Theorem pw2dvdseulemle 12408

Description: Lemma for pw2dvdseu 12409. 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 9331	. 2 |- ( ph -> A e. RR )
3		pw2dvdseulemle.b	. . 3 |- ( ph -> B e. NN0 )
4	3	nn0red 9331	. 2 |- ( ph -> B e. RR )
5		pw2dvdseulemle.n2b	. . 3 |- ( ph -> -. ( 2 ^ ( B + 1 ) ) || N )
6		2cnd 9091	. . . . . 6 |- ( ( ph /\ B < A ) -> 2 e. CC )
7	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ B < A ) -> B e. NN0 )
8		peano2nn0 9317	. . . . . . . 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 9417	. . . . . . . . 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 9428	. . . . . . 7 |- ( ( ( B + 1 ) e. NN0 /\ A e. NN0 /\ ( B + 1 ) <_ A ) -> ( A - ( B + 1 ) ) e. NN0 )
16	9, 10, 14, 15	syl3anc 1249	. . . . . 6 |- ( ( ph /\ B < A ) -> ( A - ( B + 1 ) ) e. NN0 )
17	6, 16, 9	expaddd 10801	. . . . 5 |- ( ( ph /\ B < A ) -> ( 2 ^ ( ( B + 1 ) + ( A - ( B + 1 ) ) ) ) = ( ( 2 ^ ( B + 1 ) ) x. ( 2 ^ ( A - ( B + 1 ) ) ) ) )
18	9	nn0cnd 9332	. . . . . . . 8 |- ( ( ph /\ B < A ) -> ( B + 1 ) e. CC )
19	10	nn0cnd 9332	. . . . . . . 8 |- ( ( ph /\ B < A ) -> A e. CC )
20	18, 19	pncan3d 8368	. . . . . . 7 |- ( ( ph /\ B < A ) -> ( ( B + 1 ) + ( A - ( B + 1 ) ) ) = A )
21	20	oveq2d 5950	. . . . . 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 4065	. . . . 5 |- ( ( ph /\ B < A ) -> ( 2 ^ ( ( B + 1 ) + ( A - ( B + 1 ) ) ) ) || N )
25	17, 24	eqbrtrrd 4067	. . . 4 |- ( ( ph /\ B < A ) -> ( ( 2 ^ ( B + 1 ) ) x. ( 2 ^ ( A - ( B + 1 ) ) ) ) || N )
26		2nn 9180	. . . . . . . 8 |- 2 e. NN
27	26	a1i 9	. . . . . . 7 |- ( ( ph /\ B < A ) -> 2 e. NN )
28	27, 9	nnexpcld 10821	. . . . . 6 |- ( ( ph /\ B < A ) -> ( 2 ^ ( B + 1 ) ) e. NN )
29	28	nnzd 9476	. . . . 5 |- ( ( ph /\ B < A ) -> ( 2 ^ ( B + 1 ) ) e. ZZ )
30	27, 16	nnexpcld 10821	. . . . . 6 |- ( ( ph /\ B < A ) -> ( 2 ^ ( A - ( B + 1 ) ) ) e. NN )
31	30	nnzd 9476	. . . . 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 9476	. . . . 5 |- ( ( ph /\ B < A ) -> N e. ZZ )
35		muldvds1 12046	. . . . 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 1249	. . . 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 666	. 2 |- ( ph -> -. B < A )
39	2, 4, 38	nltled 8175	1 |- ( ph -> A <_ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2175   class class class wbr 4043  (class class class)co 5934   1c1 7908   + caddc 7910   x. cmul 7912   < clt 8089   <_ cle 8090   - cmin 8225   NNcn 9018   2c2 9069   NN0cn0 9277   ZZcz 9354   ^cexp 10664   || cdvds 12017

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024  ax-pre-mulext 8025

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-ilim 4414  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-frec 6467  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637  df-div 8728  df-inn 9019  df-2 9077  df-n0 9278  df-z 9355  df-uz 9631  df-seqfrec 10574  df-exp 10665  df-dvds 12018

This theorem is referenced by:  pw2dvdseu  12409