Theorem pw2dvdseulemle 12802

Description: Lemma for pw2dvdseu 12803. 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 9500	. 2 |- ( ph -> A e. RR )
3		pw2dvdseulemle.b	. . 3 |- ( ph -> B e. NN0 )
4	3	nn0red 9500	. 2 |- ( ph -> B e. RR )
5		pw2dvdseulemle.n2b	. . 3 |- ( ph -> -. ( 2 ^ ( B + 1 ) ) || N )
6		2cnd 9258	. . . . . 6 |- ( ( ph /\ B < A ) -> 2 e. CC )
7	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ B < A ) -> B e. NN0 )
8		peano2nn0 9484	. . . . . . . 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 9586	. . . . . . . . 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 9597	. . . . . . 7 |- ( ( ( B + 1 ) e. NN0 /\ A e. NN0 /\ ( B + 1 ) <_ A ) -> ( A - ( B + 1 ) ) e. NN0 )
16	9, 10, 14, 15	syl3anc 1274	. . . . . 6 |- ( ( ph /\ B < A ) -> ( A - ( B + 1 ) ) e. NN0 )
17	6, 16, 9	expaddd 10983	. . . . 5 |- ( ( ph /\ B < A ) -> ( 2 ^ ( ( B + 1 ) + ( A - ( B + 1 ) ) ) ) = ( ( 2 ^ ( B + 1 ) ) x. ( 2 ^ ( A - ( B + 1 ) ) ) ) )
18	9	nn0cnd 9501	. . . . . . . 8 |- ( ( ph /\ B < A ) -> ( B + 1 ) e. CC )
19	10	nn0cnd 9501	. . . . . . . 8 |- ( ( ph /\ B < A ) -> A e. CC )
20	18, 19	pncan3d 8535	. . . . . . 7 |- ( ( ph /\ B < A ) -> ( ( B + 1 ) + ( A - ( B + 1 ) ) ) = A )
21	20	oveq2d 6044	. . . . . 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 4115	. . . . 5 |- ( ( ph /\ B < A ) -> ( 2 ^ ( ( B + 1 ) + ( A - ( B + 1 ) ) ) ) || N )
25	17, 24	eqbrtrrd 4117	. . . 4 |- ( ( ph /\ B < A ) -> ( ( 2 ^ ( B + 1 ) ) x. ( 2 ^ ( A - ( B + 1 ) ) ) ) || N )
26		2nn 9347	. . . . . . . 8 |- 2 e. NN
27	26	a1i 9	. . . . . . 7 |- ( ( ph /\ B < A ) -> 2 e. NN )
28	27, 9	nnexpcld 11003	. . . . . 6 |- ( ( ph /\ B < A ) -> ( 2 ^ ( B + 1 ) ) e. NN )
29	28	nnzd 9645	. . . . 5 |- ( ( ph /\ B < A ) -> ( 2 ^ ( B + 1 ) ) e. ZZ )
30	27, 16	nnexpcld 11003	. . . . . 6 |- ( ( ph /\ B < A ) -> ( 2 ^ ( A - ( B + 1 ) ) ) e. NN )
31	30	nnzd 9645	. . . . 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 9645	. . . . 5 |- ( ( ph /\ B < A ) -> N e. ZZ )
35		muldvds1 12440	. . . . 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 1274	. . . 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 671	. 2 |- ( ph -> -. B < A )
39	2, 4, 38	nltled 8342	1 |- ( ph -> A <_ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   1c1 8076   + caddc 8078   x. cmul 8080   < clt 8256   <_ cle 8257   - cmin 8392   NNcn 9185   2c2 9236   NN0cn0 9444   ZZcz 9523   ^cexp 10846   || cdvds 12411

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-n0 9445  df-z 9524  df-uz 9800  df-seqfrec 10756  df-exp 10847  df-dvds 12412

This theorem is referenced by:  pw2dvdseu  12803