Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11420 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
2 | zleltp1 11428 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑀 ≤ (𝑁 − 1) ↔ 𝑀 < ((𝑁 − 1) + 1))) | |
3 | 1, 2 | sylan2 491 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ (𝑁 − 1) ↔ 𝑀 < ((𝑁 − 1) + 1))) |
4 | zcn 11382 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
5 | ax-1cn 9994 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | npcan 10290 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
7 | 4, 5, 6 | sylancl 694 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑁 − 1) + 1) = 𝑁) |
8 | 7 | adantl 482 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 − 1) + 1) = 𝑁) |
9 | 8 | breq2d 4665 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < ((𝑁 − 1) + 1) ↔ 𝑀 < 𝑁)) |
10 | 3, 9 | bitr2d 269 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 ℤcz 11377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 |
This theorem is referenced by: nn0ltlem1 11437 nn0lt2 11440 nn0le2is012 11441 nnltlem1 11444 nnm1ge0 11445 zextlt 11451 uzm1 11718 elfzm11 12411 preduz 12461 predfz 12464 elfzo 12472 fzosplitprm1 12578 intfracq 12658 seqf1olem1 12840 seqcoll 13248 isercolllem1 14395 fzm1ndvds 15044 bitscmp 15160 nn0seqcvgd 15283 isprm3 15396 ncoprmlnprm 15436 prmdiveq 15491 4sqlem12 15660 degltlem1 23832 dgreq0 24021 wilthlem1 24794 lgseisenlem2 25101 lgsquadlem1 25105 2lgslem1a1 25114 2sqlem8 25151 crctcshwlkn0lem4 26705 clwlkclwwlklem2a4 26898 clwlkclwwlklem2a 26899 frgrreggt1 27251 bcm1n 29554 smatrcl 29862 ballotlemimin 30567 ballotlemfrcn0 30591 knoppndvlem2 32504 poimirlem2 33411 poimirlem24 33433 fmul01lt1lem2 39817 fourierdlem41 40365 fourierdlem42 40366 fourierdlem50 40373 fourierdlem64 40387 fourierdlem79 40402 etransclem44 40495 etransclem48 40499 pw2m1lepw2m1 42310 fllog2 42362 |
Copyright terms: Public domain | W3C validator |