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Theorem 0nodd 41810
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 11455 . . . . . . . . . . 11 |- -. ( 1 / 2 ) e. ZZ
2 eleq1 2689 . . . . . . . . . . 11 |- ( ( 1 / 2 ) = -u x -> ( ( 1 / 2 ) e. ZZ <-> -u x e. ZZ ) )
31, 2mtbii 316 . . . . . . . . . 10 |- ( ( 1 / 2 ) = -u x -> -. -u x e. ZZ )
4 znegcl 11412 . . . . . . . . . 10 |- ( x e. ZZ -> -u x e. ZZ )
53, 4nsyl3 133 . . . . . . . . 9 |- ( x e. ZZ -> -. ( 1 / 2 ) = -u x )
6 eqcom 2629 . . . . . . . . 9 |- ( -u x = ( 1 / 2 ) <-> ( 1 / 2 ) = -u x )
75, 6sylnibr 319 . . . . . . . 8 |- ( x e. ZZ -> -. -u x = ( 1 / 2 ) )
8 ax-1cn 9994 . . . . . . . . . . . 12 |- 1 e. CC
9 2cn 11091 . . . . . . . . . . . 12 |- 2 e. CC
10 2ne0 11113 . . . . . . . . . . . 12 |- 2 =/= 0
11 divneg 10719 . . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> -u ( 1 / 2 ) = ( -u 1 / 2 ) )
1211eqcomd 2628 . . . . . . . . . . . 12 |- ( ( 1 e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( -u 1 / 2 ) = -u ( 1 / 2 ) )
138, 9, 10, 12mp3an 1424 . . . . . . . . . . 11 |- ( -u 1 / 2 ) = -u ( 1 / 2 )
1413a1i 11 . . . . . . . . . 10 |- ( x e. ZZ -> ( -u 1 / 2 ) = -u ( 1 / 2 ) )
1514eqeq1d 2624 . . . . . . . . 9 |- ( x e. ZZ -> ( ( -u 1 / 2 ) = x <-> -u ( 1 / 2 ) = x ) )
16 halfcn 11247 . . . . . . . . . . 11 |- ( 1 / 2 ) e. CC
1716a1i 11 . . . . . . . . . 10 |- ( x e. ZZ -> ( 1 / 2 ) e. CC )
18 zcn 11382 . . . . . . . . . 10 |- ( x e. ZZ -> x e. CC )
1917, 18negcon1d 10386 . . . . . . . . 9 |- ( x e. ZZ -> ( -u ( 1 / 2 ) = x <-> -u x = ( 1 / 2 ) ) )
2015, 19bitrd 268 . . . . . . . 8 |- ( x e. ZZ -> ( ( -u 1 / 2 ) = x <-> -u x = ( 1 / 2 ) ) )
217, 20mtbird 315 . . . . . . 7 |- ( x e. ZZ -> -. ( -u 1 / 2 ) = x )
22 neg1cn 11124 . . . . . . . . 9 |- -u 1 e. CC
2322a1i 11 . . . . . . . 8 |- ( x e. ZZ -> -u 1 e. CC )
24 2cnd 11093 . . . . . . . 8 |- ( x e. ZZ -> 2 e. CC )
2510a1i 11 . . . . . . . 8 |- ( x e. ZZ -> 2 =/= 0 )
2623, 18, 24, 25divmul2d 10834 . . . . . . 7 |- ( x e. ZZ -> ( ( -u 1 / 2 ) = x <-> -u 1 = ( 2 x. x ) ) )
2721, 26mtbid 314 . . . . . 6 |- ( x e. ZZ -> -. -u 1 = ( 2 x. x ) )
28 eqcom 2629 . . . . . . . 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 10033 . . . . . . . 8 |- ( x e. ZZ -> 0 e. CC )
31 1cnd 10056 . . . . . . . 8 |- ( x e. ZZ -> 1 e. CC )
3224, 18mulcld 10060 . . . . . . . 8 |- ( x e. ZZ -> ( 2 x. x ) e. CC )
33 subadd2 10285 . . . . . . . . 9 |- ( ( 0 e. CC /\ 1 e. CC /\ ( 2 x. x ) e. CC ) -> ( ( 0 - 1 ) = ( 2 x. x ) <-> ( ( 2 x. x ) + 1 ) = 0 ) )
3433bicomd 213 . . . . . . . 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 1326 . . . . . . 7 |- ( x e. ZZ -> ( ( ( 2 x. x ) + 1 ) = 0 <-> ( 0 - 1 ) = ( 2 x. x ) ) )
36 df-neg 10269 . . . . . . . . . 10 |- -u 1 = ( 0 - 1 )
3736eqcomi 2631 . . . . . . . . 9 |- ( 0 - 1 ) = -u 1
3837a1i 11 . . . . . . . 8 |- ( x e. ZZ -> ( 0 - 1 ) = -u 1 )
3938eqeq1d 2624 . . . . . . 7 |- ( x e. ZZ -> ( ( 0 - 1 ) = ( 2 x. x ) <-> -u 1 = ( 2 x. x ) ) )
4029, 35, 393bitrd 294 . . . . . 6 |- ( x e. ZZ -> ( 0 = ( ( 2 x. x ) + 1 ) <-> -u 1 = ( 2 x. x ) ) )
4127, 40mtbird 315 . . . . 5 |- ( x e. ZZ -> -. 0 = ( ( 2 x. x ) + 1 ) )
4241nrex 3000 . . . 4 |- -. E. x e. ZZ 0 = ( ( 2 x. x ) + 1 )
4342intnan 960 . . 3 |- -. ( 0 e. ZZ /\ E. x e. ZZ 0 = ( ( 2 x. x ) + 1 ) )
44 eqeq1 2626 . . . . 5 |- ( z = 0 -> ( z = ( ( 2 x. x ) + 1 ) <-> 0 = ( ( 2 x. x ) + 1 ) ) )
4544rexbidv 3052 . . . 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 3366 . . 3 |- ( 0 e. O <-> ( 0 e. ZZ /\ E. x e. ZZ 0 = ( ( 2 x. x ) + 1 ) ) )
4843, 47mtbir 313 . 2 |- -. 0 e. O
4948nelir 2900 1 |- 0 e/ O
Colors of variables: wff setvar class
Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   E.wrex 2913   {crab 2916  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   ZZcz 11377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378
This theorem is referenced by: (None)
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