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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 12617 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
| 2 | zre 12617 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
| 3 | 1re 11261 | . . . 4 ⊢ 1 ∈ ℝ | |
| 4 | leadd1 11731 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) | |
| 5 | 3, 4 | mp3an3 1452 | . . 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 12658 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
| 8 | zltp1le 12667 | . . 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 282 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 ℤcz 12613 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 |
| This theorem is referenced by: zltlem1 12670 nnleltp1 12673 nn0leltp1 12677 suprzcl 12698 le9lt10 12760 uzwo 12953 flge 13845 flhalf 13870 om2uzlti 13991 seqf1olem1 14082 fz1isolem 14500 hashtpg 14524 ltoddhalfle 16398 prmind2 16722 prm23lt5 16852 prmreclem2 16955 prmgaplem8 17096 chfacfisf 22860 chfacfisfcpmat 22861 chfacfscmulgsum 22866 chfacfpmmulgsum 22870 plyco0 26231 plydivex 26339 logf1o2 26692 ang180lem3 26854 basellem3 27126 ppieq0 27219 chpeq0 27252 bposlem1 27328 bposlem6 27333 dchrvmasumiflem1 27545 mulog2sumlem2 27579 dp2lt10 32866 1smat1 33803 ballotlemfc0 34495 ballotlemfcc 34496 poimirlem24 37651 poimirlem28 37655 fdc 37752 sticksstones10 42156 sticksstones12a 42158 sticksstones12 42159 sticksstones22 42169 irrapxlem1 42833 pellexlem5 42844 jm2.24 42975 zltlesub 45297 dvnxpaek 45957 fourierdlem50 46171 zgeltp1eq 47321 odz2prm2pw 47550 fmtno4prmfac 47559 2pwp1prm 47576 nnsum3primesle9 47781 |
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