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Theorem zneo 9973
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 9969 . . 3  |-  -.  (
1  /  2 )  e.  ZZ
2 2cn 9696 . . . . . . 7  |-  2  e.  CC
3 zcn 9908 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  CC )
43adantr 453 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  CC )
5 mulcl 8701 . . . . . . 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 9908 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
87adantl 454 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  CC )
9 mulcl 8701 . . . . . . 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 8675 . . . . . . 7  |-  1  e.  CC
1211a1i 12 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  1  e.  CC )
136, 10, 12subaddd 9055 . . . . 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 9115 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  x.  ( A  -  B )
)  =  ( ( 2  x.  A )  -  ( 2  x.  B ) ) )
1615oveq1d 5725 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  ( A  -  B
) )  /  2
)  =  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  /  2 ) )
17 zsubcl 9940 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
18 zcn 9908 . . . . . . . . . 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 9709 . . . . . . . . . 10  |-  2  =/=  0
2120a1i 12 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  2  =/=  0 )
2219, 14, 21divcan3d 9421 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  x.  ( A  -  B
) )  /  2
)  =  ( A  -  B ) )
2316, 22eqtr3d 2287 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  =  ( A  -  B ) )
2423, 17eqeltrd 2327 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( 2  x.  A )  -  ( 2  x.  B
) )  /  2
)  e.  ZZ )
25 oveq1 5717 . . . . . . 7  |-  ( ( ( 2  x.  A
)  -  ( 2  x.  B ) )  =  1  ->  (
( ( 2  x.  A )  -  (
2  x.  B ) )  /  2 )  =  ( 1  / 
2 ) )
2625eleq1d 2319 . . . . . 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 2449 . . 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 2495 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 2412  (class class class)co 5710   CCcc 8615   0cc0 8617   1c1 8618    + caddc 8620    x. cmul 8622    - cmin 8917    / cdiv 9303   2c2 9675   ZZcz 9903
This theorem is referenced by:  nneo  9974  zeo2  9977  znnenlem  12364
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-n0 9845  df-z 9904
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