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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 12185 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ) | |
2 | zltp1le 12192 | . . 3 ⊢ (((𝑀 − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 − 1) < 𝑁 ↔ ((𝑀 − 1) + 1) ≤ 𝑁)) | |
3 | 1, 2 | sylan 583 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 − 1) < 𝑁 ↔ ((𝑀 − 1) + 1) ≤ 𝑁)) |
4 | zcn 12146 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
5 | ax-1cn 10752 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | npcan 11052 | . . . . 5 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 − 1) + 1) = 𝑀) | |
7 | 4, 5, 6 | sylancl 589 | . . . 4 ⊢ (𝑀 ∈ ℤ → ((𝑀 − 1) + 1) = 𝑀) |
8 | 7 | adantr 484 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 − 1) + 1) = 𝑀) |
9 | 8 | breq1d 5049 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑀 − 1) + 1) ≤ 𝑁 ↔ 𝑀 ≤ 𝑁)) |
10 | 3, 9 | bitr2d 283 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 (class class class)co 7191 ℂcc 10692 1c1 10695 + caddc 10697 < clt 10832 ≤ cle 10833 − cmin 11027 ℤcz 12141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 |
This theorem is referenced by: nn0lem1lt 12207 nnlem1lt 12208 zbtwnre 12507 uzdisj 13150 nn0disj 13193 fzon 13228 ssfzo12 13300 ceile 13387 cshwidxn 14339 bitsfzolem 15956 bitscmp 15960 bitsinv1lem 15963 hashdvds 16291 logf1o2 25492 ang180lem3 25648 lgsquadlem1 26215 fzsplit3 30789 ballotlemfc0 32125 ballotlemfcc 32126 ballotlemimin 32138 ballotlemfrceq 32161 ballotlemfrcn0 32162 0nn0m1nnn0 32738 poimirlem23 35486 poimirlem24 35487 sticksstones10 39780 metakunt7 39794 metakunt18 39805 metakunt30 39817 irrapxlem3 40290 hashnzfz2 41553 fzdifsuc2 42463 stoweidlem26 43185 fourierdlem12 43278 fzoopth 44435 fpprel2 44809 nnsum3primesle9 44862 evengpop3 44866 zgtp1leeq 45478 m1modmmod 45483 nnolog2flm1 45552 |
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