gcddvds - Intuitionistic Logic Explorer

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Theorem gcddvds 10587

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 8520	. . . . . 6 ⊢ 0 ∈ ℤ
2		dvds0 10443	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∥ 0)
3	1, 2	ax-mp 7	. . . . 5 ⊢ 0 ∥ 0
4		breq2 3810	. . . . . . 7 ⊢ (𝑀 = 0 → (0 ∥ 𝑀 ↔ 0 ∥ 0))
5		breq2 3810	. . . . . . 7 ⊢ (𝑁 = 0 → (0 ∥ 𝑁 ↔ 0 ∥ 0))
6	4, 5	bi2anan9 571	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) ↔ (0 ∥ 0 ∧ 0 ∥ 0)))
7		anidm 388	. . . . . 6 ⊢ ((0 ∥ 0 ∧ 0 ∥ 0) ↔ 0 ∥ 0)
8	6, 7	syl6bb 194	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((0 ∥ 𝑀 ∧ 0 ∥ 𝑁) ↔ 0 ∥ 0))
9	3, 8	mpbiri 166	. . . 4 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (0 ∥ 𝑀 ∧ 0 ∥ 𝑁))
10		oveq12 5574	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
11		gcd0val 10584	. . . . . . 7 ⊢ (0 gcd 0) = 0
12	10, 11	syl6eq 2131	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = 0)
13	12	breq1d 3816	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 gcd 𝑁) ∥ 𝑀 ↔ 0 ∥ 𝑀))
14	12	breq1d 3816	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ 0 ∥ 𝑁))
15	13, 14	anbi12d 457	. . . 4 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ↔ (0 ∥ 𝑀 ∧ 0 ∥ 𝑁)))
16	9, 15	mpbird 165	. . 3 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
17	16	adantl 271	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
18		gcdn0val 10585	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))
19		zssre 8516	. . . . . 6 ⊢ ℤ ⊆ ℝ
20		gcdsupex 10581	. . . . . 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 3069	. . . . . 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 64	. . . . 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 3089	. . . . . 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 8603	. . . 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 2159	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)})
27		gcdn0cl 10586	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ ℕ)
28	27	nnzd 8626	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ ℤ)
29		breq1 3809	. . . . . 6 ⊢ (𝑛 = (𝑀 gcd 𝑁) → (𝑛 ∥ 𝑀 ↔ (𝑀 gcd 𝑁) ∥ 𝑀))
30		breq1 3809	. . . . . 6 ⊢ (𝑛 = (𝑀 gcd 𝑁) → (𝑛 ∥ 𝑁 ↔ (𝑀 gcd 𝑁) ∥ 𝑁))
31	29, 30	anbi12d 457	. . . . 5 ⊢ (𝑛 = (𝑀 gcd 𝑁) → ((𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁) ↔ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁)))
32	31	elrab3 2759	. . . 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 145	. 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
35		gcdmndc 10572	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∧ 𝑁 = 0))
36		exmiddc 778	. . 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 745	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 102 ↔ wb 103 ∨ wo 662 DECID wdc 776 = wceq 1285 ∈ wcel 1434 ∀wral 2353 ∃wrex 2354 {crab 2357 ⊆ wss 2983 class class class wbr 3806 (class class class)co 5565 supcsup 6491 ℝcr 7119 0cc0 7120 < clt 7292 ℤcz 8509 ∥ cdvds 10428 gcd cgcd 10570

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3914 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-iinf 4358 ax-cnex 7206 ax-resscn 7207 ax-1cn 7208 ax-1re 7209 ax-icn 7210 ax-addcl 7211 ax-addrcl 7212 ax-mulcl 7213 ax-mulrcl 7214 ax-addcom 7215 ax-mulcom 7216 ax-addass 7217 ax-mulass 7218 ax-distr 7219 ax-i2m1 7220 ax-0lt1 7221 ax-1rid 7222 ax-0id 7223 ax-rnegex 7224 ax-precex 7225 ax-cnre 7226 ax-pre-ltirr 7227 ax-pre-ltwlin 7228 ax-pre-lttrn 7229 ax-pre-apti 7230 ax-pre-ltadd 7231 ax-pre-mulgt0 7232 ax-pre-mulext 7233 ax-arch 7234 ax-caucvg 7235

This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2613 df-sbc 2826 df-csb 2919 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-if 3370 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-int 3658 df-iun 3701 df-br 3807 df-opab 3861 df-mpt 3862 df-tr 3897 df-id 4077 df-po 4080 df-iso 4081 df-iord 4150 df-on 4152 df-ilim 4153 df-suc 4155 df-iom 4361 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-f1 4958 df-fo 4959 df-f1o 4960 df-fv 4961 df-riota 5521 df-ov 5568 df-oprab 5569 df-mpt2 5570 df-1st 5820 df-2nd 5821 df-recs 5976 df-frec 6062 df-sup 6493 df-pnf 7294 df-mnf 7295 df-xr 7296 df-ltxr 7297 df-le 7298 df-sub 7425 df-neg 7426 df-reap 7819 df-ap 7826 df-div 7905 df-inn 8184 df-2 8242 df-3 8243 df-4 8244 df-n0 8433 df-z 8510 df-uz 8778 df-q 8863 df-rp 8893 df-fz 9183 df-fzo 9307 df-fl 9429 df-mod 9482 df-iseq 9599 df-iexp 9650 df-cj 9955 df-re 9956 df-im 9957 df-rsqrt 10110 df-abs 10111 df-dvds 10429 df-gcd 10571

This theorem is referenced by: zeqzmulgcd 10594 divgcdz 10595 divgcdnn 10598 gcd0id 10602 gcdneg 10605 gcdaddm 10607 gcd1 10610 dvdsgcdb 10634 dfgcd2 10635 mulgcd 10637 gcdzeq 10643 dvdsmulgcd 10646 sqgcd 10650 dvdssqlem 10651 bezoutr 10653 gcddvdslcm 10687 lcmgcdlem 10691 lcmgcdeq 10697 coprmgcdb 10702 ncoprmgcdne1b 10703 mulgcddvds 10708 rpmulgcd2 10709 qredeu 10711 rpdvds 10713 divgcdcoprm0 10715 divgcdodd 10754 coprm 10755 rpexp 10764 divnumden 10806 phimullem 10833 hashgcdlem 10835 hashgcdeq 10836

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