Theorem pw2dvdseu 12122

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 12120	. 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 12121	. . . . . 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 12121	. . . . . 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 9189	. . . . . . 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 9189	. . . . . . 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 8035	. . . . . 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 5861	. . . . . 6 |- ( m = x -> ( 2 ^ m ) = ( 2 ^ x ) )
18	17	breq1d 3999	. . . . 5 |- ( m = x -> ( ( 2 ^ m ) || N <-> ( 2 ^ x ) || N ) )
19		oveq1 5860	. . . . . . . 8 |- ( m = x -> ( m + 1 ) = ( x + 1 ) )
20	19	oveq2d 5869	. . . . . . 7 |- ( m = x -> ( 2 ^ ( m + 1 ) ) = ( 2 ^ ( x + 1 ) ) )
21	20	breq1d 3999	. . . . . 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 3989  (class class class)co 5853   1c1 7775   + caddc 7777   <_ cle 7955   NNcn 8878   2c2 8929   NN0cn0 9135   ^cexp 10475   || cdvds 11749

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 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

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 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-dvds 11750

This theorem is referenced by:  oddpwdclemxy  12123  oddpwdclemdvds  12124  oddpwdclemndvds  12125  oddpwdclemodd  12126  oddpwdclemdc  12127  oddpwdc  12128