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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcm2un | Structured version Visualization version GIF version |
Description: Least common multiple of natural numbers up to 2 equals 2. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
lcm2un | ⊢ (lcm‘(1...2)) = 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 12292 | . . . 4 ⊢ 2 ∈ ℕ | |
2 | id 22 | . . . . 5 ⊢ (2 ∈ ℕ → 2 ∈ ℕ) | |
3 | 2 | lcmfunnnd 41343 | . . . 4 ⊢ (2 ∈ ℕ → (lcm‘(1...2)) = ((lcm‘(1...(2 − 1))) lcm 2)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ (lcm‘(1...2)) = ((lcm‘(1...(2 − 1))) lcm 2) |
5 | 2m1e1 12345 | . . . . . 6 ⊢ (2 − 1) = 1 | |
6 | 5 | oveq2i 7423 | . . . . 5 ⊢ (1...(2 − 1)) = (1...1) |
7 | 6 | fveq2i 6894 | . . . 4 ⊢ (lcm‘(1...(2 − 1))) = (lcm‘(1...1)) |
8 | 7 | oveq1i 7422 | . . 3 ⊢ ((lcm‘(1...(2 − 1))) lcm 2) = ((lcm‘(1...1)) lcm 2) |
9 | 4, 8 | eqtri 2759 | . 2 ⊢ (lcm‘(1...2)) = ((lcm‘(1...1)) lcm 2) |
10 | lcm1un 41344 | . . . 4 ⊢ (lcm‘(1...1)) = 1 | |
11 | 10 | oveq1i 7422 | . . 3 ⊢ ((lcm‘(1...1)) lcm 2) = (1 lcm 2) |
12 | 1z 12599 | . . . . 5 ⊢ 1 ∈ ℤ | |
13 | 2z 12601 | . . . . 5 ⊢ 2 ∈ ℤ | |
14 | lcmcom 16537 | . . . . 5 ⊢ ((1 ∈ ℤ ∧ 2 ∈ ℤ) → (1 lcm 2) = (2 lcm 1)) | |
15 | 12, 13, 14 | mp2an 689 | . . . 4 ⊢ (1 lcm 2) = (2 lcm 1) |
16 | lcm1 16554 | . . . . . 6 ⊢ (2 ∈ ℤ → (2 lcm 1) = (abs‘2)) | |
17 | 13, 16 | ax-mp 5 | . . . . 5 ⊢ (2 lcm 1) = (abs‘2) |
18 | 2re 12293 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
19 | 0le2 12321 | . . . . . . 7 ⊢ 0 ≤ 2 | |
20 | 18, 19 | pm3.2i 470 | . . . . . 6 ⊢ (2 ∈ ℝ ∧ 0 ≤ 2) |
21 | absid 15250 | . . . . . 6 ⊢ ((2 ∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) | |
22 | 20, 21 | ax-mp 5 | . . . . 5 ⊢ (abs‘2) = 2 |
23 | 17, 22 | eqtri 2759 | . . . 4 ⊢ (2 lcm 1) = 2 |
24 | 15, 23 | eqtri 2759 | . . 3 ⊢ (1 lcm 2) = 2 |
25 | 11, 24 | eqtri 2759 | . 2 ⊢ ((lcm‘(1...1)) lcm 2) = 2 |
26 | 9, 25 | eqtri 2759 | 1 ⊢ (lcm‘(1...2)) = 2 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ℝcr 11115 0cc0 11116 1c1 11117 ≤ cle 11256 − cmin 11451 ℕcn 12219 2c2 12274 ℤcz 12565 ...cfz 13491 abscabs 15188 lcm clcm 16532 lcmclcmf 16533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-prod 15857 df-dvds 16205 df-gcd 16443 df-lcm 16534 df-lcmf 16535 |
This theorem is referenced by: lcm3un 41346 |
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