![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 3lcm2e6 | Structured version Visualization version GIF version |
Description: The least common multiple of three and two is six. The operands are unequal primes and thus coprime, so the result is (the absolute value of) their product. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 27-Aug-2020.) |
Ref | Expression |
---|---|
3lcm2e6 | ⊢ (3 lcm 2) = 6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 12286 | . . . . . 6 ⊢ 2 ∈ ℝ | |
2 | 2lt3 12384 | . . . . . 6 ⊢ 2 < 3 | |
3 | 1, 2 | gtneii 11326 | . . . . 5 ⊢ 3 ≠ 2 |
4 | 3prm 16631 | . . . . . 6 ⊢ 3 ∈ ℙ | |
5 | 2prm 16629 | . . . . . 6 ⊢ 2 ∈ ℙ | |
6 | prmrp 16649 | . . . . . 6 ⊢ ((3 ∈ ℙ ∧ 2 ∈ ℙ) → ((3 gcd 2) = 1 ↔ 3 ≠ 2)) | |
7 | 4, 5, 6 | mp2an 691 | . . . . 5 ⊢ ((3 gcd 2) = 1 ↔ 3 ≠ 2) |
8 | 3, 7 | mpbir 230 | . . . 4 ⊢ (3 gcd 2) = 1 |
9 | 8 | oveq2i 7420 | . . 3 ⊢ ((3 lcm 2) · (3 gcd 2)) = ((3 lcm 2) · 1) |
10 | 3nn 12291 | . . . 4 ⊢ 3 ∈ ℕ | |
11 | 2nn 12285 | . . . 4 ⊢ 2 ∈ ℕ | |
12 | lcmgcdnn 16548 | . . . 4 ⊢ ((3 ∈ ℕ ∧ 2 ∈ ℕ) → ((3 lcm 2) · (3 gcd 2)) = (3 · 2)) | |
13 | 10, 11, 12 | mp2an 691 | . . 3 ⊢ ((3 lcm 2) · (3 gcd 2)) = (3 · 2) |
14 | 10 | nnzi 12586 | . . . . . 6 ⊢ 3 ∈ ℤ |
15 | 11 | nnzi 12586 | . . . . . 6 ⊢ 2 ∈ ℤ |
16 | lcmcl 16538 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → (3 lcm 2) ∈ ℕ0) | |
17 | 14, 15, 16 | mp2an 691 | . . . . 5 ⊢ (3 lcm 2) ∈ ℕ0 |
18 | 17 | nn0cni 12484 | . . . 4 ⊢ (3 lcm 2) ∈ ℂ |
19 | 18 | mulridi 11218 | . . 3 ⊢ ((3 lcm 2) · 1) = (3 lcm 2) |
20 | 9, 13, 19 | 3eqtr3ri 2770 | . 2 ⊢ (3 lcm 2) = (3 · 2) |
21 | 3t2e6 12378 | . 2 ⊢ (3 · 2) = 6 | |
22 | 20, 21 | eqtri 2761 | 1 ⊢ (3 lcm 2) = 6 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 (class class class)co 7409 1c1 11111 · cmul 11115 ℕcn 12212 2c2 12267 3c3 12268 6c6 12271 ℕ0cn0 12472 ℤcz 12558 gcd cgcd 16435 lcm clcm 16525 ℙcprime 16608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198 df-gcd 16436 df-lcm 16527 df-prm 16609 |
This theorem is referenced by: lcm3un 40880 |
Copyright terms: Public domain | W3C validator |