2sqcoprm - Metamath Proof Explorer

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Theorem 2sqcoprm 27174

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 27173	. 2 ⊢ (𝜑 → 𝐴 ≠ 0)
6	2, 3	gcdcld 16453	. . . 4 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ₀)
7	6	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴 gcd 𝐵) ∈ ℕ₀)
8	2	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℤ)
9	3	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐵 ∈ ℤ)
10		simpr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ≠ 0)
11	10	neneqd 2943	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ¬ 𝐴 = 0)
12	11	intnanrd 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
13		gcdn0cl 16447	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
14	8, 9, 12, 13	syl21anc 834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴 gcd 𝐵) ∈ ℕ)
15	14	nnsqcld 14211	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝐴 gcd 𝐵)↑2) ∈ ℕ)
16	6	nn0zd 12588	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ)
17		sqnprm 16643	. . . . . . . . . . 11 ⊢ ((𝐴 gcd 𝐵) ∈ ℤ → ¬ ((𝐴 gcd 𝐵)↑2) ∈ ℙ)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ¬ ((𝐴 gcd 𝐵)↑2) ∈ ℙ)
19		zsqcl 14098	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 gcd 𝐵) ∈ ℤ → ((𝐴 gcd 𝐵)↑2) ∈ ℤ)
20	16, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∈ ℤ)
21		zsqcl 14098	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
22	2, 21	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴↑2) ∈ ℤ)
23		zsqcl 14098	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ)
24	3, 23	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵↑2) ∈ ℤ)
25		gcddvds 16448	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
26	2, 3, 25	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
27	26	simpld 493	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐴)
28		dvdssqim 16500	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2)))
29	28	imp 405	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝐴 gcd 𝐵) ∥ 𝐴) → ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2))
30	16, 2, 27, 29	syl21anc 834	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐴↑2))
31	26	simprd 494	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐵)
32		dvdssqim 16500	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐵 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2)))
33	32	imp 405	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 gcd 𝐵) ∥ 𝐵) → ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2))
34	16, 3, 31, 33	syl21anc 834	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ (𝐵↑2))
35	20, 22, 24, 30, 34	dvds2addd 16239	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2)))
36	35, 4	breqtrd 5173	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ 𝑃)
37	36	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → ((𝐴 gcd 𝐵)↑2) ∥ 𝑃)
38		simpr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2))
39	1	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → 𝑃 ∈ ℙ)
40		dvdsprm 16644	. . . . . . . . . . . . 13 ⊢ ((((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (((𝐴 gcd 𝐵)↑2) ∥ 𝑃 ↔ ((𝐴 gcd 𝐵)↑2) = 𝑃))
41	38, 39, 40	syl2anc 582	. . . . . . . . . . . 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 2831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2)) → ((𝐴 gcd 𝐵)↑2) ∈ ℙ)
44	18, 43	mtand 812	. . . . . . . . 9 ⊢ (𝜑 → ¬ ((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2))
45		eluz2b3 12910	. . . . . . . . 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 398	. . . . . . . 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 479	. . . . . 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 2939	. . . . 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 15227	. . 3 ⊢ (((𝐴 gcd 𝐵) ∈ ℕ₀ ∧ ((𝐴 gcd 𝐵)↑2) = 1) → (𝐴 gcd 𝐵) = 1)
55	7, 53, 54	syl2anc 582	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴 gcd 𝐵) = 1)
56	5, 55	mpdan 683	1 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 0cc0 11112 1c1 11113 + caddc 11115 ℕcn 12216 2c2 12271 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ↑cexp 14031 ∥ cdvds 16201 gcd cgcd 16439 ℙcprime 16612

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-dvds 16202 df-gcd 16440 df-prm 16613

This theorem is referenced by: 2sqmod 27175

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