Mathbox for Stefan O'Rear |
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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 7166 | . . . . 5 ⊢ (𝑎 = 0 → (𝐴 Yrm 𝑎) = (𝐴 Yrm 0)) | |
3 | 1, 2 | breq12d 5081 | . . . 4 ⊢ (𝑎 = 0 → (𝑎 ≤ (𝐴 Yrm 𝑎) ↔ 0 ≤ (𝐴 Yrm 0))) |
4 | 3 | imbi2d 343 | . . 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 7166 | . . . . 5 ⊢ (𝑎 = 𝑏 → (𝐴 Yrm 𝑎) = (𝐴 Yrm 𝑏)) | |
7 | 5, 6 | breq12d 5081 | . . . 4 ⊢ (𝑎 = 𝑏 → (𝑎 ≤ (𝐴 Yrm 𝑎) ↔ 𝑏 ≤ (𝐴 Yrm 𝑏))) |
8 | 7 | imbi2d 343 | . . 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 7166 | . . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝐴 Yrm 𝑎) = (𝐴 Yrm (𝑏 + 1))) | |
11 | 9, 10 | breq12d 5081 | . . . 4 ⊢ (𝑎 = (𝑏 + 1) → (𝑎 ≤ (𝐴 Yrm 𝑎) ↔ (𝑏 + 1) ≤ (𝐴 Yrm (𝑏 + 1)))) |
12 | 11 | imbi2d 343 | . . 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 7166 | . . . . 5 ⊢ (𝑎 = 𝑁 → (𝐴 Yrm 𝑎) = (𝐴 Yrm 𝑁)) | |
15 | 13, 14 | breq12d 5081 | . . . 4 ⊢ (𝑎 = 𝑁 → (𝑎 ≤ (𝐴 Yrm 𝑎) ↔ 𝑁 ≤ (𝐴 Yrm 𝑁))) |
16 | 15 | imbi2d 343 | . . 3 ⊢ (𝑎 = 𝑁 → ((𝐴 ∈ (ℤ≥‘2) → 𝑎 ≤ (𝐴 Yrm 𝑎)) ↔ (𝐴 ∈ (ℤ≥‘2) → 𝑁 ≤ (𝐴 Yrm 𝑁)))) |
17 | 0le0 11741 | . . . 4 ⊢ 0 ≤ 0 | |
18 | rmy0 39533 | . . . 4 ⊢ (𝐴 ∈ (ℤ≥‘2) → (𝐴 Yrm 0) = 0) | |
19 | 17, 18 | breqtrrid 5106 | . . 3 ⊢ (𝐴 ∈ (ℤ≥‘2) → 0 ≤ (𝐴 Yrm 0)) |
20 | nn0z 12008 | . . . . . . . . 9 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℤ) | |
21 | 20 | 3ad2ant1 1129 | . . . . . . . 8 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 𝑏 ∈ ℤ) |
22 | 21 | peano2zd 12093 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝑏 + 1) ∈ ℤ) |
23 | 22 | zred 12090 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝑏 + 1) ∈ ℝ) |
24 | simp2 1133 | . . . . . . . . 9 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 𝐴 ∈ (ℤ≥‘2)) | |
25 | frmy 39518 | . . . . . . . . . 10 ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ | |
26 | 25 | fovcl 7281 | . . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℤ) → (𝐴 Yrm 𝑏) ∈ ℤ) |
27 | 24, 21, 26 | syl2anc 586 | . . . . . . . 8 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝐴 Yrm 𝑏) ∈ ℤ) |
28 | 27 | peano2zd 12093 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → ((𝐴 Yrm 𝑏) + 1) ∈ ℤ) |
29 | 28 | zred 12090 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → ((𝐴 Yrm 𝑏) + 1) ∈ ℝ) |
30 | 25 | fovcl 7281 | . . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑏 + 1) ∈ ℤ) → (𝐴 Yrm (𝑏 + 1)) ∈ ℤ) |
31 | 24, 22, 30 | syl2anc 586 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝐴 Yrm (𝑏 + 1)) ∈ ℤ) |
32 | 31 | zred 12090 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝐴 Yrm (𝑏 + 1)) ∈ ℝ) |
33 | nn0re 11909 | . . . . . . . 8 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℝ) | |
34 | 33 | 3ad2ant1 1129 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 𝑏 ∈ ℝ) |
35 | 27 | zred 12090 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝐴 Yrm 𝑏) ∈ ℝ) |
36 | 1red 10644 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 1 ∈ ℝ) | |
37 | simp3 1134 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 𝑏 ≤ (𝐴 Yrm 𝑏)) | |
38 | 34, 35, 36, 37 | leadd1dd 11256 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝑏 + 1) ≤ ((𝐴 Yrm 𝑏) + 1)) |
39 | 34 | ltp1d 11572 | . . . . . . . 8 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → 𝑏 < (𝑏 + 1)) |
40 | ltrmy 39556 | . . . . . . . . 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 1367 | . . . . . . . 8 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝑏 < (𝑏 + 1) ↔ (𝐴 Yrm 𝑏) < (𝐴 Yrm (𝑏 + 1)))) |
42 | 39, 41 | mpbid 234 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝐴 Yrm 𝑏) < (𝐴 Yrm (𝑏 + 1))) |
43 | zltp1le 12035 | . . . . . . . 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 586 | . . . . . . 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 234 | . . . . . 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 10799 | . . . . 5 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ≤ (𝐴 Yrm 𝑏)) → (𝑏 + 1) ≤ (𝐴 Yrm (𝑏 + 1))) |
47 | 46 | 3exp 1115 | . . . 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 12080 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐴 ∈ (ℤ≥‘2) → 𝑁 ≤ (𝐴 Yrm 𝑁))) |
50 | 49 | impcom 410 | 1 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝐴 Yrm 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 2c2 11695 ℕ0cn0 11900 ℤcz 11984 ℤ≥cuz 12246 Yrm crmy 39505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-dvds 15610 df-gcd 15846 df-numer 16077 df-denom 16078 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-squarenn 39445 df-pell1qr 39446 df-pell14qr 39447 df-pell1234qr 39448 df-pellfund 39449 df-rmx 39506 df-rmy 39507 |
This theorem is referenced by: jm2.27a 39609 jm2.27c 39611 expdiophlem1 39625 |
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