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Mirrors > Home > MPE Home > Th. List > zltp1le | Structured version Visualization version GIF version |
Description: Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
zltp1le | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1 11668 | . . . 4 ⊢ ((𝑁 − 𝑀) ∈ ℕ → 1 ≤ (𝑁 − 𝑀)) | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 − 𝑀) ∈ ℕ → 1 ≤ (𝑁 − 𝑀))) |
3 | znnsub 12031 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ)) | |
4 | zre 11988 | . . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
5 | zre 11988 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | 1re 10643 | . . . . 5 ⊢ 1 ∈ ℝ | |
7 | leaddsub2 11119 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 + 1) ≤ 𝑁 ↔ 1 ≤ (𝑁 − 𝑀))) | |
8 | 6, 7 | mp3an2 1445 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 + 1) ≤ 𝑁 ↔ 1 ≤ (𝑁 − 𝑀))) |
9 | 4, 5, 8 | syl2an 597 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 + 1) ≤ 𝑁 ↔ 1 ≤ (𝑁 − 𝑀))) |
10 | 2, 3, 9 | 3imtr4d 296 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 → (𝑀 + 1) ≤ 𝑁)) |
11 | 4 | adantr 483 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ) |
12 | 11 | ltp1d 11572 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 < (𝑀 + 1)) |
13 | peano2re 10815 | . . . . 5 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ) | |
14 | 11, 13 | syl 17 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 1) ∈ ℝ) |
15 | 5 | adantl 484 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ) |
16 | ltletr 10734 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 < (𝑀 + 1) ∧ (𝑀 + 1) ≤ 𝑁) → 𝑀 < 𝑁)) | |
17 | 11, 14, 15, 16 | syl3anc 1367 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 < (𝑀 + 1) ∧ (𝑀 + 1) ≤ 𝑁) → 𝑀 < 𝑁)) |
18 | 12, 17 | mpand 693 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 + 1) ≤ 𝑁 → 𝑀 < 𝑁)) |
19 | 10, 18 | impbid 214 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 − cmin 10872 ℕcn 11640 ℤcz 11984 |
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-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 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-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-iun 4923 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-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-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 |
This theorem is referenced by: zleltp1 12036 zlem1lt 12037 zgt0ge1 12039 nnltp1le 12041 nn0ltp1le 12043 btwnnz 12061 uzind2 12078 fzind 12083 eluzp1l 12272 eluz2b1 12322 zltaddlt1le 12893 fzsplit2 12935 m1modge3gt1 13289 bcval5 13681 seqcoll 13825 hashge2el2dif 13841 hashge2el2difr 13842 swrd2lsw 14316 2swrd2eqwrdeq 14317 isercoll 15026 nn0o1gt2 15734 divalglem6 15751 isprm3 16029 dvdsnprmd 16036 2mulprm 16039 oddprmge3 16046 ge2nprmge4 16047 hashdvds 16114 prmreclem5 16258 prmgaplem3 16391 prmgaplem5 16393 prmgaplem6 16394 prmgaplem8 16396 sylow1lem3 18727 chfacfscmul0 21468 chfacfscmulfsupp 21469 chfacfpmmul0 21472 chfacfpmmulfsupp 21473 dyaddisjlem 24198 plyeq0lem 24802 basellem2 25661 chtub 25790 bposlem9 25870 lgsdilem2 25911 lgsquadlem1 25958 2lgslem1a 25969 pntpbnd1 26164 pntpbnd2 26165 tgldimor 26290 eucrct2eupth 28026 konigsberglem5 28037 nndiffz1 30511 ltesubnnd 30540 dp2ltc 30565 smatrcl 31063 breprexplemc 31905 zltp1ne 32350 dnibndlem13 33831 knoppndvlem6 33858 poimirlem3 34897 poimirlem4 34898 poimirlem15 34909 poimirlem17 34911 poimirlem28 34922 ellz1 39371 lzunuz 39372 rmygeid 39568 jm3.1lem2 39622 bccbc 40684 elfzop1le2 41563 monoords 41571 fmul01lt1lem1 41872 dvnxpaek 42234 iblspltprt 42265 itgspltprt 42271 fourierdlem6 42405 fourierdlem12 42411 fourierdlem19 42418 fourierdlem42 42441 fourierdlem48 42446 fourierdlem49 42447 fourierdlem79 42477 iccpartiltu 43589 iccpartgt 43594 icceuelpartlem 43602 iccpartnel 43605 lighneallem4b 43781 evenltle 43889 gbowge7 43935 gbege6 43937 stgoldbwt 43948 sbgoldbwt 43949 sbgoldbalt 43953 sbgoldbm 43956 bgoldbtbndlem1 43977 tgblthelfgott 43987 elfzolborelfzop1 44581 |
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