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Theorem gcddvds 12450

Description: The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)

**Assertion**
Ref	Expression
gcddvds	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))

Proof of Theorem gcddvds Dummy variables 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0z 9425	. . . . . 6 ⊢ 0 ∈ ℤ
2		dvds0 12283	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∥ 0)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 0 ∥ 0
4		breq2 4066	. . . . . . 7 ⊢ (𝑀 = 0 → (0 ∥ 𝑀 ↔ 0 ∥ 0))
5		breq2 4066	. . . . . . 7 ⊢ (𝑁 = 0 → (0 ∥ 𝑁 ↔ 0 ∥ 0))
6	4, 5	bi2anan9 608	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) ↔ (0 ∥ 0 ∧ 0 ∥ 0)))
7		anidm 396	. . . . . 6 ⊢ ((0 ∥ 0 ∧ 0 ∥ 0) ↔ 0 ∥ 0)
8	6, 7	bitrdi 196	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) ↔ 0 ∥ 0))
9	3, 8	mpbiri 168	. . . 4 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (0 ∥ 𝑀 ∧ 0 ∥ 𝑁))
10		oveq12 5983	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
11		gcd0val 12447	. . . . . . 7 ⊢ (0 gcd 0) = 0
12	10, 11	eqtrdi 2258	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = 0)
13	12	breq1d 4072	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 gcd 𝑁) ∥ 𝑀 ↔ 0 ∥ 𝑀))
14	12	breq1d 4072	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ 0 ∥ 𝑁))
15	13, 14	anbi12d 473	. . . 4 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ↔ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁)))
16	9, 15	mpbird 167	. . 3 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
17	16	adantl 277	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
18		gcdn0val 12448	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))
19		zssre 9421	. . . . . 6 ⊢ ℤ ⊆ ℝ
20		gcdsupex 12444	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}𝑦 < 𝑧)))
21		ssrexv 3269	. . . . . 6 ⊢ (ℤ ⊆ ℝ → (∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}𝑦 < 𝑧))))
22	19, 20, 21	mpsyl 65	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}𝑦 < 𝑧)))
23		ssrab2 3289	. . . . . 6 ⊢ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} ⊆ ℤ
24	23	a1i 9	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} ⊆ ℤ)
25	22, 24	suprzclex 9513	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ) ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)})
26	18, 25	eqeltrd 2286	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)})
27		gcdn0cl 12449	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ ℕ)
28	27	nnzd 9536	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ ℤ)
29		breq1 4065	. . . . . 6 ⊢ (𝑛 = (𝑀 gcd 𝑁) → (𝑛 ∥ 𝑀 ↔ (𝑀 gcd 𝑁) ∥ 𝑀))
30		breq1 4065	. . . . . 6 ⊢ (𝑛 = (𝑀 gcd 𝑁) → (𝑛 ∥ 𝑁 ↔ (𝑀 gcd 𝑁) ∥ 𝑁))
31	29, 30	anbi12d 473	. . . . 5 ⊢ (𝑛 = (𝑀 gcd 𝑁) → ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁) ↔ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)))
32	31	elrab3 2940	. . . 4 ⊢ ((𝑀 gcd 𝑁) ∈ ℤ → ((𝑀 gcd 𝑁) ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} ↔ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)))
33	28, 32	syl 14	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((𝑀 gcd 𝑁) ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)} ↔ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)))
34	26, 33	mpbid 147	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
35		gcdmndc 12442	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∧ 𝑁 = 0))
36		exmiddc 840	. . 3 ⊢ (DECID (𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 = 0 ∧ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∧ 𝑁 = 0)))
37	35, 36	syl 14	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∧ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∧ 𝑁 = 0)))
38	17, 34, 37	mpjaodan 802	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 712 DECID wdc 838 = wceq 1375 ∈ wcel 2180 ∀wral 2488 ∃wrex 2489 {crab 2492 ⊆ wss 3177 class class class wbr 4062 (class class class)co 5974 supcsup 7117 ℝcr 7966 0cc0 7967 < clt 8149 ℤcz 9414 ∥ cdvds 12264 gcd cgcd 12440

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086 ax-caucvg 8087

This theorem depends on definitions: df-bi 117 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-ilim 4437 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-recs 6421 df-frec 6507 df-sup 7119 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-n0 9338 df-z 9415 df-uz 9691 df-q 9783 df-rp 9818 df-fz 10173 df-fzo 10307 df-fl 10457 df-mod 10512 df-seqfrec 10637 df-exp 10728 df-cj 11319 df-re 11320 df-im 11321 df-rsqrt 11475 df-abs 11476 df-dvds 12265 df-gcd 12441

This theorem is referenced by: zeqzmulgcd 12457 divgcdz 12458 divgcdnn 12462 gcd0id 12466 gcdneg 12469 gcdaddm 12471 gcd1 12474 dvdsgcdb 12500 dfgcd2 12501 mulgcd 12503 gcdzeq 12509 dvdsmulgcd 12512 sqgcd 12516 dvdssqlem 12517 bezoutr 12519 gcddvdslcm 12561 lcmgcdlem 12565 lcmgcdeq 12571 coprmgcdb 12576 ncoprmgcdne1b 12577 mulgcddvds 12582 rpmulgcd2 12583 qredeu 12585 rpdvds 12587 divgcdcoprm0 12589 divgcdodd 12631 coprm 12632 rpexp 12641 divnumden 12684 phimullem 12713 hashgcdlem 12726 hashgcdeq 12728 phisum 12729 pythagtriplem4 12757 pythagtriplem19 12771 pcgcd1 12817 pc2dvds 12819 pockthlem 12845 znunit 14588 znrrg 14589 mpodvdsmulf1o 15629 2sqlem8 15767

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