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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 12273 | . . 3 ⊢ 3 ∈ ℕ | |
2 | id 22 | . . . 4 ⊢ (3 ∈ ℕ → 3 ∈ ℕ) | |
3 | 2 | lcmfunnnd 40682 | . . 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 12322 | . . . . . . 7 ⊢ (3 − 1) = 2 | |
6 | 5 | oveq2i 7404 | . . . . . 6 ⊢ (1...(3 − 1)) = (1...2) |
7 | 6 | fveq2i 6881 | . . . . 5 ⊢ (lcm‘(1...(3 − 1))) = (lcm‘(1...2)) |
8 | lcm2un 40684 | . . . . 5 ⊢ (lcm‘(1...2)) = 2 | |
9 | 7, 8 | eqtri 2759 | . . . 4 ⊢ (lcm‘(1...(3 − 1))) = 2 |
10 | 9 | oveq1i 7403 | . . 3 ⊢ ((lcm‘(1...(3 − 1))) lcm 3) = (2 lcm 3) |
11 | 2z 12576 | . . . . . 6 ⊢ 2 ∈ ℤ | |
12 | 3z 12577 | . . . . . 6 ⊢ 3 ∈ ℤ | |
13 | 11, 12 | pm3.2i 471 | . . . . 5 ⊢ (2 ∈ ℤ ∧ 3 ∈ ℤ) |
14 | lcmcom 16512 | . . . . 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 16650 | . . . 4 ⊢ (3 lcm 2) = 6 | |
17 | 15, 16 | eqtri 2759 | . . 3 ⊢ (2 lcm 3) = 6 |
18 | 10, 17 | eqtri 2759 | . 2 ⊢ ((lcm‘(1...(3 − 1))) lcm 3) = 6 |
19 | 4, 18 | eqtri 2759 | 1 ⊢ (lcm‘(1...3)) = 6 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 1c1 11093 − cmin 11426 ℕcn 12194 2c2 12249 3c3 12250 6c6 12253 ℤcz 12540 ...cfz 13466 lcm clcm 16507 lcmclcmf 16508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-inf 9420 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-n0 12455 df-z 12541 df-uz 12805 df-rp 12957 df-fz 13467 df-fzo 13610 df-fl 13739 df-mod 13817 df-seq 13949 df-exp 14010 df-hash 14273 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-clim 15414 df-prod 15832 df-dvds 16180 df-gcd 16418 df-lcm 16509 df-lcmf 16510 df-prm 16591 |
This theorem is referenced by: lcm4un 40686 |
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