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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 11559 | . . . . . 6 ⊢ 2 ∈ ℝ | |
2 | 2lt3 11657 | . . . . . 6 ⊢ 2 < 3 | |
3 | 1, 2 | gtneii 10599 | . . . . 5 ⊢ 3 ≠ 2 |
4 | 3prm 15867 | . . . . . 6 ⊢ 3 ∈ ℙ | |
5 | 2prm 15865 | . . . . . 6 ⊢ 2 ∈ ℙ | |
6 | prmrp 15885 | . . . . . 6 ⊢ ((3 ∈ ℙ ∧ 2 ∈ ℙ) → ((3 gcd 2) = 1 ↔ 3 ≠ 2)) | |
7 | 4, 5, 6 | mp2an 688 | . . . . 5 ⊢ ((3 gcd 2) = 1 ↔ 3 ≠ 2) |
8 | 3, 7 | mpbir 232 | . . . 4 ⊢ (3 gcd 2) = 1 |
9 | 8 | oveq2i 7027 | . . 3 ⊢ ((3 lcm 2) · (3 gcd 2)) = ((3 lcm 2) · 1) |
10 | 3nn 11564 | . . . 4 ⊢ 3 ∈ ℕ | |
11 | 2nn 11558 | . . . 4 ⊢ 2 ∈ ℕ | |
12 | lcmgcdnn 15784 | . . . 4 ⊢ ((3 ∈ ℕ ∧ 2 ∈ ℕ) → ((3 lcm 2) · (3 gcd 2)) = (3 · 2)) | |
13 | 10, 11, 12 | mp2an 688 | . . 3 ⊢ ((3 lcm 2) · (3 gcd 2)) = (3 · 2) |
14 | 10 | nnzi 11855 | . . . . . 6 ⊢ 3 ∈ ℤ |
15 | 11 | nnzi 11855 | . . . . . 6 ⊢ 2 ∈ ℤ |
16 | lcmcl 15774 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → (3 lcm 2) ∈ ℕ0) | |
17 | 14, 15, 16 | mp2an 688 | . . . . 5 ⊢ (3 lcm 2) ∈ ℕ0 |
18 | 17 | nn0cni 11757 | . . . 4 ⊢ (3 lcm 2) ∈ ℂ |
19 | 18 | mulid1i 10491 | . . 3 ⊢ ((3 lcm 2) · 1) = (3 lcm 2) |
20 | 9, 13, 19 | 3eqtr3ri 2828 | . 2 ⊢ (3 lcm 2) = (3 · 2) |
21 | 3t2e6 11651 | . 2 ⊢ (3 · 2) = 6 | |
22 | 20, 21 | eqtri 2819 | 1 ⊢ (3 lcm 2) = 6 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 (class class class)co 7016 1c1 10384 · cmul 10388 ℕcn 11486 2c2 11540 3c3 11541 6c6 11544 ℕ0cn0 11745 ℤcz 11829 gcd cgcd 15676 lcm clcm 15761 ℙcprime 15844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-fz 12743 df-fl 13012 df-mod 13088 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-dvds 15441 df-gcd 15677 df-lcm 15763 df-prm 15845 |
This theorem is referenced by: (None) |
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