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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrmxnn0 | Structured version Visualization version GIF version |
Description: The X-sequence is strictly monotonic on ℕ0. (Contributed by Stefan O'Rear, 4-Oct-2014.) |
Ref | Expression |
---|---|
ltrmxnn0 | ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝐴 Xrm 𝑀) < (𝐴 Xrm 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z 11993 | . . . . . 6 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℤ) | |
2 | frmx 39388 | . . . . . . 7 ⊢ Xrm :((ℤ≥‘2) × ℤ)⟶ℕ0 | |
3 | 2 | fovcl 7268 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℤ) → (𝐴 Xrm 𝑏) ∈ ℕ0) |
4 | 1, 3 | sylan2 592 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (𝐴 Xrm 𝑏) ∈ ℕ0) |
5 | 4 | nn0red 11944 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (𝐴 Xrm 𝑏) ∈ ℝ) |
6 | eluzelre 12242 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 𝐴 ∈ ℝ) | |
7 | 6 | adantr 481 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → 𝐴 ∈ ℝ) |
8 | 5, 7 | remulcld 10659 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → ((𝐴 Xrm 𝑏) · 𝐴) ∈ ℝ) |
9 | 1 | peano2zd 12078 | . . . . . 6 ⊢ (𝑏 ∈ ℕ0 → (𝑏 + 1) ∈ ℤ) |
10 | 2 | fovcl 7268 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑏 + 1) ∈ ℤ) → (𝐴 Xrm (𝑏 + 1)) ∈ ℕ0) |
11 | 9, 10 | sylan2 592 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (𝐴 Xrm (𝑏 + 1)) ∈ ℕ0) |
12 | 11 | nn0red 11944 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (𝐴 Xrm (𝑏 + 1)) ∈ ℝ) |
13 | eluz2b2 12309 | . . . . . . 7 ⊢ (𝐴 ∈ (ℤ≥‘2) ↔ (𝐴 ∈ ℕ ∧ 1 < 𝐴)) | |
14 | 13 | simprbi 497 | . . . . . 6 ⊢ (𝐴 ∈ (ℤ≥‘2) → 1 < 𝐴) |
15 | 14 | adantr 481 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → 1 < 𝐴) |
16 | rmxypos 39422 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (0 < (𝐴 Xrm 𝑏) ∧ 0 ≤ (𝐴 Yrm 𝑏))) | |
17 | 16 | simpld 495 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → 0 < (𝐴 Xrm 𝑏)) |
18 | ltmulgt11 11488 | . . . . . 6 ⊢ (((𝐴 Xrm 𝑏) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 0 < (𝐴 Xrm 𝑏)) → (1 < 𝐴 ↔ (𝐴 Xrm 𝑏) < ((𝐴 Xrm 𝑏) · 𝐴))) | |
19 | 5, 7, 17, 18 | syl3anc 1363 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (1 < 𝐴 ↔ (𝐴 Xrm 𝑏) < ((𝐴 Xrm 𝑏) · 𝐴))) |
20 | 15, 19 | mpbid 233 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (𝐴 Xrm 𝑏) < ((𝐴 Xrm 𝑏) · 𝐴)) |
21 | rmspecnonsq 39382 | . . . . . . . . . 10 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN)) | |
22 | 21 | eldifad 3945 | . . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ≥‘2) → ((𝐴↑2) − 1) ∈ ℕ) |
23 | 22 | adantr 481 | . . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → ((𝐴↑2) − 1) ∈ ℕ) |
24 | 23 | nnred 11641 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → ((𝐴↑2) − 1) ∈ ℝ) |
25 | frmy 39389 | . . . . . . . . . 10 ⊢ Yrm :((ℤ≥‘2) × ℤ)⟶ℤ | |
26 | 25 | fovcl 7268 | . . . . . . . . 9 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℤ) → (𝐴 Yrm 𝑏) ∈ ℤ) |
27 | 1, 26 | sylan2 592 | . . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (𝐴 Yrm 𝑏) ∈ ℤ) |
28 | 27 | zred 12075 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (𝐴 Yrm 𝑏) ∈ ℝ) |
