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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmygeid | Structured version Visualization version GIF version |
Description: Y(n) increases faster than n. Used implicitly without proof or comment in lemma 2.27 of [JonesMatijasevic] p. 697. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
Ref | Expression |
---|---|
rmygeid | ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝐴 Yrm 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ (𝑎 = 0 → 𝑎 = 0) | |
2 | oveq2 6698 | . . . . 5 ⊢ (𝑎 = 0 → (𝐴 Yrm 𝑎) = (𝐴 Yrm 0)) | |
3 | 1, 2 | breq12d 4698 | . . . 4 ⊢ (𝑎 = 0 → (𝑎 ≤ (𝐴 Yrm 𝑎) ↔ 0 ≤ (𝐴 Yrm 0))) |
4 | 3 | imbi2d 329 | . . 3 ⊢ (𝑎 = 0 → ((𝐴 ∈ (ℤ≥‘2) → 𝑎 ≤ (𝐴 Yrm 𝑎)) ↔ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ (𝐴 Yrm 0)))) |
5 | id 22 | . . . . 5 ⊢ (𝑎 = 𝑏 → 𝑎 = 𝑏) | |
6 | oveq2 6698 | . . . . 5 ⊢ (𝑎 = 𝑏 → (𝐴 Yrm 𝑎) = (𝐴 Yrm 𝑏)) | |
7 | 5, 6 | breq12d 4698 | . . . 4 ⊢ (𝑎 = 𝑏 → (𝑎 ≤ (𝐴 Yrm 𝑎) ↔ 𝑏 ≤ (𝐴 Yrm 𝑏))) |
8 | 7 | imbi2d 329 | . . 3 ⊢ (𝑎 = 𝑏 → ((𝐴 ∈ (ℤ≥‘2) → 𝑎 ≤ (𝐴 Yrm 𝑎)) ↔ (𝐴 ∈ (ℤ≥‘2) → 𝑏 ≤ (𝐴 Yrm 𝑏)))) |
9 | id 22 | . . . . 5 ⊢ (𝑎 = (𝑏 + 1) → 𝑎 = (𝑏 + 1)) | |
10 | oveq2 6698 | . . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝐴 Yrm 𝑎) = (𝐴 Yrm (𝑏 + 1))) | |
11 | 9, 10 | breq12d 4698 | . . . 4 ⊢ (𝑎 = (𝑏 + 1) → (𝑎 ≤ (𝐴 Yrm 𝑎) ↔ (𝑏 + 1) ≤ (𝐴 Yrm (𝑏 + 1)))) |
12 | 11 | imbi2d 329 | . . 3 ⊢ (𝑎 = (𝑏 + 1) → ((𝐴 ∈ (ℤ≥‘2) → 𝑎 ≤ (𝐴 Yrm 𝑎)) ↔ (𝐴 ∈ (ℤ≥‘2) → (𝑏 + 1) ≤ (𝐴 Yrm (𝑏 + 1))))) |
13 | id 22 | . . . . 5 ⊢ (𝑎 = 𝑁 → 𝑎 = 𝑁) | |
14 | oveq2 6698 | . . . . 5 ⊢ (𝑎 = 𝑁 → (𝐴 Yrm 𝑎) = (𝐴 Yrm 𝑁)) | |
15 | 13, 14 | breq12d 4698 | . . . 4 ⊢ (𝑎 = 𝑁 → (𝑎 ≤ (𝐴 Yrm 𝑎) ↔ 𝑁 ≤ (𝐴 Yrm 𝑁))) |
16 | 15 | imbi2d 329 | . . 3 ⊢ (𝑎 = 𝑁 → ((𝐴 ∈ (ℤ≥‘2) → 𝑎 ≤ (𝐴 Yrm 𝑎)) ↔ (𝐴 ∈ (ℤ≥‘2) → 𝑁 ≤ (𝐴 Yrm 𝑁)))) |
17 | 0le0 11148 | . . . 4 ⊢ 0 ≤ 0 | |
18 | rmy0 37811 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 Yrm 0) = 0) | |
19 | 17, 18 | syl5breqr 4723 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ (𝐴 Yrm 0)) |
20 | nn0z 11438 | . . . . . . . . 9 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℤ) | |
21 | 20 | 3ad2ant1 1102 | . . . . . . . 8 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 𝑏 ∈ ℤ) |
22 | 21 | peano2zd 11523 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝑏 + 1) ∈ ℤ) |
23 | 22 | zred 11520 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝑏 + 1) ∈ ℝ) |
24 | simp2 1082 | . . . . . . . . 9 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 𝐴 ∈ (ℤ≥‘2)) | |
25 | frmy 37796 | . . . . . . . . . 10 ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ | |
26 | 25 | fovcl 6807 | . . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℤ) → (𝐴 Yrm 𝑏) ∈ ℤ) |
27 | 24, 21, 26 | syl2anc 694 | . . . . . . . 8 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝐴 Yrm 𝑏) ∈ ℤ) |
28 | 27 | peano2zd 11523 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → ((𝐴 Yrm 𝑏) + 1) ∈ ℤ) |
29 | 28 | zred 11520 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → ((𝐴 Yrm 𝑏) + 1) ∈ ℝ) |
30 | 25 | fovcl 6807 | . . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑏 + 1) ∈ ℤ) → (𝐴 Yrm (𝑏 + 1)) ∈ ℤ) |
