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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcmeprodgcdi | Structured version Visualization version GIF version |
Description: Calculate the least common multiple of two natural numbers. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
lcmeprodgcdi.1 | ⊢ 𝑀 ∈ ℕ |
lcmeprodgcdi.2 | ⊢ 𝑁 ∈ ℕ |
lcmeprodgcdi.3 | ⊢ 𝐺 ∈ ℕ |
lcmeprodgcdi.4 | ⊢ 𝐻 ∈ ℕ |
lcmeprodgcdi.5 | ⊢ (𝑀 gcd 𝑁) = 𝐺 |
lcmeprodgcdi.6 | ⊢ (𝐺 · 𝐻) = 𝐴 |
lcmeprodgcdi.7 | ⊢ (𝑀 · 𝑁) = 𝐴 |
Ref | Expression |
---|---|
lcmeprodgcdi | ⊢ (𝑀 lcm 𝑁) = 𝐻 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmeprodgcdi.5 | . . . 4 ⊢ (𝑀 gcd 𝑁) = 𝐺 | |
2 | 1 | oveq2i 7441 | . . 3 ⊢ ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = ((𝑀 lcm 𝑁) · 𝐺) |
3 | lcmeprodgcdi.1 | . . . . . 6 ⊢ 𝑀 ∈ ℕ | |
4 | lcmeprodgcdi.2 | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
5 | lcmgcdnn 16644 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝑀 · 𝑁)) | |
6 | 3, 4, 5 | mp2an 692 | . . . . 5 ⊢ ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝑀 · 𝑁) |
7 | lcmeprodgcdi.6 | . . . . . 6 ⊢ (𝐺 · 𝐻) = 𝐴 | |
8 | lcmeprodgcdi.7 | . . . . . 6 ⊢ (𝑀 · 𝑁) = 𝐴 | |
9 | 7, 8 | eqtr4i 2765 | . . . . 5 ⊢ (𝐺 · 𝐻) = (𝑀 · 𝑁) |
10 | 6, 9 | eqtr4i 2765 | . . . 4 ⊢ ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝐺 · 𝐻) |
11 | lcmeprodgcdi.3 | . . . . 5 ⊢ 𝐺 ∈ ℕ | |
12 | lcmeprodgcdi.4 | . . . . 5 ⊢ 𝐻 ∈ ℕ | |
13 | 11, 12 | mulcomnni 41968 | . . . 4 ⊢ (𝐺 · 𝐻) = (𝐻 · 𝐺) |
14 | 10, 13 | eqtri 2762 | . . 3 ⊢ ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝐻 · 𝐺) |
15 | 2, 14 | eqtr3i 2764 | . 2 ⊢ ((𝑀 lcm 𝑁) · 𝐺) = (𝐻 · 𝐺) |
16 | 3 | nnzi 12638 | . . . . . . 7 ⊢ 𝑀 ∈ ℤ |
17 | 4 | nnzi 12638 | . . . . . . 7 ⊢ 𝑁 ∈ ℤ |
18 | 16, 17 | pm3.2i 470 | . . . . . 6 ⊢ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) |
19 | lcmcl 16634 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0) | |
20 | 18, 19 | ax-mp 5 | . . . . 5 ⊢ (𝑀 lcm 𝑁) ∈ ℕ0 |
21 | 20 | nn0cni 12535 | . . . 4 ⊢ (𝑀 lcm 𝑁) ∈ ℂ |
22 | 12 | nncni 12273 | . . . 4 ⊢ 𝐻 ∈ ℂ |
23 | 11 | nncni 12273 | . . . . 5 ⊢ 𝐺 ∈ ℂ |
24 | 11 | nnne0i 12303 | . . . . 5 ⊢ 𝐺 ≠ 0 |
25 | 23, 24 | pm3.2i 470 | . . . 4 ⊢ (𝐺 ∈ ℂ ∧ 𝐺 ≠ 0) |
26 | 21, 22, 25 | 3pm3.2i 1338 | . . 3 ⊢ ((𝑀 lcm 𝑁) ∈ ℂ ∧ 𝐻 ∈ ℂ ∧ (𝐺 ∈ ℂ ∧ 𝐺 ≠ 0)) |
27 | mulcan2 11898 | . . 3 ⊢ (((𝑀 lcm 𝑁) ∈ ℂ ∧ 𝐻 ∈ ℂ ∧ (𝐺 ∈ ℂ ∧ 𝐺 ≠ 0)) → (((𝑀 lcm 𝑁) · 𝐺) = (𝐻 · 𝐺) ↔ (𝑀 lcm 𝑁) = 𝐻)) | |
28 | 26, 27 | ax-mp 5 | . 2 ⊢ (((𝑀 lcm 𝑁) · 𝐺) = (𝐻 · 𝐺) ↔ (𝑀 lcm 𝑁) = 𝐻) |
29 | 15, 28 | mpbi 230 | 1 ⊢ (𝑀 lcm 𝑁) = 𝐻 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 (class class class)co 7430 ℂcc 11150 0cc0 11152 · cmul 11157 ℕcn 12263 ℕ0cn0 12523 ℤcz 12610 gcd cgcd 16527 lcm clcm 16621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 df-gcd 16528 df-lcm 16623 |
This theorem is referenced by: 12lcm5e60 41989 60lcm6e60 41990 60lcm7e420 41991 420lcm8e840 41992 |
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