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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcmfunnnd | Structured version Visualization version GIF version |
Description: Useful equation to calculate the least common multiple of 1 to n. (Contributed by metakunt, 29-Apr-2024.) |
Ref | Expression |
---|---|
lcmfunnnd.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
lcmfunnnd | ⊢ (𝜑 → (lcm‘(1...𝑁)) = ((lcm‘(1...(𝑁 − 1))) lcm 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmfunnnd.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
2 | 1 | nncnd 11640 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
3 | 1cnd 10622 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ) | |
4 | 2, 3 | npcand 10987 | . . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
5 | 4 | oveq2d 7158 | . . . . 5 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
6 | nnm1nn0 11925 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
7 | 1, 6 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0) |
8 | nn0uz 12267 | . . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 8 | eleq2i 2904 | . . . . . . . 8 ⊢ ((𝑁 − 1) ∈ ℕ0 ↔ (𝑁 − 1) ∈ (ℤ≥‘0)) |
10 | 7, 9 | sylib 220 | . . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘0)) |
11 | 1m1e0 11696 | . . . . . . . . . 10 ⊢ (1 − 1) = 0 | |
12 | 11 | fveq2i 6659 | . . . . . . . . 9 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0) |
13 | 12 | eleq2i 2904 | . . . . . . . 8 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(1 − 1)) ↔ (𝑁 − 1) ∈ (ℤ≥‘0)) |
14 | 13 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ((𝑁 − 1) ∈ (ℤ≥‘(1 − 1)) ↔ (𝑁 − 1) ∈ (ℤ≥‘0))) |
15 | 10, 14 | mpbird 259 | . . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘(1 − 1))) |
16 | 1z 11999 | . . . . . . 7 ⊢ 1 ∈ ℤ | |
17 | fzsuc2 12955 | . . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (𝑁 − 1) ∈ (ℤ≥‘(1 − 1))) → (1...((𝑁 − 1) + 1)) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) | |
18 | 16, 17 | mpan 688 | . . . . . 6 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(1 − 1)) → (1...((𝑁 − 1) + 1)) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
19 | 15, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
20 | 5, 19 | eqtr3d 2858 | . . . 4 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)})) |
21 | 4 | sneqd 4565 | . . . . 5 ⊢ (𝜑 → {((𝑁 − 1) + 1)} = {𝑁}) |
22 | 21 | uneq2d 4127 | . . . 4 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {((𝑁 − 1) + 1)}) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
23 | 20, 22 | eqtrd 2856 | . . 3 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁})) |
24 | 23 | fveq2d 6660 | . 2 ⊢ (𝜑 → (lcm‘(1...𝑁)) = (lcm‘((1...(𝑁 − 1)) ∪ {𝑁}))) |
25 | fzssz 12899 | . . . . 5 ⊢ (1...(𝑁 − 1)) ⊆ ℤ | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ ℤ) |
27 | fzfi 13330 | . . . . 5 ⊢ (1...(𝑁 − 1)) ∈ Fin | |
28 | 27 | a1i 11 | . . . 4 ⊢ (𝜑 → (1...(𝑁 − 1)) ∈ Fin) |
29 | nnz 11991 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
30 | 1, 29 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
31 | 26, 28, 30 | 3jca 1124 | . . 3 ⊢ (𝜑 → ((1...(𝑁 − 1)) ⊆ ℤ ∧ (1...(𝑁 − 1)) ∈ Fin ∧ 𝑁 ∈ ℤ)) |
32 | lcmfunsn 15971 | . . 3 ⊢ (((1...(𝑁 − 1)) ⊆ ℤ ∧ (1...(𝑁 − 1)) ∈ Fin ∧ 𝑁 ∈ ℤ) → (lcm‘((1...(𝑁 − 1)) ∪ {𝑁})) = ((lcm‘(1...(𝑁 − 1))) lcm 𝑁)) | |
33 | 31, 32 | syl 17 | . 2 ⊢ (𝜑 → (lcm‘((1...(𝑁 − 1)) ∪ {𝑁})) = ((lcm‘(1...(𝑁 − 1))) lcm 𝑁)) |
34 | 24, 33 | eqtrd 2856 | 1 ⊢ (𝜑 → (lcm‘(1...𝑁)) = ((lcm‘(1...(𝑁 − 1))) lcm 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∪ cun 3922 ⊆ wss 3924 {csn 4553 ‘cfv 6341 (class class class)co 7142 Fincfn 8495 0cc0 10523 1c1 10524 + caddc 10526 − cmin 10856 ℕcn 11624 ℕ0cn0 11884 ℤcz 11968 ℤ≥cuz 12230 ...cfz 12882 lcm clcm 15915 lcmclcmf 15916 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-inf2 9090 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-se 5501 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-isom 6350 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-sup 8892 df-inf 8893 df-oi 8960 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-n0 11885 df-z 11969 df-uz 12231 df-rp 12377 df-fz 12883 df-fzo 13024 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-clim 14830 df-prod 15245 df-dvds 15593 df-gcd 15827 df-lcm 15917 df-lcmf 15918 |
This theorem is referenced by: lcm1un 39160 lcm2un 39161 lcm3un 39162 lcm4un 39163 lcm5un 39164 lcm6un 39165 lcm7un 39166 lcm8un 39167 |
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