Theorem pw2dvdseu 12924

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 12922	. 2 |- ( N e. NN -> E. m e. NN0 ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) )
2		simpll 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 ) ) ) -> N e. NN )
3		simplrl 541	. . . . . . 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 542	. . . . . . 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 543	. . . . . . 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 546	. . . . . . 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 12923	. . . . . 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 545	. . . . . . 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 544	. . . . . . 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 12923	. . . . . 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 9600	. . . . . . 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 9600	. . . . . . 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 8431	. . . . . 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 957	. . . . 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 2632	. . 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 6083	. . . . . 6 |- ( m = x -> ( 2 ^ m ) = ( 2 ^ x ) )
18	17	breq1d 4135	. . . . 5 |- ( m = x -> ( ( 2 ^ m ) || N <-> ( 2 ^ x ) || N ) )
19		oveq1 6082	. . . . . . . 8 |- ( m = x -> ( m + 1 ) = ( x + 1 ) )
20	19	oveq2d 6091	. . . . . . 7 |- ( m = x -> ( 2 ^ ( m + 1 ) ) = ( 2 ^ ( x + 1 ) ) )
21	20	breq1d 4135	. . . . . 6 |- ( m = x -> ( ( 2 ^ ( m + 1 ) ) || N <-> ( 2 ^ ( x + 1 ) ) || N ) )
22	21	notbid 677	. . . . 5 |- ( m = x -> ( -. ( 2 ^ ( m + 1 ) ) || N <-> -. ( 2 ^ ( x + 1 ) ) || N ) )
23	18, 22	anbi12d 477	. . . 4 |- ( m = x -> ( ( ( 2 ^ m ) || N /\ -. ( 2 ^ ( m + 1 ) ) || N ) <-> ( ( 2 ^ x ) || N /\ -. ( 2 ^ ( x + 1 ) ) || N ) ) )
24	23	rmo4 3019	. . 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 2770	. 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 421	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 2209   A.wral 2528   E.wrex 2529   E!wreu 2530   E*wrmo 2531   class class class wbr 4125  (class class class)co 6075   1c1 8170   + caddc 8172   <_ cle 8351   NNcn 9283   2c2 9334   NN0cn0 9542   ^cexp 10953   || cdvds 12532

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-dvds 12533

This theorem is referenced by:  oddpwdclemxy  12925  oddpwdclemdvds  12926  oddpwdclemndvds  12927  oddpwdclemodd  12928  oddpwdclemdc  12929  oddpwdc  12930