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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcm6un | Structured version Visualization version GIF version |
Description: Least common multiple of natural numbers up to 6 equals 60. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
lcm6un | ⊢ (lcm‘(1...6)) = ;60 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 12332 | . . 3 ⊢ 6 ∈ ℕ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (6 ∈ ℕ → 6 ∈ ℕ) |
3 | 2 | lcmfunnnd 41483 | . . 3 ⊢ (6 ∈ ℕ → (lcm‘(1...6)) = ((lcm‘(1...(6 − 1))) lcm 6)) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (lcm‘(1...6)) = ((lcm‘(1...(6 − 1))) lcm 6) |
5 | 6m1e5 12374 | . . . . . 6 ⊢ (6 − 1) = 5 | |
6 | 5 | oveq2i 7431 | . . . . 5 ⊢ (1...(6 − 1)) = (1...5) |
7 | 6 | fveq2i 6900 | . . . 4 ⊢ (lcm‘(1...(6 − 1))) = (lcm‘(1...5)) |
8 | 7 | oveq1i 7430 | . . 3 ⊢ ((lcm‘(1...(6 − 1))) lcm 6) = ((lcm‘(1...5)) lcm 6) |
9 | lcm5un 41488 | . . . 4 ⊢ (lcm‘(1...5)) = ;60 | |
10 | 9 | oveq1i 7430 | . . 3 ⊢ ((lcm‘(1...5)) lcm 6) = (;60 lcm 6) |
11 | 8, 10 | eqtri 2756 | . 2 ⊢ ((lcm‘(1...(6 − 1))) lcm 6) = (;60 lcm 6) |
12 | 60lcm6e60 41480 | . 2 ⊢ (;60 lcm 6) = ;60 | |
13 | 4, 11, 12 | 3eqtri 2760 | 1 ⊢ (lcm‘(1...6)) = ;60 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 0cc0 11139 1c1 11140 − cmin 11475 ℕcn 12243 5c5 12301 6c6 12302 ;cdc 12708 ...cfz 13517 lcm clcm 16559 lcmclcmf 16560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-rp 13008 df-fz 13518 df-fzo 13661 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-prod 15883 df-dvds 16232 df-gcd 16470 df-lcm 16561 df-lcmf 16562 df-prm 16643 |
This theorem is referenced by: lcm7un 41490 |
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