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

Theorem zneo 10341
Description: No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)
Assertion
Ref Expression
zneo  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A
)  =/=  ( ( 2  x.  B )  +  1 ) )

Proof of Theorem zneo
StepHypRef Expression
1 halfnz 10337 . . 3  |-  -.  (
1  /  2 )  e.  ZZ
2 2cn 10059 . . . . . . 7  |-  2  e.  CC
3 zcn 10276 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  CC )
43adantr 452 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  CC )
5 mulcl 9063 . . . . . . 7  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
62, 4, 5sylancr 645 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A
)  e.  CC )
7 zcn 10276 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
87adantl 453 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
9 mulcl 9063 . . . . . . 7  |-  ( ( 2  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  e.  CC )
102, 8, 9sylancr 645 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  B
)  e.  CC )
11 ax-1cn 9037 . . . . . . 7  |-  1  e.  CC
1211a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  1  e.  CC )
136, 10, 12subaddd 9418 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  =  1  <-> 
( ( 2  x.  B )  +  1 )  =  ( 2  x.  A ) ) )
142a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  2  e.  CC )
1514, 4, 8subdid 9478 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  ( A  -  B )
)  =  ( ( 2  x.  A )  -  ( 2  x.  B ) ) )
1615oveq1d 6087 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  ( A  -  B
) )  /  2
)  =  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  /  2 ) )
17 zsubcl 10308 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
18 zcn 10276 . . . . . . . . . 10  |-  ( ( A  -  B )  e.  ZZ  ->  ( A  -  B )  e.  CC )
1917, 18syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  CC )
20 2ne0 10072 . . . . . . . . . 10  |-  2  =/=  0
2120a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  2  =/=  0 )
2219, 14, 21divcan3d 9784 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  ( A  -  B
) )  /  2
)  =  ( A  -  B ) )
2316, 22eqtr3d 2469 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  =  ( A  -  B ) )
2423, 17eqeltrd 2509 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  e.  ZZ )
25 oveq1 6079 . . . . . . 7  |-  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  =  1  ->  (
( ( 2  x.  A )  -  (
2  x.  B ) )  /  2 )  =  ( 1  / 
2 ) )
2625eleq1d 2501 . . . . . 6  |-  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  =  1  ->  (
( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  e.  ZZ  <->  ( 1  /  2 )  e.  ZZ ) )
2724, 26syl5ibcom 212 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  =  1  ->  ( 1  / 
2 )  e.  ZZ ) )
2813, 27sylbird 227 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  B )  +  1 )  =  ( 2  x.  A )  ->  ( 1  / 
2 )  e.  ZZ ) )
2928necon3bd 2635 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( -.  ( 1  /  2 )  e.  ZZ  ->  ( (
2  x.  B )  +  1 )  =/=  ( 2  x.  A
) ) )
301, 29mpi 17 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  B )  +  1 )  =/=  ( 2  x.  A ) )
3130necomd 2681 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A
)  =/=  ( ( 2  x.  B )  +  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6072   CCcc 8977   0cc0 8979   1c1 8980    + caddc 8982    x. cmul 8984    - cmin 9280    / cdiv 9666   2c2 10038   ZZcz 10271
This theorem is referenced by:  nneo  10342  zeo2  10345  znnenlem  12799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-n0 10211  df-z 10272
  Copyright terms: Public domain W3C validator