MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nn0sub Unicode version

Theorem nn0sub 10030
Description: Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
Assertion
Ref Expression
nn0sub  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  <_  N  <->  ( N  -  M )  e.  NN0 ) )

Proof of Theorem nn0sub
StepHypRef Expression
1 nn0re 9990 . . . 4  |-  ( M  e.  NN0  ->  M  e.  RR )
2 nn0re 9990 . . . 4  |-  ( N  e.  NN0  ->  N  e.  RR )
3 leloe 8924 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <_  N  <->  ( M  <  N  \/  M  =  N )
) )
41, 2, 3syl2an 463 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  <_  N  <->  ( M  <  N  \/  M  =  N )
) )
5 elnn0 9983 . . . . . 6  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
6 elnn0 9983 . . . . . . . 8  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
7 nnsub 9800 . . . . . . . . . 10  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  <  N  <->  ( N  -  M )  e.  NN ) )
87ex 423 . . . . . . . . 9  |-  ( M  e.  NN  ->  ( N  e.  NN  ->  ( M  <  N  <->  ( N  -  M )  e.  NN ) ) )
9 nngt0 9791 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  0  <  N )
10 nncn 9770 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  N  e.  CC )
1110subid1d 9162 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  ( N  -  0 )  =  N )
12 id 19 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  NN )
1311, 12eqeltrd 2370 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  -  0 )  e.  NN )
149, 132thd 231 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
0  <  N  <->  ( N  -  0 )  e.  NN ) )
15 breq1 4042 . . . . . . . . . . 11  |-  ( M  =  0  ->  ( M  <  N  <->  0  <  N ) )
16 oveq2 5882 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( N  -  M )  =  ( N  - 
0 ) )
1716eleq1d 2362 . . . . . . . . . . 11  |-  ( M  =  0  ->  (
( N  -  M
)  e.  NN  <->  ( N  -  0 )  e.  NN ) )
1815, 17bibi12d 312 . . . . . . . . . 10  |-  ( M  =  0  ->  (
( M  <  N  <->  ( N  -  M )  e.  NN )  <->  ( 0  <  N  <->  ( N  -  0 )  e.  NN ) ) )
1914, 18syl5ibr 212 . . . . . . . . 9  |-  ( M  =  0  ->  ( N  e.  NN  ->  ( M  <  N  <->  ( N  -  M )  e.  NN ) ) )
208, 19jaoi 368 . . . . . . . 8  |-  ( ( M  e.  NN  \/  M  =  0 )  ->  ( N  e.  NN  ->  ( M  <  N  <->  ( N  -  M )  e.  NN ) ) )
216, 20sylbi 187 . . . . . . 7  |-  ( M  e.  NN0  ->  ( N  e.  NN  ->  ( M  <  N  <->  ( N  -  M )  e.  NN ) ) )
22 nn0nlt0 10008 . . . . . . . . . 10  |-  ( M  e.  NN0  ->  -.  M  <  0 )
2322pm2.21d 98 . . . . . . . . 9  |-  ( M  e.  NN0  ->  ( M  <  0  ->  (
0  -  M )  e.  NN ) )
24 nngt0 9791 . . . . . . . . . 10  |-  ( ( 0  -  M )  e.  NN  ->  0  <  ( 0  -  M
) )
25 0re 8854 . . . . . . . . . . 11  |-  0  e.  RR
26 posdif 9283 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  0  e.  RR )  ->  ( M  <  0  <->  0  <  ( 0  -  M ) ) )
271, 25, 26sylancl 643 . . . . . . . . . 10  |-  ( M  e.  NN0  ->  ( M  <  0  <->  0  <  ( 0  -  M ) ) )
2824, 27syl5ibr 212 . . . . . . . . 9  |-  ( M  e.  NN0  ->  ( ( 0  -  M )  e.  NN  ->  M  <  0 ) )
2923, 28impbid 183 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( M  <  0  <->  ( 0  -  M )  e.  NN ) )
30 breq2 4043 . . . . . . . . 9  |-  ( N  =  0  ->  ( M  <  N  <->  M  <  0 ) )
31 oveq1 5881 . . . . . . . . . 10  |-  ( N  =  0  ->  ( N  -  M )  =  ( 0  -  M ) )
3231eleq1d 2362 . . . . . . . . 9  |-  ( N  =  0  ->  (
( N  -  M
)  e.  NN  <->  ( 0  -  M )  e.  NN ) )
3330, 32bibi12d 312 . . . . . . . 8  |-  ( N  =  0  ->  (
( M  <  N  <->  ( N  -  M )  e.  NN )  <->  ( M  <  0  <->  ( 0  -  M )  e.  NN ) ) )
3429, 33syl5ibrcom 213 . . . . . . 7  |-  ( M  e.  NN0  ->  ( N  =  0  ->  ( M  <  N  <->  ( N  -  M )  e.  NN ) ) )
3521, 34jaod 369 . . . . . 6  |-  ( M  e.  NN0  ->  ( ( N  e.  NN  \/  N  =  0 )  ->  ( M  < 
N  <->  ( N  -  M )  e.  NN ) ) )
365, 35syl5bi 208 . . . . 5  |-  ( M  e.  NN0  ->  ( N  e.  NN0  ->  ( M  <  N  <->  ( N  -  M )  e.  NN ) ) )
3736imp 418 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  <  N  <->  ( N  -  M )  e.  NN ) )
38 nn0cn 9991 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  CC )
39 nn0cn 9991 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  CC )
40 subeq0 9089 . . . . . 6  |-  ( ( N  e.  CC  /\  M  e.  CC )  ->  ( ( N  -  M )  =  0  <-> 
N  =  M ) )
4138, 39, 40syl2anr 464 . . . . 5  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( N  -  M )  =  0  <-> 
N  =  M ) )
42 eqcom 2298 . . . . 5  |-  ( N  =  M  <->  M  =  N )
4341, 42syl6rbb 253 . . . 4  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  =  N  <-> 
( N  -  M
)  =  0 ) )
4437, 43orbi12d 690 . . 3  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( M  < 
N  \/  M  =  N )  <->  ( ( N  -  M )  e.  NN  \/  ( N  -  M )  =  0 ) ) )
454, 44bitrd 244 . 2  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  <_  N  <->  ( ( N  -  M
)  e.  NN  \/  ( N  -  M
)  =  0 ) ) )
46 elnn0 9983 . 2  |-  ( ( N  -  M )  e.  NN0  <->  ( ( N  -  M )  e.  NN  \/  ( N  -  M )  =  0 ) )
4745, 46syl6bbr 254 1  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  <_  N  <->  ( N  -  M )  e.  NN0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   NN0cn0 9981
This theorem is referenced by:  elz2  10056  nn0sub2  10093  psrbagcon  16133  coe1tmmul2  16368  aaliou3lem6  19744  basellem5  20338  jm2.27c  27203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982
  Copyright terms: Public domain W3C validator