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Mirrors > Home > MPE Home > Th. List > zleltp1 | Structured version Visualization version GIF version |
Description: Integer ordering relation. (Contributed by NM, 10-May-2004.) |
Ref | Expression |
---|---|
zleltp1 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 11258 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
2 | zre 11258 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
3 | 1re 9918 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | leadd1 10375 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) | |
5 | 3, 4 | mp3an3 1405 | . . 3 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
6 | 1, 2, 5 | syl2an 493 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
7 | peano2z 11295 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
8 | zltp1le 11304 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → (𝑀 < (𝑁 + 1) ↔ (𝑀 + 1) ≤ (𝑁 + 1))) | |
9 | 7, 8 | sylan2 490 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < (𝑁 + 1) ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
10 | 6, 9 | bitr4d 270 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 |
This theorem is referenced by: zltlem1 11307 nnleltp1 11309 nn0leltp1 11313 suprzcl 11333 le9lt10 11405 declecOLD 11420 uzwo 11627 flge 12468 flhalf 12493 om2uzlti 12611 seqf1olem1 12702 fz1isolem 13102 hashtpg 13121 ltoddhalfle 14923 prmind2 15236 prm23lt5 15357 prmreclem2 15459 prmgaplem8 15600 chfacfisf 20478 chfacfisfcpmat 20479 chfacfscmulgsum 20484 chfacfpmmulgsum 20488 plyco0 23752 plydivex 23856 logf1o2 24196 ang180lem3 24341 basellem3 24609 ppieq0 24702 chpeq0 24733 bposlem1 24809 bposlem6 24814 dchrvmasumiflem1 24990 mulog2sumlem2 25024 1smat1 29198 ballotlemfc0 29881 ballotlemfcc 29882 poimirlem24 32603 poimirlem28 32607 fdc 32711 irrapxlem1 36404 pellexlem5 36415 jm2.24 36548 zltlesub 38438 dvnxpaek 38832 fourierdlem50 39049 zgeltp1eq 39943 odz2prm2pw 40013 fmtno4prmfac 40022 2pwp1prm 40041 nnsum3primesle9 40210 |
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