Mathbox for metakunt |
< Previous
Next >
Nearby theorems |
||
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 12058 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
2 | 1 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ) |
3 | 2pos 12087 | . . . . . . . . 9 ⊢ 0 < 2 | |
4 | 3 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 0 < 2) |
5 | aks4d1lem1.1 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) | |
6 | eluzelz 12603 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
7 | 5, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
8 | 7 | zred 12437 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
9 | 0red 10989 | . . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ) | |
10 | 3re 12064 | . . . . . . . . . 10 ⊢ 3 ∈ ℝ | |
11 | 10 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 3 ∈ ℝ) |
12 | 3pos 12089 | . . . . . . . . . 10 ⊢ 0 < 3 | |
13 | 12 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → 0 < 3) |
14 | eluzle 12606 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ≤ 𝑁) | |
15 | 5, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 3 ≤ 𝑁) |
16 | 9, 11, 8, 13, 15 | ltletrd 11146 | . . . . . . . 8 ⊢ (𝜑 → 0 < 𝑁) |
17 | 1red 10987 | . . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ) | |
18 | 1lt2 12155 | . . . . . . . . . . 11 ⊢ 1 < 2 | |
19 | 18 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → 1 < 2) |
20 | 17, 19 | ltned 11122 | . . . . . . . . 9 ⊢ (𝜑 → 1 ≠ 2) |
21 | 20 | necomd 3001 | . . . . . . . 8 ⊢ (𝜑 → 2 ≠ 1) |
22 | 2, 4, 8, 16, 21 | relogbcld 39990 | . . . . . . 7 ⊢ (𝜑 → (2 logb 𝑁) ∈ ℝ) |
23 | 5nn0 12264 | . . . . . . . 8 ⊢ 5 ∈ ℕ0 | |
24 | 23 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 5 ∈ ℕ0) |
25 | 22, 24 | reexpcld 13892 | . . . . . 6 ⊢ (𝜑 → ((2 logb 𝑁)↑5) ∈ ℝ) |
26 | 25 | ceilcld 13574 | . . . . 5 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) ∈ ℤ) |
27 | 9re 12083 | . . . . . . 7 ⊢ 9 ∈ ℝ | |
28 | 27 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 9 ∈ ℝ) |
29 | 26 | zred 12437 | . . . . . 6 ⊢ (𝜑 → (⌈‘((2 logb 𝑁)↑5)) ∈ ℝ) |
30 | 9pos 12097 | . . . . . . 7 ⊢ 0 < 9 | |
31 | 30 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 < 9) |
32 | 8, 15 | 3lexlogpow5ineq4 40073 | . . . . . . 7 ⊢ (𝜑 → 9 < ((2 logb 𝑁)↑5)) |
33 | ceilge 13576 | . . . . . . . 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 11146 | . . . . . 6 ⊢ (𝜑 → 9 < (⌈‘((2 logb 𝑁)↑5))) |
36 | 9, 28, 29, 31, 35 | lttrd 11147 | . . . . 5 ⊢ (𝜑 → 0 < (⌈‘((2 logb 𝑁)↑5))) |
37 | 26, 36 | jca 512 | . . . 4 ⊢ (𝜑 → ((⌈‘((2 logb 𝑁)↑5)) ∈ ℤ ∧ 0 < (⌈‘((2 logb 𝑁)↑5)))) |
38 | elnnz 12340 | . . . 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 2831 | . . 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 5107 | . 2 ⊢ (𝜑 → 9 < 𝐵) |
45 | 42, 44 | jca 512 | 1 ⊢ (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 class class class wbr 5079 ‘cfv 6432 (class class class)co 7272 ℝcr 10881 0cc0 10882 1c1 10883 < clt 11020 ≤ cle 11021 ℕcn 11984 2c2 12039 3c3 12040 5c5 12042 9c9 12046 ℕ0cn0 12244 ℤcz 12330 ℤ≥cuz 12593 ⌈cceil 13522 ↑cexp 13793 logb clogb 25925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-inf2 9387 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 ax-addf 10961 ax-mulf 10962 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-of 7528 df-om 7708 df-1st 7825 df-2nd 7826 df-supp 7970 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-2o 8290 df-er 8490 df-map 8609 df-pm 8610 df-ixp 8678 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-uz 12594 df-q 12700 df-rp 12742 df-xneg 12859 df-xadd 12860 df-xmul 12861 df-ioo 13094 df-ioc 13095 df-ico 13096 df-icc 13097 df-fz 13251 df-fzo 13394 df-fl 13523 df-ceil 13524 df-mod 13601 df-seq 13733 df-exp 13794 df-fac 13999 df-bc 14028 df-hash 14056 df-shft 14789 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-limsup 15191 df-clim 15208 df-rlim 15209 df-sum 15409 df-ef 15788 df-sin 15790 df-cos 15791 df-pi 15793 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-starv 16988 df-sca 16989 df-vsca 16990 df-ip 16991 df-tset 16992 df-ple 16993 df-ds 16995 df-unif 16996 df-hom 16997 df-cco 16998 df-rest 17144 df-topn 17145 df-0g 17163 df-gsum 17164 df-topgen 17165 df-pt 17166 df-prds 17169 df-xrs 17224 df-qtop 17229 df-imas 17230 df-xps 17232 df-mre 17306 df-mrc 17307 df-acs 17309 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 df-mulg 18712 df-cntz 18934 df-cmn 19399 df-psmet 20600 df-xmet 20601 df-met 20602 df-bl 20603 df-mopn 20604 df-fbas 20605 df-fg 20606 df-cnfld 20609 df-top 22054 df-topon 22071 df-topsp 22093 df-bases 22107 df-cld 22181 df-ntr 22182 df-cls 22183 df-nei 22260 df-lp 22298 df-perf 22299 df-cn 22389 df-cnp 22390 df-haus 22477 df-tx 22724 df-hmeo 22917 df-fil 23008 df-fm 23100 df-flim 23101 df-flf 23102 df-xms 23484 df-ms 23485 df-tms 23486 df-cncf 24052 df-limc 25041 df-dv 25042 df-log 25723 df-cxp 25724 df-logb 25926 |
This theorem is referenced by: aks4d1p9 40105 |
Copyright terms: Public domain | W3C validator |