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Theorem dfodd6 41550
Description: Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Assertion
Ref Expression
dfodd6 |- Odd = { z e. ZZ | E. i e. ZZ z = ( ( 2 x. i ) + 1 ) }
Distinct variable group:   z, i
Proof of Theorem dfodd6
StepHypRef Expression
1 dfodd2 41549 . 2 |- Odd = { z e. ZZ | ( ( z - 1 ) / 2 ) e. ZZ }
2 simpr 477 . . . . . 6 |- ( ( z e. ZZ /\ ( ( z - 1 ) / 2 ) e. ZZ ) -> ( ( z - 1 ) / 2 ) e. ZZ )
3 oveq2 6658 . . . . . . . . . 10 |- ( i = ( ( z - 1 ) / 2 ) -> ( 2 x. i ) = ( 2 x. ( ( z - 1 ) / 2 ) ) )
4 peano2zm 11420 . . . . . . . . . . . . . 14 |- ( z e. ZZ -> ( z - 1 ) e. ZZ )
54zcnd 11483 . . . . . . . . . . . . 13 |- ( z e. ZZ -> ( z - 1 ) e. CC )
6 2cnd 11093 . . . . . . . . . . . . 13 |- ( z e. ZZ -> 2 e. CC )
7 2ne0 11113 . . . . . . . . . . . . . 14 |- 2 =/= 0
87a1i 11 . . . . . . . . . . . . 13 |- ( z e. ZZ -> 2 =/= 0 )
95, 6, 83jca 1242 . . . . . . . . . . . 12 |- ( z e. ZZ -> ( ( z - 1 ) e. CC /\ 2 e. CC /\ 2 =/= 0 ) )
109adantr 481 . . . . . . . . . . 11 |- ( ( z e. ZZ /\ ( ( z - 1 ) / 2 ) e. ZZ ) -> ( ( z - 1 ) e. CC /\ 2 e. CC /\ 2 =/= 0 ) )
11 divcan2 10693 . . . . . . . . . . 11 |- ( ( ( z - 1 ) e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( 2 x. ( ( z - 1 ) / 2 ) ) = ( z - 1 ) )
1210, 11syl 17 . . . . . . . . . 10 |- ( ( z e. ZZ /\ ( ( z - 1 ) / 2 ) e. ZZ ) -> ( 2 x. ( ( z - 1 ) / 2 ) ) = ( z - 1 ) )
133, 12sylan9eqr 2678 . . . . . . . . 9 |- ( ( ( z e. ZZ /\ ( ( z - 1 ) / 2 ) e. ZZ ) /\ i = ( ( z - 1 ) / 2 ) ) -> ( 2 x. i ) = ( z - 1 ) )
1413oveq1d 6665 . . . . . . . 8 |- ( ( ( z e. ZZ /\ ( ( z - 1 ) / 2 ) e. ZZ ) /\ i = ( ( z - 1 ) / 2 ) ) -> ( ( 2 x. i ) + 1 ) = ( ( z - 1 ) + 1 ) )
15 zcn 11382 . . . . . . . . . . 11 |- ( z e. ZZ -> z e. CC )
16 npcan1 10455 . . . . . . . . . . 11 |- ( z e. CC -> ( ( z - 1 ) + 1 ) = z )
1715, 16syl 17 . . . . . . . . . 10 |- ( z e. ZZ -> ( ( z - 1 ) + 1 ) = z )
1817adantr 481 . . . . . . . . 9 |- ( ( z e. ZZ /\ ( ( z - 1 ) / 2 ) e. ZZ ) -> ( ( z - 1 ) + 1 ) = z )
1918adantr 481 . . . . . . . 8 |- ( ( ( z e. ZZ /\ ( ( z - 1 ) / 2 ) e. ZZ ) /\ i = ( ( z - 1 ) / 2 ) ) -> ( ( z - 1 ) + 1 ) = z )
2014, 19eqtrd 2656 . . . . . . 7 |- ( ( ( z e. ZZ /\ ( ( z - 1 ) / 2 ) e. ZZ ) /\ i = ( ( z - 1 ) / 2 ) ) -> ( ( 2 x. i ) + 1 ) = z )
2120eqeq2d 2632 . . . . . 6 |- ( ( ( z e. ZZ /\ ( ( z - 1 ) / 2 ) e. ZZ ) /\ i = ( ( z - 1 ) / 2 ) ) -> ( z = ( ( 2 x. i ) + 1 ) <-> z = z ) )
22 eqidd 2623 . . . . . 6 |- ( ( z e. ZZ /\ ( ( z - 1 ) / 2 ) e. ZZ ) -> z = z )
232, 21, 22rspcedvd 3317 . . . . 5 |- ( ( z e. ZZ /\ ( ( z - 1 ) / 2 ) e. ZZ ) -> E. i e. ZZ z = ( ( 2 x. i ) + 1 ) )
