Theorem pw2dvdseu 12199

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 12197	. 2 |- ( N e. NN -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
2		simpll 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 ) ) ) -> N e. NN )
3		simplrl 535	. . . . . . 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 536	. . . . . . 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 537	. . . . . . 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 540	. . . . . . 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 12198	. . . . . 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 539	. . . . . . 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 538	. . . . . . 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 12198	. . . . . 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 9259	. . . . . . 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 9259	. . . . . . 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 8102	. . . . . 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 946	. . . . 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 115	. . . 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 2572	. . 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 5903	. . . . . 6 |- ( m = x -> ( 2 ^ m ) = ( 2 ^ x ) )
18	17	breq1d 4028	. . . . 5 |- ( m = x -> ( ( 2 ^ m ) || N <-> ( 2 ^ x ) || N ) )
19		oveq1 5902	. . . . . . . 8 |- ( m = x -> ( m + 1 ) = ( x + 1 ) )
20	19	oveq2d 5911	. . . . . . 7 |- ( m = x -> ( 2 ^ ( m + 1 ) ) = ( 2 ^ ( x + 1 ) ) )
21	20	breq1d 4028	. . . . . 6 |- ( m = x -> ( ( 2 ^ ( m + 1 ) ) || N <-> ( 2 ^ ( x + 1 ) ) || N ) )
22	21	notbid 668	. . . . 5 |- ( m = x -> ( -. ( 2 ^ ( m + 1 ) ) || N <-> -. ( 2 ^ ( x + 1 ) ) || N ) )
23	18, 22	anbi12d 473	. . . 4 |- ( m = x -> ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) <-> ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) )
24	23	rmo4 2945	. . 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 134	. 2 |- ( N e. NN -> E* m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
26		reu5 2703	. 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 417	1 |- ( N e. NN -> E! m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2160   A.wral 2468   E.wrex 2469   E!wreu 2470   E*wrmo 2471   class class class wbr 4018  (class class class)co 5895   1c1 7841   + caddc 7843   <_ cle 8022   NNcn 8948   2c2 8999   NN0cn0 9205   ^cexp 10549   || cdvds 11825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-frec 6415  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-2 9007  df-n0 9206  df-z 9283  df-uz 9558  df-q 9649  df-rp 9683  df-fz 10038  df-fl 10300  df-mod 10353  df-seqfrec 10476  df-exp 10550  df-dvds 11826

This theorem is referenced by:  oddpwdclemxy  12200  oddpwdclemdvds  12201  oddpwdclemndvds  12202  oddpwdclemodd  12203  oddpwdclemdc  12204  oddpwdc  12205