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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 12499 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
2 | zre 12499 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
3 | 1re 11151 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | leadd1 11619 | . . . 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 12540 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
8 | zltp1le 12549 | . . 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 7353 ℝcr 11046 1c1 11048 + caddc 11050 < clt 11185 ≤ cle 11186 ℤcz 12495 |
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 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 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 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-n0 12410 df-z 12496 |
This theorem is referenced by: zltlem1 12552 nnleltp1 12554 nn0leltp1 12558 suprzcl 12579 le9lt10 12641 uzwo 12828 flge 13702 flhalf 13727 om2uzlti 13847 seqf1olem1 13939 fz1isolem 14352 hashtpg 14376 ltoddhalfle 16235 prmind2 16553 prm23lt5 16678 prmreclem2 16781 prmgaplem8 16922 chfacfisf 22187 chfacfisfcpmat 22188 chfacfscmulgsum 22193 chfacfpmmulgsum 22197 plyco0 25537 plydivex 25641 logf1o2 25989 ang180lem3 26145 basellem3 26416 ppieq0 26509 chpeq0 26540 bposlem1 26616 bposlem6 26621 dchrvmasumiflem1 26833 mulog2sumlem2 26867 dp2lt10 31623 1smat1 32254 ballotlemfc0 32961 ballotlemfcc 32962 poimirlem24 36069 poimirlem28 36073 fdc 36171 sticksstones10 40530 sticksstones12a 40532 sticksstones12 40533 sticksstones22 40543 irrapxlem1 41083 pellexlem5 41094 jm2.24 41225 zltlesub 43455 dvnxpaek 44115 fourierdlem50 44329 zgeltp1eq 45473 odz2prm2pw 45687 fmtno4prmfac 45696 2pwp1prm 45713 nnsum3primesle9 45918 |
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