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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 11986 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
2 | zre 11986 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
3 | 1re 10641 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | leadd1 11108 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) | |
5 | 3, 4 | mp3an3 1446 | . . 3 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
6 | 1, 2, 5 | syl2an 597 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
7 | peano2z 12024 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℤ) | |
8 | zltp1le 12033 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 + 1) ∈ ℤ) → (𝑀 < (𝑁 + 1) ↔ (𝑀 + 1) ≤ (𝑁 + 1))) | |
9 | 7, 8 | sylan2 594 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < (𝑁 + 1) ↔ (𝑀 + 1) ≤ (𝑁 + 1))) |
10 | 6, 9 | bitr4d 284 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ 𝑀 < (𝑁 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 ℤcz 11982 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 |
This theorem is referenced by: zltlem1 12036 nnleltp1 12038 nn0leltp1 12042 suprzcl 12063 le9lt10 12126 uzwo 12312 flge 13176 flhalf 13201 om2uzlti 13319 seqf1olem1 13410 fz1isolem 13820 hashtpg 13844 ltoddhalfle 15710 prmind2 16029 prm23lt5 16151 prmreclem2 16253 prmgaplem8 16394 chfacfisf 21462 chfacfisfcpmat 21463 chfacfscmulgsum 21468 chfacfpmmulgsum 21472 plyco0 24782 plydivex 24886 logf1o2 25233 ang180lem3 25389 basellem3 25660 ppieq0 25753 chpeq0 25784 bposlem1 25860 bposlem6 25865 dchrvmasumiflem1 26077 mulog2sumlem2 26111 dp2lt10 30560 1smat1 31069 ballotlemfc0 31750 ballotlemfcc 31751 poimirlem24 34931 poimirlem28 34935 fdc 35035 irrapxlem1 39439 pellexlem5 39450 jm2.24 39580 zltlesub 41571 dvnxpaek 42247 fourierdlem50 42461 zgeltp1eq 43529 odz2prm2pw 43745 fmtno4prmfac 43754 2pwp1prm 43771 nnsum3primesle9 43979 |
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