gcdmultipled - Metamath Proof Explorer

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

Theorem gcdmultipled 16170

Description: The greatest common divisor of a nonnegative integer 𝑀 and a multiple of it is 𝑀 itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)

**Hypotheses**
Ref	Expression
gcdmultipled.1	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
gcdmultipled.2	⊢ (𝜑 → 𝑁 ∈ ℤ)

**Assertion**
Ref	Expression
gcdmultipled	⊢ (𝜑 → (𝑀 gcd (𝑁 · 𝑀)) = 𝑀)

**Proof of Theorem gcdmultipled**
Step	Hyp	Ref	Expression
1		gcdmultipled.2	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
2		gcdmultipled.1	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
3	2	nn0zd 12353	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4		0zd 12261	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
5		gcdaddm 16160	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑀 gcd 0) = (𝑀 gcd (0 + (𝑁 · 𝑀))))
6	1, 3, 4, 5	syl3anc 1369	. 2 ⊢ (𝜑 → (𝑀 gcd 0) = (𝑀 gcd (0 + (𝑁 · 𝑀))))
7		nn0gcdid0 16156	. . 3 ⊢ (𝑀 ∈ ℕ₀ → (𝑀 gcd 0) = 𝑀)
8	2, 7	syl 17	. 2 ⊢ (𝜑 → (𝑀 gcd 0) = 𝑀)
9	1, 3	zmulcld 12361	. . . . 5 ⊢ (𝜑 → (𝑁 · 𝑀) ∈ ℤ)
10	9	zcnd 12356	. . . 4 ⊢ (𝜑 → (𝑁 · 𝑀) ∈ ℂ)
11	10	addid2d 11106	. . 3 ⊢ (𝜑 → (0 + (𝑁 · 𝑀)) = (𝑁 · 𝑀))
12	11	oveq2d 7271	. 2 ⊢ (𝜑 → (𝑀 gcd (0 + (𝑁 · 𝑀))) = (𝑀 gcd (𝑁 · 𝑀)))
13	6, 8, 12	3eqtr3rd 2787	1 ⊢ (𝜑 → (𝑀 gcd (𝑁 · 𝑀)) = 𝑀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 (class class class)co 7255 0cc0 10802 + caddc 10805 · cmul 10807 ℕ₀cn0 12163 ℤcz 12249 gcd cgcd 16129

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130

This theorem is referenced by: gcdmultiplez 16171 dvdsgcdidd 16173

W3C validator