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Mirrors > Home > MPE Home > Th. List > gcdmultipled | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
gcdmultipled.1 | ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
gcdmultipled.2 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
Ref | Expression |
---|---|
gcdmultipled | ⊢ (𝜑 → (𝑀 gcd (𝑁 · 𝑀)) = 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdmultipled.2 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
2 | gcdmultipled.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | |
3 | 2 | nn0zd 12661 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
4 | 0zd 12647 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
5 | gcdaddm 16565 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑀 gcd 0) = (𝑀 gcd (0 + (𝑁 · 𝑀)))) | |
6 | 1, 3, 4, 5 | syl3anc 1371 | . 2 ⊢ (𝜑 → (𝑀 gcd 0) = (𝑀 gcd (0 + (𝑁 · 𝑀)))) |
7 | nn0gcdid0 16561 | . . 3 ⊢ (𝑀 ∈ ℕ0 → (𝑀 gcd 0) = 𝑀) | |
8 | 2, 7 | syl 17 | . 2 ⊢ (𝜑 → (𝑀 gcd 0) = 𝑀) |
9 | 1, 3 | zmulcld 12749 | . . . . 5 ⊢ (𝜑 → (𝑁 · 𝑀) ∈ ℤ) |
10 | 9 | zcnd 12744 | . . . 4 ⊢ (𝜑 → (𝑁 · 𝑀) ∈ ℂ) |
11 | 10 | addlidd 11487 | . . 3 ⊢ (𝜑 → (0 + (𝑁 · 𝑀)) = (𝑁 · 𝑀)) |
12 | 11 | oveq2d 7461 | . 2 ⊢ (𝜑 → (𝑀 gcd (0 + (𝑁 · 𝑀))) = (𝑀 gcd (𝑁 · 𝑀))) |
13 | 6, 8, 12 | 3eqtr3rd 2783 | 1 ⊢ (𝜑 → (𝑀 gcd (𝑁 · 𝑀)) = 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2103 (class class class)co 7445 0cc0 11180 + caddc 11183 · cmul 11185 ℕ0cn0 12549 ℤcz 12635 gcd cgcd 16534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-sup 9507 df-inf 9508 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-n0 12550 df-z 12636 df-uz 12900 df-rp 13054 df-seq 14049 df-exp 14109 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-dvds 16297 df-gcd 16535 |
This theorem is referenced by: gcdmultiplez 16576 dvdsgcdidd 16578 unitscyglem2 42101 unitscyglem4 42103 |
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