29 | 23 | nnnn0d 11943 | . . . . . . . 8 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → ((𝐴↑2) − 1) ∈ ℕ0) |
30 | 29 | nn0ge0d 11946 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → 0 ≤ ((𝐴↑2) − 1)) |
31 | 16 | simprd 496 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → 0 ≤ (𝐴 Yrm 𝑏)) |
32 | 24, 28, 30, 31 | mulge0d 11205 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → 0 ≤ (((𝐴↑2) − 1) · (𝐴 Yrm 𝑏))) |
33 | 24, 28 | remulcld 10659 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (((𝐴↑2) − 1) · (𝐴 Yrm 𝑏)) ∈ ℝ) |
34 | 8, 33 | addge01d 11216 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (0 ≤ (((𝐴↑2) − 1) · (𝐴 Yrm 𝑏)) ↔ ((𝐴 Xrm 𝑏) · 𝐴) ≤ (((𝐴 Xrm 𝑏) · 𝐴) + (((𝐴↑2) − 1) · (𝐴 Yrm 𝑏))))) |
35 | 32, 34 | mpbid 233 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → ((𝐴 Xrm 𝑏) · 𝐴) ≤ (((𝐴 Xrm 𝑏) · 𝐴) + (((𝐴↑2) − 1) · (𝐴 Yrm 𝑏)))) |
36 | rmxp1 39407 | . . . . . 6 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℤ) → (𝐴 Xrm (𝑏 + 1)) = (((𝐴 Xrm 𝑏) · 𝐴) + (((𝐴↑2) − 1) · (𝐴 Yrm 𝑏)))) | |
37 | 1, 36 | sylan2 592 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (𝐴 Xrm (𝑏 + 1)) = (((𝐴 Xrm 𝑏) · 𝐴) + (((𝐴↑2) − 1) · (𝐴 Yrm 𝑏)))) |
38 | 35, 37 | breqtrrd 5085 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → ((𝐴 Xrm 𝑏) · 𝐴) ≤ (𝐴 Xrm (𝑏 + 1))) |
39 | 5, 8, 12, 20, 38 | ltletrd 10788 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑏 ∈ ℕ0) → (𝐴 Xrm 𝑏) < (𝐴 Xrm (𝑏 + 1))) |
40 | nn0z 11993 | . . . . 5 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℤ) | |
41 | 2 | fovcl 7268 | . . . . 5 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑎 ∈ ℤ) → (𝐴 Xrm 𝑎) ∈ ℕ0) |
42 | 40, 41 | sylan2 592 | . . . 4 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑎 ∈ ℕ0) → (𝐴 Xrm 𝑎) ∈ ℕ0) |
43 | 42 | nn0red 11944 | . . 3 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑎 ∈ ℕ0) → (𝐴 Xrm 𝑎) ∈ ℝ) |
44 | nn0uz 12268 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
45 | oveq2 7153 | . . 3 ⊢ (𝑎 = (𝑏 + 1) → (𝐴 Xrm 𝑎) = (𝐴 Xrm (𝑏 + 1))) | |
46 | oveq2 7153 | . . 3 ⊢ (𝑎 = 𝑏 → (𝐴 Xrm 𝑎) = (𝐴 Xrm 𝑏)) | |
47 | oveq2 7153 | . . 3 ⊢ (𝑎 = 𝑀 → (𝐴 Xrm 𝑎) = (𝐴 Xrm 𝑀)) | |
48 | oveq2 7153 | . . 3 ⊢ (𝑎 = 𝑁 → (𝐴 Xrm 𝑎) = (𝐴 Xrm 𝑁)) | |
49 | 39, 43, 44, 45, 46, 47, 48 | monotuz 39416 | . 2 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → (𝑀 < 𝑁 ↔ (𝐴 Xrm 𝑀) < (𝐴 Xrm 𝑁))) |
50 | 49 | 3impb 1107 | 1 ⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ (𝐴 Xrm 𝑀) < (𝐴 Xrm 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 < clt 10663 ≤ cle 10664 − cmin 10858 ℕcn 11626 2c2 11680 ℕ0cn0 11885 ℤcz 11969 ℤ≥cuz 12231 ↑cexp 13417 ◻NNcsquarenn 39311 Xrm crmx 39375 Yrm crmy 39376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-acn 9359 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-dvds 15596 df-gcd 15832 df-numer 16063 df-denom 16064 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 df-squarenn 39316 df-pell1qr 39317 df-pell14qr 39318 df-pell1234qr 39319 df-pellfund 39320 df-rmx 39377 df-rmy 39378 |
This theorem is referenced by: lermxnn0 39425 |
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