Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nnge1 | Structured version Visualization version GIF version |
Description: A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
nnge1 | ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5061 | . 2 ⊢ (𝑥 = 1 → (1 ≤ 𝑥 ↔ 1 ≤ 1)) | |
2 | breq2 5061 | . 2 ⊢ (𝑥 = 𝑦 → (1 ≤ 𝑥 ↔ 1 ≤ 𝑦)) | |
3 | breq2 5061 | . 2 ⊢ (𝑥 = (𝑦 + 1) → (1 ≤ 𝑥 ↔ 1 ≤ (𝑦 + 1))) | |
4 | breq2 5061 | . 2 ⊢ (𝑥 = 𝐴 → (1 ≤ 𝑥 ↔ 1 ≤ 𝐴)) | |
5 | 1le1 11256 | . 2 ⊢ 1 ≤ 1 | |
6 | nnre 11633 | . . 3 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
7 | recn 10615 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
8 | 7 | addid1d 10828 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) = 𝑦) |
9 | 8 | breq2d 5069 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) ↔ 1 ≤ 𝑦)) |
10 | 0lt1 11150 | . . . . . . . 8 ⊢ 0 < 1 | |
11 | 0re 10631 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
12 | 1re 10629 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
13 | axltadd 10702 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 < 1 → (𝑦 + 0) < (𝑦 + 1))) | |
14 | 11, 12, 13 | mp3an12 1442 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (0 < 1 → (𝑦 + 0) < (𝑦 + 1))) |
15 | 10, 14 | mpi 20 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) < (𝑦 + 1)) |
16 | readdcl 10608 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑦 + 0) ∈ ℝ) | |
17 | 11, 16 | mpan2 687 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) ∈ ℝ) |
18 | peano2re 10801 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ) | |
19 | lttr 10705 | . . . . . . . . 9 ⊢ (((𝑦 + 0) ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ ∧ 1 ∈ ℝ) → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) | |
20 | 12, 19 | mp3an3 1441 | . . . . . . . 8 ⊢ (((𝑦 + 0) ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) |
21 | 17, 18, 20 | syl2anc 584 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) |
22 | 15, 21 | mpand 691 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((𝑦 + 1) < 1 → (𝑦 + 0) < 1)) |
23 | 22 | con3d 155 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (¬ (𝑦 + 0) < 1 → ¬ (𝑦 + 1) < 1)) |
24 | lenlt 10707 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ (𝑦 + 0) ∈ ℝ) → (1 ≤ (𝑦 + 0) ↔ ¬ (𝑦 + 0) < 1)) | |
25 | 12, 17, 24 | sylancr 587 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) ↔ ¬ (𝑦 + 0) < 1)) |
26 | lenlt 10707 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) → (1 ≤ (𝑦 + 1) ↔ ¬ (𝑦 + 1) < 1)) | |
27 | 12, 18, 26 | sylancr 587 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 1) ↔ ¬ (𝑦 + 1) < 1)) |
28 | 23, 25, 27 | 3imtr4d 295 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) → 1 ≤ (𝑦 + 1))) |
29 | 9, 28 | sylbird 261 | . . 3 ⊢ (𝑦 ∈ ℝ → (1 ≤ 𝑦 → 1 ≤ (𝑦 + 1))) |
30 | 6, 29 | syl 17 | . 2 ⊢ (𝑦 ∈ ℕ → (1 ≤ 𝑦 → 1 ≤ (𝑦 + 1))) |
31 | 1, 2, 3, 4, 5, 30 | nnind 11644 | 1 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 < clt 10663 ≤ cle 10664 ℕcn 11626 |
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-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 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-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-iun 4912 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-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-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 |
This theorem is referenced by: nngt1ne1 11654 nnle1eq1 11655 nngt0 11656 nnnlt1 11657 nnrecgt0 11668 nnge1d 11673 elnnnn0c 11930 zle0orge1 11986 elnnz1 11996 zltp1le 12020 nn0ledivnn 12490 fzo1fzo0n0 13076 elfzom1elp1fzo 13092 fzo0sn0fzo1 13114 addmodlteq 13302 nnlesq 13556 digit1 13586 expnngt1 13590 faclbnd 13638 faclbnd3 13640 faclbnd4lem1 13641 faclbnd4lem4 13644 len0nnbi 13891 fstwrdne0 13896 divalglem1 15733 coprmgcdb 15981 isprm3 16015 pockthg 16230 infpn2 16237 setsstruct 16511 chfacfpmmulgsum2 21401 dscmet 23109 ovolunlem1a 24024 vitali 24141 plyeq0lem 24727 logtayllem 25169 leibpi 25447 vmalelog 25708 chtublem 25714 logfaclbnd 25725 bposlem1 25787 gausslemma2dlem1a 25868 dchrisum0lem1 26019 logdivbnd 26059 pntlemn 26103 ostth2lem3 26138 clwwisshclwwslem 27719 clwlknf1oclwwlknlem2 27788 clwlknf1oclwwlknlem3 27789 clwlknf1oclwwlkn 27790 nnmulge 30400 lmatfvlem 30979 eulerpartlems 31517 eulerpartlemb 31525 ballotlem2 31645 reprlt 31789 fz0n 32859 nndivlub 33703 knoppndvlem1 33748 knoppndvlem2 33749 knoppndvlem7 33754 knoppndvlem11 33758 knoppndvlem14 33761 fzsplit1nn0 39229 pell1qrgaplem 39348 pellqrex 39354 monotoddzzfi 39417 jm2.23 39471 sumnnodd 41787 dvnmul 42104 wallispilem4 42230 wallispilem5 42231 wallispi 42232 wallispi2lem1 42233 stirlinglem5 42240 stirlinglem13 42248 dirkertrigeqlem1 42260 fouriersw 42393 etransclem24 42420 iccpartigtl 43460 fmtnodvds 43583 lighneallem2 43648 logbpw2m1 44555 blennnelnn 44564 blenpw2m1 44567 dignnld 44591 |
Copyright terms: Public domain | W3C validator |