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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcm1un | Structured version Visualization version GIF version |
Description: Least common multiple of natural numbers up to 1 equals 1. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
lcm1un | ⊢ (lcm‘(1...1)) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 12306 | . . 3 ⊢ 1 ∈ ℕ | |
2 | id 22 | . . . 4 ⊢ (1 ∈ ℕ → 1 ∈ ℕ) | |
3 | 2 | lcmfunnnd 41971 | . . 3 ⊢ (1 ∈ ℕ → (lcm‘(1...1)) = ((lcm‘(1...(1 − 1))) lcm 1)) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (lcm‘(1...1)) = ((lcm‘(1...(1 − 1))) lcm 1) |
5 | 1m1e0 12367 | . . . . . . . 8 ⊢ (1 − 1) = 0 | |
6 | 5 | oveq2i 7461 | . . . . . . 7 ⊢ (1...(1 − 1)) = (1...0) |
7 | fz10 13607 | . . . . . . 7 ⊢ (1...0) = ∅ | |
8 | 6, 7 | eqtri 2768 | . . . . . 6 ⊢ (1...(1 − 1)) = ∅ |
9 | 8 | fveq2i 6925 | . . . . 5 ⊢ (lcm‘(1...(1 − 1))) = (lcm‘∅) |
10 | lcmf0 16683 | . . . . 5 ⊢ (lcm‘∅) = 1 | |
11 | 9, 10 | eqtri 2768 | . . . 4 ⊢ (lcm‘(1...(1 − 1))) = 1 |
12 | 11 | oveq1i 7460 | . . 3 ⊢ ((lcm‘(1...(1 − 1))) lcm 1) = (1 lcm 1) |
13 | 1z 12675 | . . . . 5 ⊢ 1 ∈ ℤ | |
14 | lcmid 16658 | . . . . 5 ⊢ (1 ∈ ℤ → (1 lcm 1) = (abs‘1)) | |
15 | 13, 14 | ax-mp 5 | . . . 4 ⊢ (1 lcm 1) = (abs‘1) |
16 | abs1 15348 | . . . 4 ⊢ (abs‘1) = 1 | |
17 | 15, 16 | eqtri 2768 | . . 3 ⊢ (1 lcm 1) = 1 |
18 | 12, 17 | eqtri 2768 | . 2 ⊢ ((lcm‘(1...(1 − 1))) lcm 1) = 1 |
19 | 4, 18 | eqtri 2768 | 1 ⊢ (lcm‘(1...1)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2108 ∅c0 4352 ‘cfv 6575 (class class class)co 7450 0cc0 11186 1c1 11187 − cmin 11522 ℕcn 12295 ℤcz 12641 ...cfz 13569 abscabs 15285 lcm clcm 16637 lcmclcmf 16638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-sup 9513 df-inf 9514 df-oi 9581 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-3 12359 df-n0 12556 df-z 12642 df-uz 12906 df-rp 13060 df-fz 13570 df-fzo 13714 df-fl 13845 df-mod 13923 df-seq 14055 df-exp 14115 df-hash 14382 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15536 df-prod 15954 df-dvds 16305 df-gcd 16543 df-lcm 16639 df-lcmf 16640 |
This theorem is referenced by: lcm2un 41973 |
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