2423ex 450 . . . 4 |- ( z e. ZZ -> ( ( ( z - 1 ) / 2 ) e. ZZ -> E. i e. ZZ z = ( ( 2 x. i ) + 1 ) ) )
25 oveq1 6657 . . . . . . . . . 10 |- ( z = ( ( 2 x. i ) + 1 ) -> ( z - 1 ) = ( ( ( 2 x. i ) + 1 ) - 1 ) )
26 zcn 11382 . . . . . . . . . . . 12 |- ( i e. ZZ -> i e. CC )
27 mulcl 10020 . . . . . . . . . . . 12 |- ( ( 2 e. CC /\ i e. CC ) -> ( 2 x. i ) e. CC )
286, 26, 27syl2an 494 . . . . . . . . . . 11 |- ( ( z e. ZZ /\ i e. ZZ ) -> ( 2 x. i ) e. CC )
29 pncan1 10454 . . . . . . . . . . 11 |- ( ( 2 x. i ) e. CC -> ( ( ( 2 x. i ) + 1 ) - 1 ) = ( 2 x. i ) )
3028, 29syl 17 . . . . . . . . . 10 |- ( ( z e. ZZ /\ i e. ZZ ) -> ( ( ( 2 x. i ) + 1 ) - 1 ) = ( 2 x. i ) )
3125, 30sylan9eqr 2678 . . . . . . . . 9 |- ( ( ( z e. ZZ /\ i e. ZZ ) /\ z = ( ( 2 x. i ) + 1 ) ) -> ( z - 1 ) = ( 2 x. i ) )
3231oveq1d 6665 . . . . . . . 8 |- ( ( ( z e. ZZ /\ i e. ZZ ) /\ z = ( ( 2 x. i ) + 1 ) ) -> ( ( z - 1 ) / 2 ) = ( ( 2 x. i ) / 2 ) )
3326adantl 482 . . . . . . . . . 10 |- ( ( z e. ZZ /\ i e. ZZ ) -> i e. CC )
34 2cnd 11093 . . . . . . . . . 10 |- ( ( z e. ZZ /\ i e. ZZ ) -> 2 e. CC )
357a1i 11 . . . . . . . . . 10 |- ( ( z e. ZZ /\ i e. ZZ ) -> 2 =/= 0 )
3633, 34, 35divcan3d 10806 . . . . . . . . 9 |- ( ( z e. ZZ /\ i e. ZZ ) -> ( ( 2 x. i ) / 2 ) = i )
3736adantr 481 . . . . . . . 8 |- ( ( ( z e. ZZ /\ i e. ZZ ) /\ z = ( ( 2 x. i ) + 1 ) ) -> ( ( 2 x. i ) / 2 ) = i )
3832, 37eqtrd 2656 . . . . . . 7 |- ( ( ( z e. ZZ /\ i e. ZZ ) /\ z = ( ( 2 x. i ) + 1 ) ) -> ( ( z - 1 ) / 2 ) = i )
39 simpr 477 . . . . . . . 8 |- ( ( z e. ZZ /\ i e. ZZ ) -> i e. ZZ )
4039adantr 481 . . . . . . 7 |- ( ( ( z e. ZZ /\ i e. ZZ ) /\ z = ( ( 2 x. i ) + 1 ) ) -> i e. ZZ )
4138, 40eqeltrd 2701 . . . . . 6 |- ( ( ( z e. ZZ /\ i e. ZZ ) /\ z = ( ( 2 x. i ) + 1 ) ) -> ( ( z - 1 ) / 2 ) e. ZZ )
4241ex 450 . . . . 5 |- ( ( z e. ZZ /\ i e. ZZ ) -> ( z = ( ( 2 x. i ) + 1 ) -> ( ( z - 1 ) / 2 ) e. ZZ ) )
4342rexlimdva 3031 . . . 4 |- ( z e. ZZ -> ( E. i e. ZZ z = ( ( 2 x. i ) + 1 ) -> ( ( z - 1 ) / 2 ) e. ZZ ) )
4424, 43impbid 202 . . 3 |- ( z e. ZZ -> ( ( ( z - 1 ) / 2 ) e. ZZ <-> E. i e. ZZ z = ( ( 2 x. i ) + 1 ) ) )
4544rabbiia 3185 . 2 |- { z e. ZZ | ( ( z - 1 ) / 2 ) e. ZZ } = { z e. ZZ | E. i e. ZZ z = ( ( 2 x. i ) + 1 ) }
461, 45eqtri 2644 1 |- Odd = { z e. ZZ | E. i e. ZZ z = ( ( 2 x. i ) + 1 ) }
Colors of variables: wff setvar class
Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   2c2 11070   ZZcz 11377   Odd codd 41538
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  df-odd 41540
This theorem is referenced by:  dfodd3  41563  odd2np1ALTV  41585  opoeALTV  41594  opeoALTV  41595
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