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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 11901 | . . . . . 6 ⊢ 2 ∈ ℝ | |
2 | 2lt3 11999 | . . . . . 6 ⊢ 2 < 3 | |
3 | 1, 2 | gtneii 10941 | . . . . 5 ⊢ 3 ≠ 2 |
4 | 3prm 16248 | . . . . . 6 ⊢ 3 ∈ ℙ | |
5 | 2prm 16246 | . . . . . 6 ⊢ 2 ∈ ℙ | |
6 | prmrp 16266 | . . . . . 6 ⊢ ((3 ∈ ℙ ∧ 2 ∈ ℙ) → ((3 gcd 2) = 1 ↔ 3 ≠ 2)) | |
7 | 4, 5, 6 | mp2an 692 | . . . . 5 ⊢ ((3 gcd 2) = 1 ↔ 3 ≠ 2) |
8 | 3, 7 | mpbir 234 | . . . 4 ⊢ (3 gcd 2) = 1 |
9 | 8 | oveq2i 7221 | . . 3 ⊢ ((3 lcm 2) · (3 gcd 2)) = ((3 lcm 2) · 1) |
10 | 3nn 11906 | . . . 4 ⊢ 3 ∈ ℕ | |
11 | 2nn 11900 | . . . 4 ⊢ 2 ∈ ℕ | |
12 | lcmgcdnn 16165 | . . . 4 ⊢ ((3 ∈ ℕ ∧ 2 ∈ ℕ) → ((3 lcm 2) · (3 gcd 2)) = (3 · 2)) | |
13 | 10, 11, 12 | mp2an 692 | . . 3 ⊢ ((3 lcm 2) · (3 gcd 2)) = (3 · 2) |
14 | 10 | nnzi 12198 | . . . . . 6 ⊢ 3 ∈ ℤ |
15 | 11 | nnzi 12198 | . . . . . 6 ⊢ 2 ∈ ℤ |
16 | lcmcl 16155 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → (3 lcm 2) ∈ ℕ0) | |
17 | 14, 15, 16 | mp2an 692 | . . . . 5 ⊢ (3 lcm 2) ∈ ℕ0 |
18 | 17 | nn0cni 12099 | . . . 4 ⊢ (3 lcm 2) ∈ ℂ |
19 | 18 | mulid1i 10834 | . . 3 ⊢ ((3 lcm 2) · 1) = (3 lcm 2) |
20 | 9, 13, 19 | 3eqtr3ri 2774 | . 2 ⊢ (3 lcm 2) = (3 · 2) |
21 | 3t2e6 11993 | . 2 ⊢ (3 · 2) = 6 | |
22 | 20, 21 | eqtri 2765 | 1 ⊢ (3 lcm 2) = 6 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2110 ≠ wne 2939 (class class class)co 7210 1c1 10727 · cmul 10731 ℕcn 11827 2c2 11882 3c3 11883 6c6 11886 ℕ0cn0 12087 ℤcz 12173 gcd cgcd 16050 lcm clcm 16142 ℙcprime 16225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-2o 8200 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-sup 9055 df-inf 9056 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-5 11893 df-6 11894 df-n0 12088 df-z 12174 df-uz 12436 df-rp 12584 df-fz 13093 df-fl 13364 df-mod 13440 df-seq 13572 df-exp 13633 df-cj 14659 df-re 14660 df-im 14661 df-sqrt 14795 df-abs 14796 df-dvds 15813 df-gcd 16051 df-lcm 16144 df-prm 16226 |
This theorem is referenced by: lcm3un 39755 |
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