Theorem pw2dvdseulemle 12684

Description: Lemma for pw2dvdseu 12685. Powers of two which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.)

Hypotheses

Ref	Expression
pw2dvdseulemle.n	|- ( ph -> N e. NN )
pw2dvdseulemle.a	|- ( ph -> A e. NN0 )
pw2dvdseulemle.b	|- ( ph -> B e. NN0 )
pw2dvdseulemle.2a	|- ( ph -> ( 2 ^ A ) || N )
pw2dvdseulemle.n2b	|- ( ph -> -. ( 2 ^ ( B + 1 ) ) || N )

Assertion

Ref	Expression
pw2dvdseulemle	|- ( ph -> A <_ B )

Proof of Theorem pw2dvdseulemle

Step	Hyp	Ref	Expression
1		pw2dvdseulemle.a	. . 3 |- ( ph -> A e. NN0 )
2	1	nn0red 9419	. 2 |- ( ph -> A e. RR )
3		pw2dvdseulemle.b	. . 3 |- ( ph -> B e. NN0 )
4	3	nn0red 9419	. 2 |- ( ph -> B e. RR )
5		pw2dvdseulemle.n2b	. . 3 |- ( ph -> -. ( 2 ^ ( B + 1 ) ) || N )
6		2cnd 9179	. . . . . 6 |- ( ( ph /\ B < A ) -> 2 e. CC )
7	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ B < A ) -> B e. NN0 )
8		peano2nn0 9405	. . . . . . . 8 |- ( B e. NN0 -> ( B + 1 ) e. NN0 )
9	7, 8	syl 14	. . . . . . 7 |- ( ( ph /\ B < A ) -> ( B + 1 ) e. NN0 )
10	1	adantr 276	. . . . . . 7 |- ( ( ph /\ B < A ) -> A e. NN0 )
11		simpr 110	. . . . . . . 8 |- ( ( ph /\ B < A ) -> B < A )
12		nn0ltp1le 9505	. . . . . . . . 9 |- ( ( B e. NN0 /\ A e. NN0 ) -> ( B < A <-> ( B + 1 ) <_ A ) )
13	7, 10, 12	syl2anc 411	. . . . . . . 8 |- ( ( ph /\ B < A ) -> ( B < A <-> ( B + 1 ) <_ A ) )
14	11, 13	mpbid 147	. . . . . . 7 |- ( ( ph /\ B < A ) -> ( B + 1 ) <_ A )
15		nn0sub2 9516	. . . . . . 7 |- ( ( ( B + 1 ) e. NN0 /\ A e. NN0 /\ ( B + 1 ) <_ A ) -> ( A - ( B + 1 ) ) e. NN0 )
16	9, 10, 14, 15	syl3anc 1271	. . . . . 6 |- ( ( ph /\ B < A ) -> ( A - ( B + 1 ) ) e. NN0 )
17	6, 16, 9	expaddd 10892	. . . . 5 |- ( ( ph /\ B < A ) -> ( 2 ^ ( ( B + 1 ) + ( A - ( B + 1 ) ) ) ) = ( ( 2 ^ ( B + 1 ) ) x. ( 2 ^ ( A - ( B + 1 ) ) ) ) )
18	9	nn0cnd 9420	. . . . . . . 8 |- ( ( ph /\ B < A ) -> ( B + 1 ) e. CC )
19	10	nn0cnd 9420	. . . . . . . 8 |- ( ( ph /\ B < A ) -> A e. CC )
20	18, 19	pncan3d 8456	. . . . . . 7 |- ( ( ph /\ B < A ) -> ( ( B + 1 ) + ( A - ( B + 1 ) ) ) = A )
21	20	oveq2d 6016	. . . . . 6 |- ( ( ph /\ B < A ) -> ( 2 ^ ( ( B + 1 ) + ( A - ( B + 1 ) ) ) ) = ( 2 ^ A ) )
22		pw2dvdseulemle.2a	. . . . . . 7 |- ( ph -> ( 2 ^ A ) || N )
23	22	adantr 276	. . . . . 6 |- ( ( ph /\ B < A ) -> ( 2 ^ A ) || N )
24	21, 23	eqbrtrd 4104	. . . . 5 |- ( ( ph /\ B < A ) -> ( 2 ^ ( ( B + 1 ) + ( A - ( B + 1 ) ) ) ) || N )
25	17, 24	eqbrtrrd 4106	. . . 4 |- ( ( ph /\ B < A ) -> ( ( 2 ^ ( B + 1 ) ) x. ( 2 ^ ( A - ( B + 1 ) ) ) ) || N )
26		2nn 9268	. . . . . . . 8 |- 2 e. NN
27	26	a1i 9	. . . . . . 7 |- ( ( ph /\ B < A ) -> 2 e. NN )
28	27, 9	nnexpcld 10912	. . . . . 6 |- ( ( ph /\ B < A ) -> ( 2 ^ ( B + 1 ) ) e. NN )
29	28	nnzd 9564	. . . . 5 |- ( ( ph /\ B < A ) -> ( 2 ^ ( B + 1 ) ) e. ZZ )
30	27, 16	nnexpcld 10912	. . . . . 6 |- ( ( ph /\ B < A ) -> ( 2 ^ ( A - ( B + 1 ) ) ) e. NN )
31	30	nnzd 9564	. . . . 5 |- ( ( ph /\ B < A ) -> ( 2 ^ ( A - ( B + 1 ) ) ) e. ZZ )
32		pw2dvdseulemle.n	. . . . . . 7 |- ( ph -> N e. NN )
33	32	adantr 276	. . . . . 6 |- ( ( ph /\ B < A ) -> N e. NN )
34	33	nnzd 9564	. . . . 5 |- ( ( ph /\ B < A ) -> N e. ZZ )
35		muldvds1 12322	. . . . 5 |- ( ( ( 2 ^ ( B + 1 ) ) e. ZZ /\ ( 2 ^ ( A - ( B + 1 ) ) ) e. ZZ /\ N e. ZZ ) -> ( ( ( 2 ^ ( B + 1 ) ) x. ( 2 ^ ( A - ( B + 1 ) ) ) ) || N -> ( 2 ^ ( B + 1 ) ) || N ) )
36	29, 31, 34, 35	syl3anc 1271	. . . 4 |- ( ( ph /\ B < A ) -> ( ( ( 2 ^ ( B + 1 ) ) x. ( 2 ^ ( A - ( B + 1 ) ) ) ) || N -> ( 2 ^ ( B + 1 ) ) || N ) )
37	25, 36	mpd 13	. . 3 |- ( ( ph /\ B < A ) -> ( 2 ^ ( B + 1 ) ) || N )
38	5, 37	mtand 669	. 2 |- ( ph -> -. B < A )
39	2, 4, 38	nltled 8263	1 |- ( ph -> A <_ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   class class class wbr 4082  (class class class)co 6000   1c1 7996   + caddc 7998   x. cmul 8000   < clt 8177   <_ cle 8178   - cmin 8313   NNcn 9106   2c2 9157   NN0cn0 9365   ZZcz 9442   ^cexp 10755   || cdvds 12293

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-n0 9366  df-z 9443  df-uz 9719  df-seqfrec 10665  df-exp 10756  df-dvds 12294

This theorem is referenced by:  pw2dvdseu  12685