gcddvds - Intuitionistic Logic Explorer

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Theorem gcddvds 11958

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 9262	. . . . . 6 ⊢ 0 ∈ ℤ
2		dvds0 11808	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∥ 0)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 0 ∥ 0
4		breq2 4007	. . . . . . 7 ⊢ (𝑀 = 0 → (0 ∥ 𝑀 ↔ 0 ∥ 0))
5		breq2 4007	. . . . . . 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 5883	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
11		gcd0val 11955	. . . . . . 7 ⊢ (0 gcd 0) = 0
12	10, 11	eqtrdi 2226	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = 0)
13	12	breq1d 4013	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 gcd 𝑁) ∥ 𝑀 ↔ 0 ∥ 𝑀))
14	12	breq1d 4013	. . . . 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 11956	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑀 ∧ 𝑛 ∥ 𝑁)}, ℝ, < ))
19		zssre 9258	. . . . . 6 ⊢ ℤ ⊆ ℝ
20		gcdsupex 11952	. . . . . 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 3220	. . . . . 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 3240	. . . . . 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 9349	. . . 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 11957	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ ℕ)
28	27	nnzd 9372	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ∈ ℤ)
29		breq1 4006	. . . . . 6 ⊢ (𝑛 = (𝑀 gcd 𝑁) → (𝑛 ∥ 𝑀 ↔ (𝑀 gcd 𝑁) ∥ 𝑀))
30		breq1 4006	. . . . . 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 2894	. . . 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 11939	. . 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 3129 class class class wbr 4003 (class class class)co 5874 supcsup 6980 ℝcr 7809 0cc0 7810 < clt 7990 ℤcz 9251 ∥ cdvds 11789 gcd cgcd 11937

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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930

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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-sup 6982 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-n0 9175 df-z 9252 df-uz 9527 df-q 9618 df-rp 9652 df-fz 10007 df-fzo 10140 df-fl 10267 df-mod 10320 df-seqfrec 10443 df-exp 10517 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-dvds 11790 df-gcd 11938

This theorem is referenced by: zeqzmulgcd 11965 divgcdz 11966 divgcdnn 11970 gcd0id 11974 gcdneg 11977 gcdaddm 11979 gcd1 11982 dvdsgcdb 12008 dfgcd2 12009 mulgcd 12011 gcdzeq 12017 dvdsmulgcd 12020 sqgcd 12024 dvdssqlem 12025 bezoutr 12027 gcddvdslcm 12067 lcmgcdlem 12071 lcmgcdeq 12077 coprmgcdb 12082 ncoprmgcdne1b 12083 mulgcddvds 12088 rpmulgcd2 12089 qredeu 12091 rpdvds 12093 divgcdcoprm0 12095 divgcdodd 12137 coprm 12138 rpexp 12147 divnumden 12190 phimullem 12219 hashgcdlem 12232 hashgcdeq 12233 phisum 12234 pythagtriplem4 12262 pythagtriplem19 12276 pcgcd1 12321 pc2dvds 12323 pockthlem 12348 2sqlem8 14390

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