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Mirrors > Home > MPE Home > Th. List > Mathboxes > aks4d1p2 | Structured version Visualization version GIF version |
Description: Technical lemma for existence of non-divisor. (Contributed by metakunt, 27-Oct-2024.) |
Ref | Expression |
---|---|
aks4d1p2.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) |
aks4d1p2.2 | ⊢ 𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1)) |
aks4d1p2.3 | ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) |
Ref | Expression |
---|---|
aks4d1p2 | ⊢ (𝜑 → (2↑𝐵) ≤ (lcm‘(1...𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aks4d1p2.3 | . . . . . 6 ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐵 = (⌈‘((2 logb 𝑁)↑5))) |
3 | 2re 12324 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ) |
5 | 2pos 12353 | . . . . . . . . 9 ⊢ 0 < 2 | |
6 | 5 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 0 < 2) |
7 | aks4d1p2.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) | |
8 | eluzelz 12870 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
9 | 7, 8 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
10 | 9 | zred 12704 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
11 | 0red 11255 | . . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ) | |
12 | 3re 12330 | . . . . . . . . . 10 ⊢ 3 ∈ ℝ | |
13 | 12 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 3 ∈ ℝ) |
14 | 3pos 12355 | . . . . . . . . . 10 ⊢ 0 < 3 | |
15 | 14 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 0 < 3) |
16 | eluzle 12873 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ≤ 𝑁) | |
17 | 7, 16 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 3 ≤ 𝑁) |
18 | 11, 13, 10, 15, 17 | ltletrd 11412 | . . . . . . . 8 ⊢ (𝜑 → 0 < 𝑁) |
19 | 1red 11253 | . . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ) | |
20 | 1lt2 12421 | . . . . . . . . . . 11 ⊢ 1 < 2 | |
21 | 20 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → 1 < 2) |
22 | 19, 21 | ltned 11388 | . . . . . . . . 9 ⊢ (𝜑 → 1 ≠ 2) |
23 | 22 | necomd 2993 | . . . . . . . 8 ⊢ (𝜑 → 2 ≠ 1) |
24 | 4, 6, 10, 18, 23 | relogbcld 41475 | . . . . . . 7 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ) |
25 | 5nn0 12530 | . . . . . . . 8 ⊢ 5 ∈ ℕ0 | |
26 | 25 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 5 ∈ ℕ0) |
27 | 24, 26 | reexpcld 14167 | . . . . . 6 ⊢ (𝜑 → ((2 logb 𝑁)↑5) ∈ ℝ) |
28 | ceilcl 13847 | . . . . . 6 ⊢ (((2 logb 𝑁)↑5) ∈ ℝ → (⌈‘((2 logb 𝑁)↑5)) ∈ ℤ) | |
29 | 27, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) ∈ ℤ) |
30 | 2, 29 | eqeltrd 2829 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
31 | 29 | zred 12704 | . . . . . 6 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) ∈ ℝ) |
32 | 7re 12343 | . . . . . . . 8 ⊢ 7 ∈ ℝ | |
33 | 32 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 7 ∈ ℝ) |
34 | 7pos 12361 | . . . . . . . 8 ⊢ 0 < 7 | |
35 | 34 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 < 7) |
36 | 10, 17 | 3lexlogpow5ineq3 41560 | . . . . . . 7 ⊢ (𝜑 → 7 < ((2 logb 𝑁)↑5)) |
37 | 11, 33, 27, 35, 36 | lttrd 11413 | . . . . . 6 ⊢ (𝜑 → 0 < ((2 logb 𝑁)↑5)) |
38 | ceilge 13850 | . . . . . . 7 ⊢ (((2 logb 𝑁)↑5) ∈ ℝ → ((2 logb 𝑁)↑5) ≤ (⌈‘((2 logb 𝑁)↑5))) | |
39 | 27, 38 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((2 logb 𝑁)↑5) ≤ (⌈‘((2 logb 𝑁)↑5))) |
40 | 11, 27, 31, 37, 39 | ltletrd 11412 | . . . . 5 ⊢ (𝜑 → 0 < (⌈‘((2 logb 𝑁)↑5))) |
41 | 40, 2 | breqtrrd 5180 | . . . 4 ⊢ (𝜑 → 0 < 𝐵) |
42 | 30, 41 | jca 510 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ 0 < 𝐵)) |
43 | elnnz 12606 | . . 3 ⊢ (𝐵 ∈ ℕ ↔ (𝐵 ∈ ℤ ∧ 0 < 𝐵)) | |
44 | 42, 43 | sylibr 233 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℕ) |
45 | 33, 27, 36 | ltled 11400 | . . . 4 ⊢ (𝜑 → 7 ≤ ((2 logb 𝑁)↑5)) |
46 | 33, 27, 31, 45, 39 | letrd 11409 | . . 3 ⊢ (𝜑 → 7 ≤ (⌈‘((2 logb 𝑁)↑5))) |
47 | 46, 2 | breqtrrd 5180 | . 2 ⊢ (𝜑 → 7 ≤ 𝐵) |
48 | 44, 47 | lcmineqlem 41555 | 1 ⊢ (𝜑 → (2↑𝐵) ≤ (lcm‘(1...𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℝcr 11145 0cc0 11146 1c1 11147 · cmul 11151 < clt 11286 ≤ cle 11287 − cmin 11482 ℕcn 12250 2c2 12305 3c3 12306 5c5 12308 7c7 12310 ℕ0cn0 12510 ℤcz 12596 ℤ≥cuz 12860 ...cfz 13524 ⌊cfl 13795 ⌈cceil 13796 ↑cexp 14066 ∏cprod 15889 lcmclcmf 16567 logb clogb 26716 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cc 10466 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
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 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-symdif 4245 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 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 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-omul 8498 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-acn 9973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-ceil 13798 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-prod 15890 df-ef 16051 df-sin 16053 df-cos 16054 df-pi 16056 df-dvds 16239 df-gcd 16477 df-lcm 16568 df-lcmf 16569 df-prm 16650 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-cmp 23311 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-ovol 25413 df-vol 25414 df-mbf 25568 df-itg1 25569 df-itg2 25570 df-ibl 25571 df-itg 25572 df-0p 25619 df-limc 25815 df-dv 25816 df-log 26510 df-cxp 26511 df-logb 26717 |
This theorem is referenced by: aks4d1p3 41581 |
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