Theorem pw2dvdseu 12109

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 12107	. 2 |- ( N e. NN -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
2		simpll 524	. . . . . . 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 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 ) ) ) -> m e. NN0 )
4		simplrr 531	. . . . . . 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 532	. . . . . . 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 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 ) ) ) -> -. ( 2 ^ ( x + 1 ) ) || N )
7	2, 3, 4, 5, 6	pw2dvdseulemle 12108	. . . . . 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 534	. . . . . . 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 533	. . . . . . 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 12108	. . . . . 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 9176	. . . . . . 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 9176	. . . . . . 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 8022	. . . . . 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 939	. . . . 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 2552	. . 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 5858	. . . . . 6 |- ( m = x -> ( 2 ^ m ) = ( 2 ^ x ) )
18	17	breq1d 3997	. . . . 5 |- ( m = x -> ( ( 2 ^ m ) || N <-> ( 2 ^ x ) || N ) )
19		oveq1 5857	. . . . . . . 8 |- ( m = x -> ( m + 1 ) = ( x + 1 ) )
20	19	oveq2d 5866	. . . . . . 7 |- ( m = x -> ( 2 ^ ( m + 1 ) ) = ( 2 ^ ( x + 1 ) ) )
21	20	breq1d 3997	. . . . . 6 |- ( m = x -> ( ( 2 ^ ( m + 1 ) ) || N <-> ( 2 ^ ( x + 1 ) ) || N ) )
22	21	notbid 662	. . . . 5 |- ( m = x -> ( -. ( 2 ^ ( m + 1 ) ) || N <-> -. ( 2 ^ ( x + 1 ) ) || N ) )
23	18, 22	anbi12d 470	. . . 4 |- ( m = x -> ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) <-> ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) )
24	23	rmo4 2923	. . 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 2682	. 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 415	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 2141   A.wral 2448   E.wrex 2449   E!wreu 2450   E*wrmo 2451   class class class wbr 3987  (class class class)co 5850   1c1 7762   + caddc 7764   <_ cle 7942   NNcn 8865   2c2 8916   NN0cn0 9122   ^cexp 10462   || cdvds 11736

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fz 9953  df-fl 10213  df-mod 10266  df-seqfrec 10389  df-exp 10463  df-dvds 11737

This theorem is referenced by:  oddpwdclemxy  12110  oddpwdclemdvds  12111  oddpwdclemndvds  12112  oddpwdclemodd  12113  oddpwdclemdc  12114  oddpwdc  12115