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Theorem zneo 10026
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 10022 . . 3  |-  -.  (
1  /  2 )  e.  ZZ
2 2cn 9749 . . . . . . 7  |-  2  e.  CC
3 zcn 9961 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  CC )
43adantr 453 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  CC )
5 mulcl 8754 . . . . . . 7  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
62, 4, 5sylancr 647 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A
)  e.  CC )
7 zcn 9961 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
87adantl 454 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
9 mulcl 8754 . . . . . . 7  |-  ( ( 2  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  e.  CC )
102, 8, 9sylancr 647 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  B
)  e.  CC )
11 ax-1cn 8728 . . . . . . 7  |-  1  e.  CC
1211a1i 12 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  1  e.  CC )
136, 10, 12subaddd 9108 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  =  1  <-> 
( ( 2  x.  B )  +  1 )  =  ( 2  x.  A ) ) )
142a1i 12 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  2  e.  CC )
1514, 4, 8subdid 9168 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  ( A  -  B )
)  =  ( ( 2  x.  A )  -  ( 2  x.  B ) ) )
1615oveq1d 5772 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  ( A  -  B
) )  /  2
)  =  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  /  2 ) )
17 zsubcl 9993 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
18 zcn 9961 . . . . . . . . . 10  |-  ( ( A  -  B )  e.  ZZ  ->  ( A  -  B )  e.  CC )
1917, 18syl 17 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  CC )
20 2ne0 9762 . . . . . . . . . 10  |-  2  =/=  0
2120a1i 12 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  2  =/=  0 )
2219, 14, 21divcan3d 9474 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  ( A  -  B
) )  /  2
)  =  ( A  -  B ) )
2316, 22eqtr3d 2290 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  =  ( A  -  B ) )
2423, 17eqeltrd 2330 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  e.  ZZ )
25 oveq1 5764 . . . . . . 7  |-  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  =  1  ->  (
( ( 2  x.  A )  -  (
2  x.  B ) )  /  2 )  =  ( 1  / 
2 ) )
2625eleq1d 2322 . . . . . 6  |-  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  =  1  ->  (
( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  e.  ZZ  <->  ( 1  /  2 )  e.  ZZ ) )
2724, 26syl5ibcom 213 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  =  1  ->  ( 1  / 
2 )  e.  ZZ ) )
2813, 27sylbird 228 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  B )  +  1 )  =  ( 2  x.  A )  ->  ( 1  / 
2 )  e.  ZZ ) )
2928necon3bd 2456 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( -.  ( 1  /  2 )  e.  ZZ  ->  ( (
2  x.  B )  +  1 )  =/=  ( 2  x.  A
) ) )
301, 29mpi 18 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  B )  +  1 )  =/=  ( 2  x.  A ) )
3130necomd 2502 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A
)  =/=  ( ( 2  x.  B )  +  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419  (class class class)co 5757   CCcc 8668   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    - cmin 8970    / cdiv 9356   2c2 9728   ZZcz 9956
This theorem is referenced by:  nneo  10027  zeo2  10030  znnenlem  12417
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-n0 9898  df-z 9957
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