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Mirrors > Home > MPE Home > Th. List > prmgaplcmlem1 | Structured version Visualization version GIF version |
Description: Lemma for prmgaplcm 17057: 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 13550 | . . 3 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∈ ℤ) | |
2 | 1 | adantl 480 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∈ ℤ) |
3 | fzssz 13552 | . . . 4 ⊢ (1...𝑁) ⊆ ℤ | |
4 | fzfi 13987 | . . . 4 ⊢ (1...𝑁) ∈ Fin | |
5 | 3, 4 | pm3.2i 469 | . . 3 ⊢ ((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) |
6 | lcmfcl 16624 | . . . 4 ⊢ (((1...𝑁) ⊆ ℤ ∧ (1...𝑁) ∈ Fin) → (lcm‘(1...𝑁)) ∈ ℕ0) | |
7 | 6 | nn0zd 12631 | . . 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 5155 | . . 3 ⊢ (𝑥 = 𝐼 → (𝑥 ∥ (lcm‘(1...𝑁)) ↔ 𝐼 ∥ (lcm‘(1...𝑁)))) | |
10 | dvdslcmf 16627 | . . . 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 12925 | . . . . 5 ⊢ 2 ∈ (ℤ≥‘1) | |
13 | fzss1 13589 | . . . . 5 ⊢ (2 ∈ (ℤ≥‘1) → (2...𝑁) ⊆ (1...𝑁)) | |
14 | 12, 13 | mp1i 13 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2...𝑁) ⊆ (1...𝑁)) |
15 | 14 | sselda 3978 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∈ (1...𝑁)) |
16 | 9, 11, 15 | rspcdva 3608 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ (lcm‘(1...𝑁))) |
17 | iddvds 16267 | . . . 4 ⊢ (𝐼 ∈ ℤ → 𝐼 ∥ 𝐼) | |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝐼 ∈ (2...𝑁) → 𝐼 ∥ 𝐼) |
19 | 18 | adantl 480 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ 𝐼) |
20 | 2, 8, 2, 16, 19 | dvds2addd 16289 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ∀wral 3050 ⊆ wss 3946 class class class wbr 5152 ‘cfv 6553 (class class class)co 7423 Fincfn 8973 1c1 11155 + caddc 11157 ℕcn 12259 2c2 12314 ℤcz 12605 ℤ≥cuz 12869 ...cfz 13533 ∥ cdvds 16251 lcmclcmf 16585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-inf2 9680 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9481 df-inf 9482 df-oi 9549 df-card 9978 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-n0 12520 df-z 12606 df-uz 12870 df-rp 13024 df-fz 13534 df-fzo 13677 df-seq 14017 df-exp 14077 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-prod 15903 df-dvds 16252 df-lcmf 16587 |
This theorem is referenced by: prmgaplcmlem2 17049 |
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