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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 5034 | . 2 ⊢ (𝑥 = 1 → (1 ≤ 𝑥 ↔ 1 ≤ 1)) | |
2 | breq2 5034 | . 2 ⊢ (𝑥 = 𝑦 → (1 ≤ 𝑥 ↔ 1 ≤ 𝑦)) | |
3 | breq2 5034 | . 2 ⊢ (𝑥 = (𝑦 + 1) → (1 ≤ 𝑥 ↔ 1 ≤ (𝑦 + 1))) | |
4 | breq2 5034 | . 2 ⊢ (𝑥 = 𝐴 → (1 ≤ 𝑥 ↔ 1 ≤ 𝐴)) | |
5 | 1le1 11257 | . 2 ⊢ 1 ≤ 1 | |
6 | nnre 11632 | . . 3 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
7 | recn 10616 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
8 | 7 | addid1d 10829 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) = 𝑦) |
9 | 8 | breq2d 5042 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) ↔ 1 ≤ 𝑦)) |
10 | 0lt1 11151 | . . . . . . . 8 ⊢ 0 < 1 | |
11 | 0re 10632 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
12 | 1re 10630 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
13 | axltadd 10703 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 < 1 → (𝑦 + 0) < (𝑦 + 1))) | |
14 | 11, 12, 13 | mp3an12 1448 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (0 < 1 → (𝑦 + 0) < (𝑦 + 1))) |
15 | 10, 14 | mpi 20 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) < (𝑦 + 1)) |
16 | readdcl 10609 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑦 + 0) ∈ ℝ) | |
17 | 11, 16 | mpan2 690 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) ∈ ℝ) |
18 | peano2re 10802 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ) | |
19 | lttr 10706 | . . . . . . . . 9 ⊢ (((𝑦 + 0) ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ ∧ 1 ∈ ℝ) → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) | |
20 | 12, 19 | mp3an3 1447 | . . . . . . . 8 ⊢ (((𝑦 + 0) ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) |
21 | 17, 18, 20 | syl2anc 587 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) |
22 | 15, 21 | mpand 694 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((𝑦 + 1) < 1 → (𝑦 + 0) < 1)) |
23 | 22 | con3d 155 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (¬ (𝑦 + 0) < 1 → ¬ (𝑦 + 1) < 1)) |
24 | lenlt 10708 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ (𝑦 + 0) ∈ ℝ) → (1 ≤ (𝑦 + 0) ↔ ¬ (𝑦 + 0) < 1)) | |
25 | 12, 17, 24 | sylancr 590 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) ↔ ¬ (𝑦 + 0) < 1)) |
26 | lenlt 10708 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) → (1 ≤ (𝑦 + 1) ↔ ¬ (𝑦 + 1) < 1)) | |
27 | 12, 18, 26 | sylancr 590 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 1) ↔ ¬ (𝑦 + 1) < 1)) |
28 | 23, 25, 27 | 3imtr4d 297 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) → 1 ≤ (𝑦 + 1))) |
29 | 9, 28 | sylbird 263 | . . 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 11643 | 1 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 ℕcn 11625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 |
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 13083 elfzom1elp1fzo 13099 fzo0sn0fzo1 13121 addmodlteq 13309 nnlesq 13564 digit1 13594 expnngt1 13598 faclbnd 13646 faclbnd3 13648 faclbnd4lem1 13649 faclbnd4lem4 13652 len0nnbi 13894 fstwrdne0 13899 divalglem1 15735 coprmgcdb 15983 isprm3 16017 pockthg 16232 infpn2 16239 setsstruct 16515 chfacfpmmulgsum2 21470 dscmet 23179 ovolunlem1a 24100 vitali 24217 plyeq0lem 24807 logtayllem 25250 leibpi 25528 vmalelog 25789 chtublem 25795 logfaclbnd 25806 bposlem1 25868 gausslemma2dlem1a 25949 dchrisum0lem1 26100 logdivbnd 26140 pntlemn 26184 ostth2lem3 26219 clwwisshclwwslem 27799 clwlknf1oclwwlknlem2 27867 clwlknf1oclwwlknlem3 27868 clwlknf1oclwwlkn 27869 nnmulge 30500 lmatfvlem 31168 eulerpartlems 31728 eulerpartlemb 31736 ballotlem2 31856 reprlt 32000 fz0n 33075 nndivlub 33919 knoppndvlem1 33964 knoppndvlem2 33965 knoppndvlem7 33970 knoppndvlem11 33974 knoppndvlem14 33977 lcmineqlem13 39329 metakunt26 39375 fzsplit1nn0 39695 pell1qrgaplem 39814 pellqrex 39820 monotoddzzfi 39883 jm2.23 39937 sumnnodd 42272 dvnmul 42585 wallispilem4 42710 wallispilem5 42711 wallispi 42712 wallispi2lem1 42713 stirlinglem5 42720 stirlinglem13 42728 dirkertrigeqlem1 42740 fouriersw 42873 etransclem24 42900 iccpartigtl 43940 fmtnodvds 44061 lighneallem2 44124 logbpw2m1 44981 blennnelnn 44990 blenpw2m1 44993 dignnld 45017 itcovalt2lem2lem1 45087 |
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