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Mirrors > Home > MPE Home > Th. List > zltlem1 | Structured version Visualization version GIF version |
Description: Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
Ref | Expression |
---|---|
zltlem1 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2zm 11874 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
2 | zleltp1 11882 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑀 ≤ (𝑁 − 1) ↔ 𝑀 < ((𝑁 − 1) + 1))) | |
3 | 1, 2 | sylan2 592 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ (𝑁 − 1) ↔ 𝑀 < ((𝑁 − 1) + 1))) |
4 | zcn 11834 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
5 | ax-1cn 10441 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | npcan 10743 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
7 | 4, 5, 6 | sylancl 586 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑁 − 1) + 1) = 𝑁) |
8 | 7 | adantl 482 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 − 1) + 1) = 𝑁) |
9 | 8 | breq2d 4974 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < ((𝑁 − 1) + 1) ↔ 𝑀 < 𝑁)) |
10 | 3, 9 | bitr2d 281 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 class class class wbr 4962 (class class class)co 7016 ℂcc 10381 1c1 10384 + caddc 10386 < clt 10521 ≤ cle 10522 − cmin 10717 ℤcz 11829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-n0 11746 df-z 11830 |
This theorem is referenced by: nn0ltlem1 11891 nn0lt2 11894 nn0le2is012 11895 nnltlem1 11898 nnm1ge0 11899 zextlt 11905 uzm1 12125 elfzm11 12828 preduz 12879 predfz 12882 elfzo 12890 fzosplitprm1 12997 intfracq 13077 seqf1olem1 13259 seqcoll 13670 isercolllem1 14855 fzm1ndvds 15505 bitscmp 15620 nn0seqcvgd 15743 isprm3 15856 ncoprmlnprm 15897 prmdiveq 15952 4sqlem12 16121 degltlem1 24349 dgreq0 24538 wilthlem1 25327 lgseisenlem2 25634 lgsquadlem1 25638 2lgslem1a1 25647 2sqlem8 25684 crctcshwlkn0lem4 27278 clwlkclwwlklem2a4 27462 clwlkclwwlklem2a 27463 frgrreggt1 27864 bcm1n 30204 smatrcl 30676 ballotlemimin 31380 ballotlemfrcn0 31404 knoppndvlem2 33461 poimirlem2 34425 poimirlem24 34447 fmul01lt1lem2 41408 fourierdlem41 41975 fourierdlem42 41976 fourierdlem50 41983 fourierdlem64 41997 fourierdlem79 42012 etransclem44 42105 etransclem48 42109 pw2m1lepw2m1 44056 fllog2 44109 |
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