31 | 24, 22, 30 | syl2anc 694 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝐴 Yrm (𝑏 + 1)) ∈ ℤ) |
32 | 31 | zred 11520 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝐴 Yrm (𝑏 + 1)) ∈ ℝ) |
33 | nn0re 11339 | . . . . . . . 8 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℝ) | |
34 | 33 | 3ad2ant1 1102 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 𝑏 ∈ ℝ) |
35 | 27 | zred 11520 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝐴 Yrm 𝑏) ∈ ℝ) |
36 | 1red 10093 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 1 ∈ ℝ) | |
37 | simp3 1083 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 𝑏 ≤ (𝐴 Yrm 𝑏)) | |
38 | 34, 35, 36, 37 | leadd1dd 10679 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝑏 + 1) ≤ ((𝐴 Yrm 𝑏) + 1)) |
39 | 34 | ltp1d 10992 | . . . . . . . 8 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 𝑏 < (𝑏 + 1)) |
40 | ltrmy 37836 | . . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℤ ∧ (𝑏 + 1) ∈ ℤ) → (𝑏 < (𝑏 + 1) ↔ (𝐴 Yrm 𝑏) < (𝐴 Yrm (𝑏 + 1)))) | |
41 | 24, 21, 22, 40 | syl3anc 1366 | . . . . . . . 8 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝑏 < (𝑏 + 1) ↔ (𝐴 Yrm 𝑏) < (𝐴 Yrm (𝑏 + 1)))) |
42 | 39, 41 | mpbid 222 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝐴 Yrm 𝑏) < (𝐴 Yrm (𝑏 + 1))) |
43 | zltp1le 11465 | . . . . . . . 8 ⊢ (((𝐴 Yrm 𝑏) ∈ ℤ ∧ (𝐴 Yrm (𝑏 + 1)) ∈ ℤ) → ((𝐴 Yrm 𝑏) < (𝐴 Yrm (𝑏 + 1)) ↔ ((𝐴 Yrm 𝑏) + 1) ≤ (𝐴 Yrm (𝑏 + 1)))) | |
44 | 27, 31, 43 | syl2anc 694 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → ((𝐴 Yrm 𝑏) < (𝐴 Yrm (𝑏 + 1)) ↔ ((𝐴 Yrm 𝑏) + 1) ≤ (𝐴 Yrm (𝑏 + 1)))) |
45 | 42, 44 | mpbid 222 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → ((𝐴 Yrm 𝑏) + 1) ≤ (𝐴 Yrm (𝑏 + 1))) |
46 | 23, 29, 32, 38, 45 | letrd 10232 | . . . . 5 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝑏 + 1) ≤ (𝐴 Yrm (𝑏 + 1))) |
47 | 46 | 3exp 1283 | . . . 4 ⊢ (𝑏 ∈ ℕ0 → (𝐴 ∈ (ℤ≥‘2) → (𝑏 ≤ (𝐴 Yrm 𝑏) → (𝑏 + 1) ≤ (𝐴 Yrm (𝑏 + 1))))) |
48 | 47 | a2d 29 | . . 3 ⊢ (𝑏 ∈ ℕ0 → ((𝐴 ∈ (ℤ≥‘2) → 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝐴 ∈ (ℤ≥‘2) → (𝑏 + 1) ≤ (𝐴 Yrm (𝑏 + 1))))) |
49 | 4, 8, 12, 16, 19, 48 | nn0ind 11510 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐴 ∈ (ℤ≥‘2) → 𝑁 ≤ (𝐴 Yrm 𝑁))) |
50 | 49 | impcom 445 | 1 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝐴 Yrm 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ℝcr 9973 0cc0 9974 1c1 9975 + caddc 9977 < clt 10112 ≤ cle 10113 2c2 11108 ℕ0cn0 11330 ℤcz 11415 ℤ≥cuz 11725 Yrm crmy 37782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-omul 7610 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-acn 8806 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-xnn0 11402 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-sin 14844 df-cos 14845 df-pi 14847 df-dvds 15028 df-gcd 15264 df-numer 15490 df-denom 15491 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676 df-log 24348 df-squarenn 37722 df-pell1qr 37723 df-pell14qr 37724 df-pell1234qr 37725 df-pellfund 37726 df-rmx 37783 df-rmy 37784 |
This theorem is referenced by: jm2.27a 37889 jm2.27c 37891 expdiophlem1 37905 |
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