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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 12461 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
2 | zre 12461 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
3 | 1re 11113 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | leadd1 11581 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) | |
5 | 3, 4 | mp3an3 1450 | . . 3 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
6 | 1, 2, 5 | syl2an 596 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
7 | peano2z 12502 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
8 | zltp1le 12511 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → (𝑀 < (𝑁 + 1) ↔ (𝑀 + 1) ≤ (𝑁 + 1))) | |
9 | 7, 8 | sylan2 593 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < (𝑁 + 1) ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
10 | 6, 9 | bitr4d 281 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5103 (class class class)co 7351 ℝcr 11008 1c1 11010 + caddc 11012 < clt 11147 ≤ cle 11148 ℤcz 12457 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 df-z 12458 |
This theorem is referenced by: zltlem1 12514 nnleltp1 12516 nn0leltp1 12520 suprzcl 12541 le9lt10 12603 uzwo 12790 flge 13664 flhalf 13689 om2uzlti 13809 seqf1olem1 13901 fz1isolem 14314 hashtpg 14338 ltoddhalfle 16203 prmind2 16521 prm23lt5 16646 prmreclem2 16749 prmgaplem8 16890 chfacfisf 22155 chfacfisfcpmat 22156 chfacfscmulgsum 22161 chfacfpmmulgsum 22165 plyco0 25505 plydivex 25609 logf1o2 25957 ang180lem3 26113 basellem3 26384 ppieq0 26477 chpeq0 26508 bposlem1 26584 bposlem6 26589 dchrvmasumiflem1 26801 mulog2sumlem2 26835 dp2lt10 31566 1smat1 32197 ballotlemfc0 32904 ballotlemfcc 32905 poimirlem24 36040 poimirlem28 36044 fdc 36142 sticksstones10 40501 sticksstones12a 40503 sticksstones12 40504 sticksstones22 40514 irrapxlem1 41054 pellexlem5 41065 jm2.24 41196 zltlesub 43424 dvnxpaek 44084 fourierdlem50 44298 zgeltp1eq 45442 odz2prm2pw 45656 fmtno4prmfac 45665 2pwp1prm 45682 nnsum3primesle9 45887 |
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