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Mirrors > Home > MPE Home > Th. List > Mathboxes > aks4d1lem1 | Structured version Visualization version GIF version |
Description: Technical lemma to reduce proof size. (Contributed by metakunt, 14-Nov-2024.) |
Ref | Expression |
---|---|
aks4d1lem1.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) |
aks4d1lem1.2 | ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) |
Ref | Expression |
---|---|
aks4d1lem1 | ⊢ (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 12314 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
2 | 1 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ) |
3 | 2pos 12343 | . . . . . . . . 9 ⊢ 0 < 2 | |
4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 0 < 2) |
5 | aks4d1lem1.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) | |
6 | eluzelz 12860 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
7 | 5, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
8 | 7 | zred 12694 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
9 | 0red 11245 | . . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ) | |
10 | 3re 12320 | . . . . . . . . . 10 ⊢ 3 ∈ ℝ | |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 3 ∈ ℝ) |
12 | 3pos 12345 | . . . . . . . . . 10 ⊢ 0 < 3 | |
13 | 12 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 0 < 3) |
14 | eluzle 12863 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ≤ 𝑁) | |
15 | 5, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 3 ≤ 𝑁) |
16 | 9, 11, 8, 13, 15 | ltletrd 11402 | . . . . . . . 8 ⊢ (𝜑 → 0 < 𝑁) |
17 | 1red 11243 | . . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ) | |
18 | 1lt2 12411 | . . . . . . . . . . 11 ⊢ 1 < 2 | |
19 | 18 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → 1 < 2) |
20 | 17, 19 | ltned 11378 | . . . . . . . . 9 ⊢ (𝜑 → 1 ≠ 2) |
21 | 20 | necomd 2986 | . . . . . . . 8 ⊢ (𝜑 → 2 ≠ 1) |
22 | 2, 4, 8, 16, 21 | relogbcld 41498 | . . . . . . 7 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ) |
23 | 5nn0 12520 | . . . . . . . 8 ⊢ 5 ∈ ℕ0 | |
24 | 23 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 5 ∈ ℕ0) |
25 | 22, 24 | reexpcld 14157 | . . . . . 6 ⊢ (𝜑 → ((2 logb 𝑁)↑5) ∈ ℝ) |
26 | 25 | ceilcld 13838 | . . . . 5 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) ∈ ℤ) |
27 | 9re 12339 | . . . . . . 7 ⊢ 9 ∈ ℝ | |
28 | 27 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 9 ∈ ℝ) |
29 | 26 | zred 12694 | . . . . . 6 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) ∈ ℝ) |
30 | 9pos 12353 | . . . . . . 7 ⊢ 0 < 9 | |
31 | 30 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 9) |
32 | 8, 15 | 3lexlogpow5ineq4 41582 | . . . . . . 7 ⊢ (𝜑 → 9 < ((2 logb 𝑁)↑5)) |
33 | ceilge 13840 | . . . . . . . 8 ⊢ (((2 logb 𝑁)↑5) ∈ ℝ → ((2 logb 𝑁)↑5) ≤ (⌈‘((2 logb 𝑁)↑5))) | |
34 | 25, 33 | syl 17 | . . . . . . 7 ⊢ (𝜑 → ((2 logb 𝑁)↑5) ≤ (⌈‘((2 logb 𝑁)↑5))) |
35 | 28, 25, 29, 32, 34 | ltletrd 11402 | . . . . . 6 ⊢ (𝜑 → 9 < (⌈‘((2 logb 𝑁)↑5))) |
36 | 9, 28, 29, 31, 35 | lttrd 11403 | . . . . 5 ⊢ (𝜑 → 0 < (⌈‘((2 logb 𝑁)↑5))) |
37 | 26, 36 | jca 510 | . . . 4 ⊢ (𝜑 → ((⌈‘((2 logb 𝑁)↑5)) ∈ ℤ ∧ 0 < (⌈‘((2 logb 𝑁)↑5)))) |
38 | elnnz 12596 | . . . 4 ⊢ ((⌈‘((2 logb 𝑁)↑5)) ∈ ℕ ↔ ((⌈‘((2 logb 𝑁)↑5)) ∈ ℤ ∧ 0 < (⌈‘((2 logb 𝑁)↑5)))) | |
39 | 37, 38 | sylibr 233 | . . 3 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) ∈ ℕ) |
40 | aks4d1lem1.2 | . . . 4 ⊢ 𝐵 = (⌈‘((2 logb 𝑁)↑5)) | |
41 | 40 | eleq1i 2816 | . . 3 ⊢ (𝐵 ∈ ℕ ↔ (⌈‘((2 logb 𝑁)↑5)) ∈ ℕ) |
42 | 39, 41 | sylibr 233 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℕ) |
43 | 40 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 = (⌈‘((2 logb 𝑁)↑5))) |
44 | 35, 43 | breqtrrd 5171 | . 2 ⊢ (𝜑 → 9 < 𝐵) |
45 | 42, 44 | jca 510 | 1 ⊢ (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5143 ‘cfv 6542 (class class class)co 7415 ℝcr 11135 0cc0 11136 1c1 11137 < clt 11276 ≤ cle 11277 ℕcn 12240 2c2 12295 3c3 12296 5c5 12298 9c9 12302 ℕ0cn0 12500 ℤcz 12586 ℤ≥cuz 12850 ⌈cceil 13786 ↑cexp 14056 logb clogb 26712 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215 |
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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ioo 13358 df-ioc 13359 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-ceil 13788 df-mod 13865 df-seq 13997 df-exp 14057 df-fac 14263 df-bc 14292 df-hash 14320 df-shft 15044 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-limsup 15445 df-clim 15462 df-rlim 15463 df-sum 15663 df-ef 16041 df-sin 16043 df-cos 16044 df-pi 16046 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-rest 17401 df-topn 17402 df-0g 17420 df-gsum 17421 df-topgen 17422 df-pt 17423 df-prds 17426 df-xrs 17481 df-qtop 17486 df-imas 17487 df-xps 17489 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-mulg 19026 df-cntz 19270 df-cmn 19739 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-fbas 21278 df-fg 21279 df-cnfld 21282 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24242 df-ms 24243 df-tms 24244 df-cncf 24814 df-limc 25811 df-dv 25812 df-log 26506 df-cxp 26507 df-logb 26713 |
This theorem is referenced by: aks4d1p9 41614 |
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