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Theorem zneo 8817
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 8812 . . 3  |-  -.  (
1  /  2 )  e.  ZZ
2 2cn 8464 . . . . . . 7  |-  2  e.  CC
3 zcn 8725 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  CC )
43adantr 270 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  CC )
5 mulcl 7448 . . . . . . 7  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
62, 4, 5sylancr 405 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A
)  e.  CC )
7 zcn 8725 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
87adantl 271 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
9 mulcl 7448 . . . . . . 7  |-  ( ( 2  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  e.  CC )
102, 8, 9sylancr 405 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  B
)  e.  CC )
11 1cnd 7483 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  1  e.  CC )
126, 10, 11subaddd 7790 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  =  1  <-> 
( ( 2  x.  B )  +  1 )  =  ( 2  x.  A ) ) )
132a1i 9 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  2  e.  CC )
1413, 4, 8subdid 7871 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  ( A  -  B )
)  =  ( ( 2  x.  A )  -  ( 2  x.  B ) ) )
1514oveq1d 5649 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  ( A  -  B
) )  /  2
)  =  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  /  2 ) )
16 zsubcl 8761 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
17 zcn 8725 . . . . . . . . . 10  |-  ( ( A  -  B )  e.  ZZ  ->  ( A  -  B )  e.  CC )
1816, 17syl 14 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  CC )
19 2ap0 8486 . . . . . . . . . 10  |-  2 #  0
2019a1i 9 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  2 #  0 )
2118, 13, 20divcanap3d 8235 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  ( A  -  B
) )  /  2
)  =  ( A  -  B ) )
2215, 21eqtr3d 2122 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  =  ( A  -  B ) )
2322, 16eqeltrd 2164 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  e.  ZZ )
24 oveq1 5641 . . . . . . 7  |-  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  =  1  ->  (
( ( 2  x.  A )  -  (
2  x.  B ) )  /  2 )  =  ( 1  / 
2 ) )
2524eleq1d 2156 . . . . . 6  |-  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  =  1  ->  (
( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  e.  ZZ  <->  ( 1  /  2 )  e.  ZZ ) )
2623, 25syl5ibcom 153 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  =  1  ->  ( 1  / 
2 )  e.  ZZ ) )
2712, 26sylbird 168 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  B )  +  1 )  =  ( 2  x.  A )  ->  ( 1  / 
2 )  e.  ZZ ) )
2827necon3bd 2298 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( -.  ( 1  /  2 )  e.  ZZ  ->  ( (
2  x.  B )  +  1 )  =/=  ( 2  x.  A
) ) )
291, 28mpi 15 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  B )  +  1 )  =/=  ( 2  x.  A ) )
3029necomd 2341 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 102    = wceq 1289    e. wcel 1438    =/= wne 2255   class class class wbr 3837  (class class class)co 5634   CCcc 7327   0cc0 7329   1c1 7330    + caddc 7332    x. cmul 7334    - cmin 7632   # cap 8034    / cdiv 8113   2c2 8444   ZZcz 8720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721
This theorem is referenced by:  nneo  8819  zeo2  8822
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