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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcm3un | Structured version Visualization version GIF version |
Description: Least common multiple of natural numbers up to 3 equals 6. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
lcm3un | ⊢ (lcm‘(1...3)) = 6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn 12290 | . . 3 ⊢ 3 ∈ ℕ | |
2 | id 22 | . . . 4 ⊢ (3 ∈ ℕ → 3 ∈ ℕ) | |
3 | 2 | lcmfunnnd 41384 | . . 3 ⊢ (3 ∈ ℕ → (lcm‘(1...3)) = ((lcm‘(1...(3 − 1))) lcm 3)) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (lcm‘(1...3)) = ((lcm‘(1...(3 − 1))) lcm 3) |
5 | 3m1e2 12339 | . . . . . . 7 ⊢ (3 − 1) = 2 | |
6 | 5 | oveq2i 7413 | . . . . . 6 ⊢ (1...(3 − 1)) = (1...2) |
7 | 6 | fveq2i 6885 | . . . . 5 ⊢ (lcm‘(1...(3 − 1))) = (lcm‘(1...2)) |
8 | lcm2un 41386 | . . . . 5 ⊢ (lcm‘(1...2)) = 2 | |
9 | 7, 8 | eqtri 2752 | . . . 4 ⊢ (lcm‘(1...(3 − 1))) = 2 |
10 | 9 | oveq1i 7412 | . . 3 ⊢ ((lcm‘(1...(3 − 1))) lcm 3) = (2 lcm 3) |
11 | 2z 12593 | . . . . . 6 ⊢ 2 ∈ ℤ | |
12 | 3z 12594 | . . . . . 6 ⊢ 3 ∈ ℤ | |
13 | 11, 12 | pm3.2i 470 | . . . . 5 ⊢ (2 ∈ ℤ ∧ 3 ∈ ℤ) |
14 | lcmcom 16533 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ) → (2 lcm 3) = (3 lcm 2)) | |
15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ (2 lcm 3) = (3 lcm 2) |
16 | 3lcm2e6 16673 | . . . 4 ⊢ (3 lcm 2) = 6 | |
17 | 15, 16 | eqtri 2752 | . . 3 ⊢ (2 lcm 3) = 6 |
18 | 10, 17 | eqtri 2752 | . 2 ⊢ ((lcm‘(1...(3 − 1))) lcm 3) = 6 |
19 | 4, 18 | eqtri 2752 | 1 ⊢ (lcm‘(1...3)) = 6 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 ‘cfv 6534 (class class class)co 7402 1c1 11108 − cmin 11443 ℕcn 12211 2c2 12266 3c3 12267 6c6 12270 ℤcz 12557 ...cfz 13485 lcm clcm 16528 lcmclcmf 16529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-fz 13486 df-fzo 13629 df-fl 13758 df-mod 13836 df-seq 13968 df-exp 14029 df-hash 14292 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-prod 15852 df-dvds 16201 df-gcd 16439 df-lcm 16530 df-lcmf 16531 df-prm 16612 |
This theorem is referenced by: lcm4un 41388 |
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