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Theorem gcdmultipled 16342

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 12526	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4		0zd 12433	. . 3 ⊢ (𝜑 → 0 ∈ ℤ)
5		gcdaddm 16332	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑀 gcd 0) = (𝑀 gcd (0 + (𝑁 · 𝑀))))
6	1, 3, 4, 5	syl3anc 1370	. 2 ⊢ (𝜑 → (𝑀 gcd 0) = (𝑀 gcd (0 + (𝑁 · 𝑀))))
7		nn0gcdid0 16328	. . 3 ⊢ (𝑀 ∈ ℕ₀ → (𝑀 gcd 0) = 𝑀)
8	2, 7	syl 17	. 2 ⊢ (𝜑 → (𝑀 gcd 0) = 𝑀)
9	1, 3	zmulcld 12534	. . . . 5 ⊢ (𝜑 → (𝑁 · 𝑀) ∈ ℤ)
10	9	zcnd 12529	. . . 4 ⊢ (𝜑 → (𝑁 · 𝑀) ∈ ℂ)
11	10	addid2d 11278	. . 3 ⊢ (𝜑 → (0 + (𝑁 · 𝑀)) = (𝑁 · 𝑀))
12	11	oveq2d 7354	. 2 ⊢ (𝜑 → (𝑀 gcd (0 + (𝑁 · 𝑀))) = (𝑀 gcd (𝑁 · 𝑀)))
13	6, 8, 12	3eqtr3rd 2785	1 ⊢ (𝜑 → (𝑀 gcd (𝑁 · 𝑀)) = 𝑀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 (class class class)co 7338 0cc0 10973 + caddc 10976 · cmul 10978 ℕ₀cn0 12335 ℤcz 12421 gcd cgcd 16301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-cnex 11029 ax-resscn 11030 ax-1cn 11031 ax-icn 11032 ax-addcl 11033 ax-addrcl 11034 ax-mulcl 11035 ax-mulrcl 11036 ax-mulcom 11037 ax-addass 11038 ax-mulass 11039 ax-distr 11040 ax-i2m1 11041 ax-1ne0 11042 ax-1rid 11043 ax-rnegex 11044 ax-rrecex 11045 ax-cnre 11046 ax-pre-lttri 11047 ax-pre-lttrn 11048 ax-pre-ltadd 11049 ax-pre-mulgt0 11050 ax-pre-sup 11051

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-tr 5211 df-id 5519 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5576 df-we 5578 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6239 df-ord 6306 df-on 6307 df-lim 6308 df-suc 6309 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-om 7782 df-2nd 7901 df-frecs 8168 df-wrecs 8199 df-recs 8273 df-rdg 8312 df-er 8570 df-en 8806 df-dom 8807 df-sdom 8808 df-sup 9300 df-inf 9301 df-pnf 11113 df-mnf 11114 df-xr 11115 df-ltxr 11116 df-le 11117 df-sub 11309 df-neg 11310 df-div 11735 df-nn 12076 df-2 12138 df-3 12139 df-n0 12336 df-z 12422 df-uz 12685 df-rp 12833 df-seq 13824 df-exp 13885 df-cj 14910 df-re 14911 df-im 14912 df-sqrt 15046 df-abs 15047 df-dvds 16064 df-gcd 16302

This theorem is referenced by: gcdmultiplez 16343 dvdsgcdidd 16345

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