gcddvds - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > gcddvds		GIF version

Theorem gcddvds 11967

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 9267	. . . . . 6 ⊢ 0 ∈ ℤ
2		dvds0 11816	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∥ 0)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 0 ∥ 0
4		breq2 4009	. . . . . . 7 ⊢ (𝑀 = 0 → (0 ∥ 𝑀 ↔ 0 ∥ 0))
5		breq2 4009	. . . . . . 7 ⊢ (𝑁 = 0 → (0 ∥ 𝑁 ↔ 0 ∥ 0))
6	4, 5	bi2anan9 606	. . . . . 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 5887	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
11		gcd0val 11964	. . . . . . 7 ⊢ (0 gcd 0) = 0
12	10, 11	eqtrdi 2226	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = 0)
13	12	breq1d 4015	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 gcd 𝑁) ∥ 𝑀 ↔ 0 ∥ 𝑀))
14	12	breq1d 4015	. . . . 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 11965	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))
19		zssre 9263	. . . . . 6 ⊢ ℤ ⊆ ℝ
20		gcdsupex 11961	. . . . . 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 3222	. . . . . 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 3242	. . . . . 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 9354	. . . 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 2254	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)})
27		gcdn0cl 11966	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ ℕ)
28	27	nnzd 9377	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ ℤ)
29		breq1 4008	. . . . . 6 ⊢ (𝑛 = (𝑀 gcd 𝑁) → (𝑛 ∥ 𝑀 ↔ (𝑀 gcd 𝑁) ∥ 𝑀))
30		breq1 4008	. . . . . 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 2896	. . . 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 11948	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∧ 𝑁 = 0))
36		exmiddc 836	. . 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 798	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 708 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 {crab 2459 ⊆ wss 3131 class class class wbr 4005 (class class class)co 5878 supcsup 6984 ℝcr 7813 0cc0 7814 < clt 7995 ℤcz 9256 ∥ cdvds 11797 gcd cgcd 11946

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 ax-pre-mulext 7932 ax-arch 7933 ax-caucvg 7934

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-frec 6395 df-sup 6986 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-reap 8535 df-ap 8542 df-div 8633 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-n0 9180 df-z 9257 df-uz 9532 df-q 9623 df-rp 9657 df-fz 10012 df-fzo 10146 df-fl 10273 df-mod 10326 df-seqfrec 10449 df-exp 10523 df-cj 10854 df-re 10855 df-im 10856 df-rsqrt 11010 df-abs 11011 df-dvds 11798 df-gcd 11947

This theorem is referenced by: zeqzmulgcd 11974 divgcdz 11975 divgcdnn 11979 gcd0id 11983 gcdneg 11986 gcdaddm 11988 gcd1 11991 dvdsgcdb 12017 dfgcd2 12018 mulgcd 12020 gcdzeq 12026 dvdsmulgcd 12029 sqgcd 12033 dvdssqlem 12034 bezoutr 12036 gcddvdslcm 12076 lcmgcdlem 12080 lcmgcdeq 12086 coprmgcdb 12091 ncoprmgcdne1b 12092 mulgcddvds 12097 rpmulgcd2 12098 qredeu 12100 rpdvds 12102 divgcdcoprm0 12104 divgcdodd 12146 coprm 12147 rpexp 12156 divnumden 12199 phimullem 12228 hashgcdlem 12241 hashgcdeq 12242 phisum 12243 pythagtriplem4 12271 pythagtriplem19 12285 pcgcd1 12330 pc2dvds 12332 pockthlem 12357 2sqlem8 14658

W3C validator