Theorem nn0onn0exALTV 41609

Description: For each odd nonnegative integer there is a nonnegative integer which, multiplied by 2 and increased by 1, results in the odd nonnegative integer. (Contributed by AV, 30-May-2020.) (Revised by AV, 22-Jun-2020.)

Assertion

Ref	Expression
nn0onn0exALTV	|- ( ( N e. NN0 /\ N e. Odd ) -> E. m e. NN0 N = ( ( 2 x. m ) + 1 ) )

Distinct variable group:   m, N

Proof of Theorem nn0onn0exALTV

Step	Hyp	Ref	Expression
1		nn0oALTV 41607	. 2 |- ( ( N e. NN0 /\ N e. Odd ) -> ( ( N - 1 ) / 2 ) e. NN0 )
2		simpr 477	. . 3 |- ( ( N e. NN0 /\ ( ( N - 1 ) / 2 ) e. NN0 ) -> ( ( N - 1 ) / 2 ) e. NN0 )
3		oveq2 6658	. . . . . 6 |- ( m = ( ( N - 1 ) / 2 ) -> ( 2 x. m ) = ( 2 x. ( ( N - 1 ) / 2 ) ) )
4	3	oveq1d 6665	. . . . 5 |- ( m = ( ( N - 1 ) / 2 ) -> ( ( 2 x. m ) + 1 ) = ( ( 2 x. ( ( N - 1 ) / 2 ) ) + 1 ) )
5	4	eqeq2d 2632	. . . 4 |- ( m = ( ( N - 1 ) / 2 ) -> ( N = ( ( 2 x. m ) + 1 ) <-> N = ( ( 2 x. ( ( N - 1 ) / 2 ) ) + 1 ) ) )
6	5	adantl 482	. . 3 |- ( ( ( N e. NN0 /\ ( ( N - 1 ) / 2 ) e. NN0 ) /\ m = ( ( N - 1 ) / 2 ) ) -> ( N = ( ( 2 x. m ) + 1 ) <-> N = ( ( 2 x. ( ( N - 1 ) / 2 ) ) + 1 ) ) )
7		nn0cn 11302	. . . . . . . 8 |- ( N e. NN0 -> N e. CC )
8		peano2cnm 10347	. . . . . . . 8 |- ( N e. CC -> ( N - 1 ) e. CC )
9	7, 8	syl 17	. . . . . . 7 |- ( N e. NN0 -> ( N - 1 ) e. CC )
10		2cnd 11093	. . . . . . 7 |- ( N e. NN0 -> 2 e. CC )
11		2ne0 11113	. . . . . . . 8 |- 2 =/= 0
12	11	a1i 11	. . . . . . 7 |- ( N e. NN0 -> 2 =/= 0 )
13	9, 10, 12	divcan2d 10803	. . . . . 6 |- ( N e. NN0 -> ( 2 x. ( ( N - 1 ) / 2 ) ) = ( N - 1 ) )
14	13	oveq1d 6665	. . . . 5 |- ( N e. NN0 -> ( ( 2 x. ( ( N - 1 ) / 2 ) ) + 1 ) = ( ( N - 1 ) + 1 ) )
15		npcan1 10455	. . . . . 6 |- ( N e. CC -> ( ( N - 1 ) + 1 ) = N )
16	7, 15	syl 17	. . . . 5 |- ( N e. NN0 -> ( ( N - 1 ) + 1 ) = N )
17	14, 16	eqtr2d 2657	. . . 4 |- ( N e. NN0 -> N = ( ( 2 x. ( ( N - 1 ) / 2 ) ) + 1 ) )
18	17	adantr 481	. . 3 |- ( ( N e. NN0 /\ ( ( N - 1 ) / 2 ) e. NN0 ) -> N = ( ( 2 x. ( ( N - 1 ) / 2 ) ) + 1 ) )
19	2, 6, 18	rspcedvd 3317	. 2 |- ( ( N e. NN0 /\ ( ( N - 1 ) / 2 ) e. NN0 ) -> E. m e. NN0 N = ( ( 2 x. m ) + 1 ) )
20	1, 19	syldan 487	1 |- ( ( N e. NN0 /\ N e. Odd ) -> E. m e. NN0 N = ( ( 2 x. m ) + 1 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   2c2 11070   NN0cn0 11292   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-even 41539  df-odd 41540

This theorem is referenced by: (None)