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Mirrors > Home > MPE Home > Th. List > prmgaplcmlem1 | Structured version Visualization version GIF version |
Description: Lemma for prmgaplcm 16866: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.) |
Ref | Expression |
---|---|
prmgaplcmlem1 | ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelz 13369 | . . 3 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∈ ℤ) | |
2 | 1 | adantl 482 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∈ ℤ) |
3 | fzssz 13371 | . . . 4 ⊢ (1...𝑁) ⊆ ℤ | |
4 | fzfi 13805 | . . . 4 ⊢ (1...𝑁) ∈ Fin | |
5 | 3, 4 | pm3.2i 471 | . . 3 ⊢ ((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) |
6 | lcmfcl 16438 | . . . 4 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → (lcm‘(1...𝑁)) ∈ ℕ0) | |
7 | 6 | nn0zd 12537 | . . 3 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → (lcm‘(1...𝑁)) ∈ ℤ) |
8 | 5, 7 | mp1i 13 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → (lcm‘(1...𝑁)) ∈ ℤ) |
9 | breq1 5106 | . . 3 ⊢ (𝑥 = 𝐼 → (𝑥 ∥ (lcm‘(1...𝑁)) ↔ 𝐼 ∥ (lcm‘(1...𝑁)))) | |
10 | dvdslcmf 16441 | . . . 4 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁))) | |
11 | 5, 10 | mp1i 13 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ∀𝑥 ∈ (1...𝑁)𝑥 ∥ (lcm‘(1...𝑁))) |
12 | 2eluzge1 12747 | . . . . 5 ⊢ 2 ∈ (ℤ≥‘1) | |
13 | fzss1 13408 | . . . . 5 ⊢ (2 ∈ (ℤ≥‘1) → (2...𝑁) ⊆ (1...𝑁)) | |
14 | 12, 13 | mp1i 13 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2...𝑁) ⊆ (1...𝑁)) |
15 | 14 | sselda 3942 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∈ (1...𝑁)) |
16 | 9, 11, 15 | rspcdva 3580 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ (lcm‘(1...𝑁))) |
17 | iddvds 16086 | . . . 4 ⊢ (𝐼 ∈ ℤ → 𝐼 ∥ 𝐼) | |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∥ 𝐼) |
19 | 18 | adantl 482 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ 𝐼) |
20 | 2, 8, 2, 16, 19 | dvds2addd 16108 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3062 ⊆ wss 3908 class class class wbr 5103 ‘cfv 6491 (class class class)co 7349 Fincfn 8816 1c1 10985 + caddc 10987 ℕcn 12086 2c2 12141 ℤcz 12432 ℤ≥cuz 12695 ...cfz 13352 ∥ cdvds 16070 lcmclcmf 16399 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-inf2 9510 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 ax-pre-sup 11062 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-se 5586 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-1st 7911 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-sup 9311 df-inf 9312 df-oi 9379 df-card 9808 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-div 11746 df-nn 12087 df-2 12149 df-3 12150 df-n0 12347 df-z 12433 df-uz 12696 df-rp 12844 df-fz 13353 df-fzo 13496 df-seq 13835 df-exp 13896 df-hash 14158 df-cj 14917 df-re 14918 df-im 14919 df-sqrt 15053 df-abs 15054 df-clim 15304 df-prod 15723 df-dvds 16071 df-lcmf 16401 |
This theorem is referenced by: prmgaplcmlem2 16858 |
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