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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 7153 | . . 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 15938 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝑀 · 𝑁)) | |
6 | 3, 4, 5 | mp2an 690 | . . . . 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 2847 | . . . . 5 ⊢ (𝐺 · 𝐻) = (𝑀 · 𝑁) |
10 | 6, 9 | eqtr4i 2847 | . . . 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 39133 | . . . 4 ⊢ (𝐺 · 𝐻) = (𝐻 · 𝐺) |
14 | 10, 13 | eqtri 2844 | . . 3 ⊢ ((𝑀 lcm 𝑁) · (𝑀 gcd 𝑁)) = (𝐻 · 𝐺) |
15 | 2, 14 | eqtr3i 2846 | . 2 ⊢ ((𝑀 lcm 𝑁) · 𝐺) = (𝐻 · 𝐺) |
16 | 3 | nnzi 11993 | . . . . . . 7 ⊢ 𝑀 ∈ ℤ |
17 | 4 | nnzi 11993 | . . . . . . 7 ⊢ 𝑁 ∈ ℤ |
18 | 16, 17 | pm3.2i 473 | . . . . . 6 ⊢ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) |
19 | lcmcl 15928 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0) | |
20 | 18, 19 | ax-mp 5 | . . . . 5 ⊢ (𝑀 lcm 𝑁) ∈ ℕ0 |
21 | 20 | nn0cni 11896 | . . . 4 ⊢ (𝑀 lcm 𝑁) ∈ ℂ |
22 | 12 | nncni 11634 | . . . 4 ⊢ 𝐻 ∈ ℂ |
23 | 11 | nncni 11634 | . . . . 5 ⊢ 𝐺 ∈ ℂ |
24 | 11 | nnne0i 11664 | . . . . 5 ⊢ 𝐺 ≠ 0 |
25 | 23, 24 | pm3.2i 473 | . . . 4 ⊢ (𝐺 ∈ ℂ ∧ 𝐺 ≠ 0) |
26 | 21, 22, 25 | 3pm3.2i 1335 | . . 3 ⊢ ((𝑀 lcm 𝑁) ∈ ℂ ∧ 𝐻 ∈ ℂ ∧ (𝐺 ∈ ℂ ∧ 𝐺 ≠ 0)) |
27 | mulcan2 11264 | . . 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 232 | 1 ⊢ (𝑀 lcm 𝑁) = 𝐻 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 (class class class)co 7142 ℂcc 10521 0cc0 10523 · cmul 10528 ℕcn 11624 ℕ0cn0 11884 ℤcz 11968 gcd cgcd 15826 lcm clcm 15915 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-sup 8892 df-inf 8893 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-n0 11885 df-z 11969 df-uz 12231 df-rp 12377 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-dvds 15593 df-gcd 15827 df-lcm 15917 |
This theorem is referenced by: 12lcm5e60 39146 60lcm6e60 39147 60lcm7e420 39148 420lcm8e840 39149 |
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