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Theorem zneo 10943
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 10939 . . 3  |-  -.  (
1  /  2 )  e.  ZZ
2 2cn 10606 . . . . . . 7  |-  2  e.  CC
3 zcn 10869 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  CC )
43adantr 465 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  CC )
5 mulcl 9576 . . . . . . 7  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
62, 4, 5sylancr 663 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A
)  e.  CC )
7 zcn 10869 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
87adantl 466 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
9 mulcl 9576 . . . . . . 7  |-  ( ( 2  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  B
)  e.  CC )
102, 8, 9sylancr 663 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  B
)  e.  CC )
11 ax-1cn 9550 . . . . . . 7  |-  1  e.  CC
1211a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  1  e.  CC )
136, 10, 12subaddd 9948 . . . . 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 10012 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  ( A  -  B )
)  =  ( ( 2  x.  A )  -  ( 2  x.  B ) ) )
1615oveq1d 6299 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  ( A  -  B
) )  /  2
)  =  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  /  2 ) )
17 zsubcl 10905 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
18 zcn 10869 . . . . . . . . . 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 10628 . . . . . . . . . 10  |-  2  =/=  0
2120a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  2  =/=  0 )
2219, 14, 21divcan3d 10325 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  ( A  -  B
) )  /  2
)  =  ( A  -  B ) )
2316, 22eqtr3d 2510 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  =  ( A  -  B ) )
2423, 17eqeltrd 2555 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  e.  ZZ )
25 oveq1 6291 . . . . . . 7  |-  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  =  1  ->  (
( ( 2  x.  A )  -  (
2  x.  B ) )  /  2 )  =  ( 1  / 
2 ) )
2625eleq1d 2536 . . . . . 6  |-  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  =  1  ->  (
( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  e.  ZZ  <->  ( 1  /  2 )  e.  ZZ ) )
2724, 26syl5ibcom 220 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  =  1  ->  ( 1  / 
2 )  e.  ZZ ) )
2813, 27sylbird 235 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  B )  +  1 )  =  ( 2  x.  A )  ->  ( 1  / 
2 )  e.  ZZ ) )
2928necon3bd 2679 . . 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 2738 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  A
)  =/=  ( ( 2  x.  B )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6284   CCcc 9490   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    - cmin 9805    / cdiv 10206   2c2 10585   ZZcz 10864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865
This theorem is referenced by:  nneo  10944  zeo2  10947  znnenlem  13806
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