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| Mirrors > Home > MPE Home > Th. List > zlem1lt | Structured version Visualization version GIF version | ||
| Description: Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
| Ref | Expression |
|---|---|
| zlem1lt | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | peano2zm 12660 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
| 2 | zltp1le 12667 | . . 3 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 − 1) < 𝑁 ↔ ((𝑀 − 1) + 1) ≤ 𝑁)) | |
| 3 | 1, 2 | sylan 580 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 − 1) < 𝑁 ↔ ((𝑀 − 1) + 1) ≤ 𝑁)) |
| 4 | zcn 12618 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
| 5 | ax-1cn 11213 | . . . . 5 ⊢ 1 ∈ ℂ | |
| 6 | npcan 11517 | . . . . 5 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀) | |
| 7 | 4, 5, 6 | sylancl 586 | . . . 4 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) + 1) = 𝑀) |
| 8 | 7 | adantr 480 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 − 1) + 1) = 𝑀) |
| 9 | 8 | breq1d 5153 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑀 − 1) + 1) ≤ 𝑁 ↔ 𝑀 ≤ 𝑁)) |
| 10 | 3, 9 | bitr2d 280 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 − cmin 11492 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: nn0lem1lt 12683 nnlem1lt 12684 zbtwnre 12988 uzdisj 13637 nn0disj 13684 fzon 13720 ssfzo12 13798 fzoopth 13801 ceile 13889 cshwidxn 14847 bitsfzolem 16471 bitscmp 16475 bitsinv1lem 16478 hashdvds 16812 logf1o2 26692 ang180lem3 26854 lgsquadlem1 27424 fzsplit3 32795 ballotlemfc0 34495 ballotlemfcc 34496 ballotlemimin 34508 ballotlemfrceq 34531 ballotlemfrcn0 34532 0nn0m1nnn0 35118 poimirlem23 37650 poimirlem24 37651 sticksstones10 42156 metakunt7 42212 metakunt18 42223 metakunt30 42235 irrapxlem3 42835 hashnzfz2 44340 fzdifsuc2 45322 stoweidlem26 46041 fourierdlem12 46134 fpprel2 47728 nnsum3primesle9 47781 evengpop3 47785 zgtp1leeq 48438 m1modmmod 48442 nnolog2flm1 48511 |
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