝐵)↑2) ∈ ℕ ∧ ((𝐴 gcd 𝐵)↑2) ≠ 1))
46	44, 45	sylnib 327	. . . . . . . 8 ⊢ (𝜑 → ¬ (((𝐴 gcd 𝐵)↑2) ∈ ℕ ∧ ((𝐴 gcd 𝐵)↑2) ≠ 1))
47		imnan 400	. . . . . . . 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 233	. . . . . . 7 ⊢ (𝜑 → (((𝐴 gcd 𝐵)↑2) ∈ ℕ → ¬ ((𝐴 gcd 𝐵)↑2) ≠ 1))
49	48	adantr 481	. . . . . 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 2944	. . . . 5 ⊢ (((𝐴 gcd 𝐵)↑2) ≠ 1 ↔ ¬ ((𝐴 gcd 𝐵)↑2) = 1)
52	50, 51	sylnib 327	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ¬ ¬ ((𝐴 gcd 𝐵)↑2) = 1)
53	52	notnotrd 133	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝐴 gcd 𝐵)↑2) = 1)
54		nn0sqeq1 15161	. . 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 685	1 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 0cc0 11051 1c1 11052 + caddc 11054 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ↑cexp 13967 ∥ cdvds 16136 gcd cgcd 16374 ℙcprime 16547

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-gcd 16375 df-prm 16548

This theorem is referenced by: 2sqmod 26784

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