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Theorem 0nodd 40318
Description: 0 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
Hypothesis
Ref Expression
oddinmgm.e  |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
2  x.  x )  +  1 ) }
Assertion
Ref Expression
0nodd  |-  0  e/  O
Distinct variable group:    x, z
Allowed substitution hints:    O( x, z)

Proof of Theorem 0nodd
StepHypRef Expression
1 halfnz 11037 . . . . . . . . . . 11  |-  -.  (
1  /  2 )  e.  ZZ
2 eleq1 2537 . . . . . . . . . . 11  |-  ( ( 1  /  2 )  =  -u x  ->  (
( 1  /  2
)  e.  ZZ  <->  -u x  e.  ZZ ) )
31, 2mtbii 309 . . . . . . . . . 10  |-  ( ( 1  /  2 )  =  -u x  ->  -.  -u x  e.  ZZ )
4 znegcl 10996 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
53, 4nsyl3 123 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  -.  ( 1  /  2
)  =  -u x
)
6 eqcom 2478 . . . . . . . . 9  |-  ( -u x  =  ( 1  /  2 )  <->  ( 1  /  2 )  = 
-u x )
75, 6sylnibr 312 . . . . . . . 8  |-  ( x  e.  ZZ  ->  -.  -u x  =  ( 1  /  2 ) )
8 ax-1cn 9615 . . . . . . . . . . . 12  |-  1  e.  CC
9 2cn 10702 . . . . . . . . . . . 12  |-  2  e.  CC
10 2ne0 10724 . . . . . . . . . . . 12  |-  2  =/=  0
11 divneg 10324 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
1  /  2 )  =  ( -u 1  /  2 ) )
1211eqcomd 2477 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  ( -u 1  /  2 )  =  -u ( 1  / 
2 ) )
138, 9, 10, 12mp3an 1390 . . . . . . . . . . 11  |-  ( -u
1  /  2 )  =  -u ( 1  / 
2 )
1413a1i 11 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  ( -u 1  /  2 )  =  -u ( 1  / 
2 ) )
1514eqeq1d 2473 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  (
( -u 1  /  2
)  =  x  <->  -u ( 1  /  2 )  =  x ) )
16 halfcn 10852 . . . . . . . . . . 11  |-  ( 1  /  2 )  e.  CC
1716a1i 11 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  (
1  /  2 )  e.  CC )
18 zcn 10966 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  x  e.  CC )
1917, 18negcon1d 9999 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  ( -u ( 1  /  2
)  =  x  <->  -u x  =  ( 1  /  2
) ) )
2015, 19bitrd 261 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
( -u 1  /  2
)  =  x  <->  -u x  =  ( 1  /  2
) ) )
217, 20mtbird 308 . . . . . . 7  |-  ( x  e.  ZZ  ->  -.  ( -u 1  /  2
)  =  x )
22 neg1cn 10735 . . . . . . . . 9  |-  -u 1  e.  CC
2322a1i 11 . . . . . . . 8  |-  ( x  e.  ZZ  ->  -u 1  e.  CC )
24 2cnd 10704 . . . . . . . 8  |-  ( x  e.  ZZ  ->  2  e.  CC )
2510a1i 11 . . . . . . . 8  |-  ( x  e.  ZZ  ->  2  =/=  0 )
2623, 18, 24, 25divmul2d 10438 . . . . . . 7  |-  ( x  e.  ZZ  ->  (
( -u 1  /  2
)  =  x  <->  -u 1  =  ( 2  x.  x
) ) )
2721, 26mtbid 307 . . . . . 6  |-  ( x  e.  ZZ  ->  -.  -u 1  =  ( 2  x.  x ) )
28 eqcom 2478 . . . . . . . 8  |-  ( 0  =  ( ( 2  x.  x )  +  1 )  <->  ( (
2  x.  x )  +  1 )  =  0 )
2928a1i 11 . . . . . . 7  |-  ( x  e.  ZZ  ->  (
0  =  ( ( 2  x.  x )  +  1 )  <->  ( (
2  x.  x )  +  1 )  =  0 ) )
30 0cnd 9654 . . . . . . . 8  |-  ( x  e.  ZZ  ->  0  e.  CC )
31 1cnd 9677 . . . . . . . 8  |-  ( x  e.  ZZ  ->  1  e.  CC )
3224, 18mulcld 9681 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
2  x.  x )  e.  CC )
33 subadd2 9899 . . . . . . . . 9  |-  ( ( 0  e.  CC  /\  1  e.  CC  /\  (
2  x.  x )  e.  CC )  -> 
( ( 0  -  1 )  =  ( 2  x.  x )  <-> 
( ( 2  x.  x )  +  1 )  =  0 ) )
3433bicomd 206 . . . . . . . 8  |-  ( ( 0  e.  CC  /\  1  e.  CC  /\  (
2  x.  x )  e.  CC )  -> 
( ( ( 2  x.  x )  +  1 )  =  0  <-> 
( 0  -  1 )  =  ( 2  x.  x ) ) )
3530, 31, 32, 34syl3anc 1292 . . . . . . 7  |-  ( x  e.  ZZ  ->  (
( ( 2  x.  x )  +  1 )  =  0  <->  (
0  -  1 )  =  ( 2  x.  x ) ) )
36 df-neg 9883 . . . . . . . . . 10  |-  -u 1  =  ( 0  -  1 )
3736eqcomi 2480 . . . . . . . . 9  |-  ( 0  -  1 )  = 
-u 1
3837a1i 11 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
0  -  1 )  =  -u 1 )
3938eqeq1d 2473 . . . . . . 7  |-  ( x  e.  ZZ  ->  (
( 0  -  1 )  =  ( 2  x.  x )  <->  -u 1  =  ( 2  x.  x
) ) )
4029, 35, 393bitrd 287 . . . . . 6  |-  ( x  e.  ZZ  ->  (
0  =  ( ( 2  x.  x )  +  1 )  <->  -u 1  =  ( 2  x.  x
) ) )
4127, 40mtbird 308 . . . . 5  |-  ( x  e.  ZZ  ->  -.  0  =  ( (
2  x.  x )  +  1 ) )
4241nrex 2841 . . . 4  |-  -.  E. x  e.  ZZ  0  =  ( ( 2  x.  x )  +  1 )
4342intnan 928 . . 3  |-  -.  (
0  e.  ZZ  /\  E. x  e.  ZZ  0  =  ( ( 2  x.  x )  +  1 ) )
44 eqeq1 2475 . . . . 5  |-  ( z  =  0  ->  (
z  =  ( ( 2  x.  x )  +  1 )  <->  0  =  ( ( 2  x.  x )  +  1 ) ) )
4544rexbidv 2892 . . . 4  |-  ( z  =  0  ->  ( E. x  e.  ZZ  z  =  ( (
2  x.  x )  +  1 )  <->  E. x  e.  ZZ  0  =  ( ( 2  x.  x
)  +  1 ) ) )
46 oddinmgm.e . . . 4  |-  O  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( (
2  x.  x )  +  1 ) }
4745, 46elrab2 3186 . . 3  |-  ( 0  e.  O  <->  ( 0  e.  ZZ  /\  E. x  e.  ZZ  0  =  ( ( 2  x.  x )  +  1 ) ) )
4843, 47mtbir 306 . 2  |-  -.  0  e.  O
4948nelir 2746 1  |-  0  e/  O
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641    e/ wnel 2642   E.wrex 2757   {crab 2760  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    - cmin 9880   -ucneg 9881    / cdiv 10291   2c2 10681   ZZcz 10961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962
This theorem is referenced by: (None)
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