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Theorem 0nodd 39914
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 11021 . . . . . . . . . . 11  |-  -.  (
1  /  2 )  e.  ZZ
2 eleq1 2519 . . . . . . . . . . 11  |-  ( ( 1  /  2 )  =  -u x  ->  (
( 1  /  2
)  e.  ZZ  <->  -u x  e.  ZZ ) )
31, 2mtbii 304 . . . . . . . . . 10  |-  ( ( 1  /  2 )  =  -u x  ->  -.  -u x  e.  ZZ )
4 znegcl 10979 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  -u x  e.  ZZ )
53, 4nsyl3 123 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  -.  ( 1  /  2
)  =  -u x
)
6 eqcom 2460 . . . . . . . . 9  |-  ( -u x  =  ( 1  /  2 )  <->  ( 1  /  2 )  = 
-u x )
75, 6sylnibr 307 . . . . . . . 8  |-  ( x  e.  ZZ  ->  -.  -u x  =  ( 1  /  2 ) )
8 ax-1cn 9602 . . . . . . . . . . . 12  |-  1  e.  CC
9 2cn 10687 . . . . . . . . . . . 12  |-  2  e.  CC
10 2ne0 10709 . . . . . . . . . . . 12  |-  2  =/=  0
11 divneg 10309 . . . . . . . . . . . . 13  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  -u (
1  /  2 )  =  ( -u 1  /  2 ) )
1211eqcomd 2459 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  ( -u 1  /  2 )  =  -u ( 1  / 
2 ) )
138, 9, 10, 12mp3an 1366 . . . . . . . . . . 11  |-  ( -u
1  /  2 )  =  -u ( 1  / 
2 )
1413a1i 11 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  ( -u 1  /  2 )  =  -u ( 1  / 
2 ) )
1514eqeq1d 2455 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  (
( -u 1  /  2
)  =  x  <->  -u ( 1  /  2 )  =  x ) )
16 halfcn 10836 . . . . . . . . . . 11  |-  ( 1  /  2 )  e.  CC
1716a1i 11 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  (
1  /  2 )  e.  CC )
18 zcn 10949 . . . . . . . . . 10  |-  ( x  e.  ZZ  ->  x  e.  CC )
1917, 18negcon1d 9985 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  ( -u ( 1  /  2
)  =  x  <->  -u x  =  ( 1  /  2
) ) )
2015, 19bitrd 257 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
( -u 1  /  2
)  =  x  <->  -u x  =  ( 1  /  2
) ) )
217, 20mtbird 303 . . . . . . 7  |-  ( x  e.  ZZ  ->  -.  ( -u 1  /  2
)  =  x )
22 neg1cn 10720 . . . . . . . . 9  |-  -u 1  e.  CC
2322a1i 11 . . . . . . . 8  |-  ( x  e.  ZZ  ->  -u 1  e.  CC )
24 2cnd 10689 . . . . . . . 8  |-  ( x  e.  ZZ  ->  2  e.  CC )
2510a1i 11 . . . . . . . 8  |-  ( x  e.  ZZ  ->  2  =/=  0 )
2623, 18, 24, 25divmul2d 10423 . . . . . . 7  |-  ( x  e.  ZZ  ->  (
( -u 1  /  2
)  =  x  <->  -u 1  =  ( 2  x.  x
) ) )
2721, 26mtbid 302 . . . . . 6  |-  ( x  e.  ZZ  ->  -.  -u 1  =  ( 2  x.  x ) )
28 eqcom 2460 . . . . . . . 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 9641 . . . . . . . 8  |-  ( x  e.  ZZ  ->  0  e.  CC )
31 1cnd 9664 . . . . . . . 8  |-  ( x  e.  ZZ  ->  1  e.  CC )
3224, 18mulcld 9668 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
2  x.  x )  e.  CC )
33 subadd2 9884 . . . . . . . . 9  |-  ( ( 0  e.  CC  /\  1  e.  CC  /\  (
2  x.  x )  e.  CC )  -> 
( ( 0  -  1 )  =  ( 2  x.  x )  <-> 
( ( 2  x.  x )  +  1 )  =  0 ) )
3433bicomd 205 . . . . . . . 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 1269 . . . . . . 7  |-  ( x  e.  ZZ  ->  (
( ( 2  x.  x )  +  1 )  =  0  <->  (
0  -  1 )  =  ( 2  x.  x ) ) )
36 df-neg 9868 . . . . . . . . . 10  |-  -u 1  =  ( 0  -  1 )
3736eqcomi 2462 . . . . . . . . 9  |-  ( 0  -  1 )  = 
-u 1
3837a1i 11 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
0  -  1 )  =  -u 1 )
3938eqeq1d 2455 . . . . . . 7  |-  ( x  e.  ZZ  ->  (
( 0  -  1 )  =  ( 2  x.  x )  <->  -u 1  =  ( 2  x.  x
) ) )
4029, 35, 393bitrd 283 . . . . . 6  |-  ( x  e.  ZZ  ->  (
0  =  ( ( 2  x.  x )  +  1 )  <->  -u 1  =  ( 2  x.  x
) ) )
4127, 40mtbird 303 . . . . 5  |-  ( x  e.  ZZ  ->  -.  0  =  ( (
2  x.  x )  +  1 ) )
4241nrex 2844 . . . 4  |-  -.  E. x  e.  ZZ  0  =  ( ( 2  x.  x )  +  1 )
4342intnan 926 . . 3  |-  -.  (
0  e.  ZZ  /\  E. x  e.  ZZ  0  =  ( ( 2  x.  x )  +  1 ) )
44 eqeq1 2457 . . . . 5  |-  ( z  =  0  ->  (
z  =  ( ( 2  x.  x )  +  1 )  <->  0  =  ( ( 2  x.  x )  +  1 ) ) )
4544rexbidv 2903 . . . 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 3200 . . 3  |-  ( 0  e.  O  <->  ( 0  e.  ZZ  /\  E. x  e.  ZZ  0  =  ( ( 2  x.  x )  +  1 ) ) )
4843, 47mtbir 301 . 2  |-  -.  0  e.  O
4948nelir 2729 1  |-  0  e/  O
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624    e/ wnel 2625   E.wrex 2740   {crab 2743  (class class class)co 6295   CCcc 9542   0cc0 9544   1c1 9545    + caddc 9547    x. cmul 9549    - cmin 9865   -ucneg 9866    / cdiv 10276   2c2 10666   ZZcz 10944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945
This theorem is referenced by: (None)
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