2sqcoprm - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 2sqcoprm		Structured version Visualization version GIF version

Theorem 2sqcoprm 27381

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 27380	. 2 ⊢ (𝜑 → 𝐴 ≠ 0)
6	2, 3	gcdcld 16483	. . . 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 2942	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ¬ 𝐴 = 0)
12	11	intnanrd 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
13		gcdn0cl 16477	. . . . . . . 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 14239	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝐴 gcd 𝐵)↑2) ∈ ℕ)
16	6	nn0zd 12615	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ)
17		sqnprm 16673	. . . . . . . . . . 11 ⊢ ((𝐴 gcd 𝐵) ∈ ℤ → ¬ ((𝐴 gcd 𝐵)↑2) ∈ ℙ)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ¬ ((𝐴 gcd 𝐵)↑2) ∈ ℙ)
19		zsqcl 14126	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 gcd 𝐵) ∈ ℤ → ((𝐴 gcd 𝐵)↑2) ∈ ℤ)
20	16, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∈ ℤ)
21		zsqcl 14126	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ)
22	2, 21	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴↑2) ∈ ℤ)
23		zsqcl 14126	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℤ → (𝐵↑2) ∈ ℤ)
24	3, 23	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵↑2) ∈ ℤ)
25		gcddvds 16478	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
26	2, 3, 25	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
27	26	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐴)
28		dvdssqim 16530	. . . . . . . . . . . . . . . . 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 16530	. . . . . . . . . . . . . . . . 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 16269	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑2) ∥ ((𝐴↑2) + (𝐵↑2)))
36	35, 4	breqtrd 5174	. . . . . . . . . . . . 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 16674	. . . . . . . . . . . . 13 ⊢ ((((𝐴 gcd 𝐵)↑2) ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (((𝐴 gcd 𝐵)↑2) ∥ 𝑃 ↔ ((𝐴 gcd 𝐵)↑2) = 𝑃))
41	38, 39, 40	syl2anc 583	. . . . . . . . . . . 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 2829	. . . . . . . . . 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 12937	. . . . . . . . 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 233	. . . . . . 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 2938	. . . . 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 15256	. . 3 ⊢ (((𝐴 gcd 𝐵) ∈ ℕ₀ ∧ ((𝐴 gcd 𝐵)↑2) = 1) → (𝐴 gcd 𝐵) = 1)
55	7, 53, 54	syl2anc 583	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝐴 gcd 𝐵) = 1)
56	5, 55	mpdan 686	1 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 0cc0 11139 1c1 11140 + caddc 11142 ℕcn 12243 2c2 12298 ℕ₀cn0 12503 ℤcz 12589 ℤ_≥cuz 12853 ↑cexp 14059 ∥ cdvds 16231 gcd cgcd 16469 ℙcprime 16642

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 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217

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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-dvds 16232 df-gcd 16470 df-prm 16643

This theorem is referenced by: 2sqmod 27382

W3C validator