![]() |
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 12643 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
2 | zleltp1 12651 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑀 ≤ (𝑁 − 1) ↔ 𝑀 < ((𝑁 − 1) + 1))) | |
3 | 1, 2 | sylan2 591 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ (𝑁 − 1) ↔ 𝑀 < ((𝑁 − 1) + 1))) |
4 | zcn 12601 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
5 | ax-1cn 11204 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | npcan 11507 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
7 | 4, 5, 6 | sylancl 584 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑁 − 1) + 1) = 𝑁) |
8 | 7 | adantl 480 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 − 1) + 1) = 𝑁) |
9 | 8 | breq2d 5164 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < ((𝑁 − 1) + 1) ↔ 𝑀 < 𝑁)) |
10 | 3, 9 | bitr2d 279 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7426 ℂcc 11144 1c1 11147 + caddc 11149 < clt 11286 ≤ cle 11287 − cmin 11482 ℤcz 12596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 |
This theorem is referenced by: nn0ltlem1 12660 nn0lt2 12663 nn0le2is012 12664 nnltlem1 12667 nnm1ge0 12668 zextlt 12674 uzm1 12898 elfzm11 13612 preduz 13663 predfz 13666 elfzo 13674 fzosplitprm1 13782 intfracq 13864 seqf1olem1 14046 seqcoll 14465 isercolllem1 15651 fzm1ndvds 16306 bitscmp 16420 nn0seqcvgd 16548 isprm3 16661 ncoprmlnprm 16707 prmdiveq 16762 4sqlem12 16932 degltlem1 26028 dgreq0 26220 wilthlem1 27020 lgseisenlem2 27329 lgsquadlem1 27333 2lgslem1a1 27342 2sqlem8 27379 crctcshwlkn0lem4 29644 clwlkclwwlklem2a4 29827 clwlkclwwlklem2a 29828 frgrreggt1 30223 bcm1n 32584 ply1degltel 33298 smatrcl 33430 ballotlemimin 34158 ballotlemfrcn0 34182 knoppndvlem2 36021 poimirlem2 37128 poimirlem24 37150 zltlem1d 41481 sticksstones10 41659 metakunt7 41695 metakunt21 41709 metakunt22 41710 metakunt24 41712 fmul01lt1lem2 45002 fourierdlem41 45565 fourierdlem42 45566 fourierdlem50 45573 fourierdlem64 45587 fourierdlem79 45602 etransclem44 45695 etransclem48 45699 pw2m1lepw2m1 47666 fllog2 47719 |
Copyright terms: Public domain | W3C validator |