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Theorem nn0sub 10014
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 9974 . . . 4  |-  ( M  e.  NN0  ->  M  e.  RR )
2 nn0re 9974 . . . 4  |-  ( N  e.  NN0  ->  N  e.  RR )
3 leloe 8908 . . . 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 9967 . . . . . 6  |-  ( N  e.  NN0  <->  ( N  e.  NN  \/  N  =  0 ) )
6 elnn0 9967 . . . . . . . 8  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
7 nnsub 9784 . . . . . . . . . 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 9775 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  0  <  N )
10 nncn 9754 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  N  e.  CC )
1110subid1d 9146 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  ( N  -  0 )  =  N )
12 id 19 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  NN )
1311, 12eqeltrd 2357 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( N  -  0 )  e.  NN )
149, 132thd 231 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
0  <  N  <->  ( N  -  0 )  e.  NN ) )
15 breq1 4026 . . . . . . . . . . 11  |-  ( M  =  0  ->  ( M  <  N  <->  0  <  N ) )
16 oveq2 5866 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( N  -  M )  =  ( N  - 
0 ) )
1716eleq1d 2349 . . . . . . . . . . 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 9992 . . . . . . . . . 10  |-  ( M  e.  NN0  ->  -.  M  <  0 )
2322pm2.21d 98 . . . . . . . . 9  |-  ( M  e.  NN0  ->  ( M  <  0  ->  (
0  -  M )  e.  NN ) )
24 nngt0 9775 . . . . . . . . . 10  |-  ( ( 0  -  M )  e.  NN  ->  0  <  ( 0  -  M
) )
25 0re 8838 . . . . . . . . . . 11  |-  0  e.  RR
26 posdif 9267 . . . . . . . . . . 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 4027 . . . . . . . . 9  |-  ( N  =  0  ->  ( M  <  N  <->  M  <  0 ) )
31 oveq1 5865 . . . . . . . . . 10  |-  ( N  =  0  ->  ( N  -  M )  =  ( 0  -  M ) )
3231eleq1d 2349 . . . . . . . . 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 9975 . . . . . 6  |-  ( N  e.  NN0  ->  N  e.  CC )
39 nn0cn 9975 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  CC )
40 subeq0 9073 . . . . . 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 2285 . . . . 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 9967 . 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 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   NN0cn0 9965
This theorem is referenced by:  elz2  10040  nn0sub2  10077  psrbagcon  16117  coe1tmmul2  16352  aaliou3lem6  19728  basellem5  20322  jm2.27c  27100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966
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