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Mirrors > Home > MPE Home > Th. List > lcmgcdnn | Structured version Visualization version GIF version |
Description: The product of two positive integers' least common multiple and greatest common divisor is the product of the two integers. (Contributed by AV, 27-Aug-2020.) |
Ref | Expression |
---|---|
lcmgcdnn | ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝑀 · 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 11601 | . . 3 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
2 | nnz 11601 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
3 | lcmgcd 15528 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) | |
4 | 1, 2, 3 | syl2an 583 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (abs‘(𝑀 · 𝑁))) |
5 | nnmulcl 11245 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) ∈ ℕ) | |
6 | 5 | nnnn0d 11553 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) ∈ ℕ0) |
7 | nn0re 11503 | . . . 4 ⊢ ((𝑀 · 𝑁) ∈ ℕ0 → (𝑀 · 𝑁) ∈ ℝ) | |
8 | nn0ge0 11520 | . . . 4 ⊢ ((𝑀 · 𝑁) ∈ ℕ0 → 0 ≤ (𝑀 · 𝑁)) | |
9 | 7, 8 | jca 501 | . . 3 ⊢ ((𝑀 · 𝑁) ∈ ℕ0 → ((𝑀 · 𝑁) ∈ ℝ ∧ 0 ≤ (𝑀 · 𝑁))) |
10 | absid 14244 | . . 3 ⊢ (((𝑀 · 𝑁) ∈ ℝ ∧ 0 ≤ (𝑀 · 𝑁)) → (abs‘(𝑀 · 𝑁)) = (𝑀 · 𝑁)) | |
11 | 6, 9, 10 | 3syl 18 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (abs‘(𝑀 · 𝑁)) = (𝑀 · 𝑁)) |
12 | 4, 11 | eqtrd 2805 | 1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝑀 · 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 ‘cfv 6031 (class class class)co 6793 ℝcr 10137 0cc0 10138 · cmul 10143 ≤ cle 10277 ℕcn 11222 ℕ0cn0 11494 ℤcz 11579 abscabs 14182 gcd cgcd 15424 lcm clcm 15509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-sup 8504 df-inf 8505 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-n0 11495 df-z 11580 df-uz 11889 df-rp 12036 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-dvds 15190 df-gcd 15425 df-lcm 15511 |
This theorem is referenced by: 3lcm2e6woprm 15536 6lcm4e12 15537 3lcm2e6 15647 ex-lcm 27657 |
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