Theorem pw2dvdseu 12100

Description: A natural number has a unique highest power of two which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.)

Assertion

Ref	Expression
pw2dvdseu	|- ( N e. NN -> E! m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )

Distinct variable group:   m, N

Proof of Theorem pw2dvdseu

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pw2dvds 12098	. 2 |- ( N e. NN -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
2		simpll 519	. . . . . . 7 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> N e. NN )
3		simplrl 525	. . . . . . 7 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> m e. NN0 )
4		simplrr 526	. . . . . . 7 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> x e. NN0 )
5		simprll 527	. . . . . . 7 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> ( 2 ^ m ) || N )
6		simprrr 530	. . . . . . 7 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> -. ( 2 ^ ( x + 1 ) ) || N )
7	2, 3, 4, 5, 6	pw2dvdseulemle 12099	. . . . . 6 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> m <_ x )
8		simprrl 529	. . . . . . 7 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> ( 2 ^ x ) || N )
9		simprlr 528	. . . . . . 7 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> -. ( 2 ^ ( m + 1 ) ) || N )
10	2, 4, 3, 8, 9	pw2dvdseulemle 12099	. . . . . 6 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> x <_ m )
11	3	nn0red 9168	. . . . . . 7 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> m e. RR )
12	4	nn0red 9168	. . . . . . 7 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> x e. RR )
13	11, 12	letri3d 8014	. . . . . 6 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> ( m = x <-> ( m <_ x /\ x <_ m ) ) )
14	7, 10, 13	mpbir2and 934	. . . . 5 |- ( ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) ) -> m = x )
15	14	ex 114	. . . 4 |- ( ( N e. NN /\ ( m e. NN0 /\ x e. NN0 ) ) -> ( ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) -> m = x ) )
16	15	ralrimivva 2548	. . 3 |- ( N e. NN -> A. m e. NN0 A. x e. NN0 ( ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) -> m = x ) )
17		oveq2 5850	. . . . . 6 |- ( m = x -> ( 2 ^ m ) = ( 2 ^ x ) )
18	17	breq1d 3992	. . . . 5 |- ( m = x -> ( ( 2 ^ m ) || N <-> ( 2 ^ x ) || N ) )
19		oveq1 5849	. . . . . . . 8 |- ( m = x -> ( m + 1 ) = ( x + 1 ) )
20	19	oveq2d 5858	. . . . . . 7 |- ( m = x -> ( 2 ^ ( m + 1 ) ) = ( 2 ^ ( x + 1 ) ) )
21	20	breq1d 3992	. . . . . 6 |- ( m = x -> ( ( 2 ^ ( m + 1 ) ) || N <-> ( 2 ^ ( x + 1 ) ) || N ) )
22	21	notbid 657	. . . . 5 |- ( m = x -> ( -. ( 2 ^ ( m + 1 ) ) || N <-> -. ( 2 ^ ( x + 1 ) ) || N ) )
23	18, 22	anbi12d 465	. . . 4 |- ( m = x -> ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) <-> ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) )
24	23	rmo4 2919	. . 3 |- ( E* m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) <-> A. m e. NN0 A. x e. NN0 ( ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) -> m = x ) )
25	16, 24	sylibr 133	. 2 |- ( N e. NN -> E* m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
26		reu5 2678	. 2 |- ( E! m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) <-> ( E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) /\ E* m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) ) )
27	1, 25, 26	sylanbrc 414	1 |- ( N e. NN -> E! m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 2136   A.wral 2444   E.wrex 2445   E!wreu 2446   E*wrmo 2447   class class class wbr 3982  (class class class)co 5842   1c1 7754   + caddc 7756   <_ cle 7934   NNcn 8857   2c2 8908   NN0cn0 9114   ^cexp 10454   || cdvds 11727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-dvds 11728

This theorem is referenced by:  oddpwdclemxy  12101  oddpwdclemdvds  12102  oddpwdclemndvds  12103  oddpwdclemodd  12104  oddpwdclemdc  12105  oddpwdc  12106