Theorem dyadss 23362

Description: Two closed dyadic rational intervals are either in a subset relationship or are almost disjoint (the interiors are disjoint). (Contributed by Mario Carneiro, 26-Mar-2015.) (Proof shortened by Mario Carneiro, 26-Apr-2016.)

Hypothesis

Ref	Expression
dyadmbl.1	|- F = ( x e. ZZ , y e. NN0 |-> <. ( x / ( 2 ^ y ) ) , ( ( x + 1 ) / ( 2 ^ y ) ) >. )

Assertion

Ref	Expression
dyadss	|- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) -> ( ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) -> D <_ C ) )

Distinct variable groups:   x, y, B   x, C, y   x, A, y   x, D, y   x, F, y

Proof of Theorem dyadss

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) )
2		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> B e. ZZ )
3		simplrr 801	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> D e. NN0 )
4		dyadmbl.1	. . . . . . . . . . 11 |- F = ( x e. ZZ , y e. NN0 |-> <. ( x / ( 2 ^ y ) ) , ( ( x + 1 ) / ( 2 ^ y ) ) >. )
5	4	dyadval 23360	. . . . . . . . . 10 |- ( ( B e. ZZ /\ D e. NN0 ) -> ( B F D ) = <. ( B / ( 2 ^ D ) ) , ( ( B + 1 ) / ( 2 ^ D ) ) >. )
6	2, 3, 5	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( B F D ) = <. ( B / ( 2 ^ D ) ) , ( ( B + 1 ) / ( 2 ^ D ) ) >. )
7	6	fveq2d 6195	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( [,] ` ( B F D ) ) = ( [,] ` <. ( B / ( 2 ^ D ) ) , ( ( B + 1 ) / ( 2 ^ D ) ) >. ) )
8		df-ov 6653	. . . . . . . 8 |- ( ( B / ( 2 ^ D ) ) [,] ( ( B + 1 ) / ( 2 ^ D ) ) ) = ( [,] ` <. ( B / ( 2 ^ D ) ) , ( ( B + 1 ) / ( 2 ^ D ) ) >. )
9	7, 8	syl6eqr 2674	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( [,] ` ( B F D ) ) = ( ( B / ( 2 ^ D ) ) [,] ( ( B + 1 ) / ( 2 ^ D ) ) ) )
10	2	zred 11482	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> B e. RR )
11		2nn 11185	. . . . . . . . . 10 |- 2 e. NN
12		nnexpcl 12873	. . . . . . . . . 10 |- ( ( 2 e. NN /\ D e. NN0 ) -> ( 2 ^ D ) e. NN )
13	11, 3, 12	sylancr 695	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( 2 ^ D ) e. NN )
14	10, 13	nndivred 11069	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( B / ( 2 ^ D ) ) e. RR )
15		peano2re 10209	. . . . . . . . . 10 |- ( B e. RR -> ( B + 1 ) e. RR )
16	10, 15	syl 17	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( B + 1 ) e. RR )
17	16, 13	nndivred 11069	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( ( B + 1 ) / ( 2 ^ D ) ) e. RR )
18		iccssre 12255	. . . . . . . 8 |- ( ( ( B / ( 2 ^ D ) ) e. RR /\ ( ( B + 1 ) / ( 2 ^ D ) ) e. RR ) -> ( ( B / ( 2 ^ D ) ) [,] ( ( B + 1 ) / ( 2 ^ D ) ) ) C_ RR )
19	14, 17, 18	syl2anc 693	. . . . . . 7 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( ( B / ( 2 ^ D ) ) [,] ( ( B + 1 ) / ( 2 ^ D ) ) ) C_ RR )
20	9, 19	eqsstrd 3639	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( [,] ` ( B F D ) ) C_ RR )
21		ovolss 23253	. . . . . 6 |- ( ( ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) /\ ( [,] ` ( B F D ) ) C_ RR ) -> ( vol* ` ( [,] ` ( A F C ) ) ) <_ ( vol* ` ( [,] ` ( B F D ) ) ) )
22	1, 20, 21	syl2anc 693	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( vol* ` ( [,] ` ( A F C ) ) ) <_ ( vol* ` ( [,] ` ( B F D ) ) ) )
23		simplll 798	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> A e. ZZ )
24		simplrl 800	. . . . . 6 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> C e. NN0 )
25	4	dyadovol 23361	. . . . . 6 |- ( ( A e. ZZ /\ C e. NN0 ) -> ( vol* ` ( [,] ` ( A F C ) ) ) = ( 1 / ( 2 ^ C ) ) )
26	23, 24, 25	syl2anc 693	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( vol* ` ( [,] ` ( A F C ) ) ) = ( 1 / ( 2 ^ C ) ) )
27	4	dyadovol 23361	. . . . . 6 |- ( ( B e. ZZ /\ D e. NN0 ) -> ( vol* ` ( [,] ` ( B F D ) ) ) = ( 1 / ( 2 ^ D ) ) )
28	2, 3, 27	syl2anc 693	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( vol* ` ( [,] ` ( B F D ) ) ) = ( 1 / ( 2 ^ D ) ) )
29	22, 26, 28	3brtr3d 4684	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( 1 / ( 2 ^ C ) ) <_ ( 1 / ( 2 ^ D ) ) )
30		nnexpcl 12873	. . . . . 6 |- ( ( 2 e. NN /\ C e. NN0 ) -> ( 2 ^ C ) e. NN )
31	11, 24, 30	sylancr 695	. . . . 5 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( 2 ^ C ) e. NN )
32		nnre 11027	. . . . . . 7 |- ( ( 2 ^ D ) e. NN -> ( 2 ^ D ) e. RR )
33		nngt0 11049	. . . . . . 7 |- ( ( 2 ^ D ) e. NN -> 0 < ( 2 ^ D ) )
34	32, 33	jca 554	. . . . . 6 |- ( ( 2 ^ D ) e. NN -> ( ( 2 ^ D ) e. RR /\ 0 < ( 2 ^ D ) ) )
35		nnre 11027	. . . . . . 7 |- ( ( 2 ^ C ) e. NN -> ( 2 ^ C ) e. RR )
36		nngt0 11049	. . . . . . 7 |- ( ( 2 ^ C ) e. NN -> 0 < ( 2 ^ C ) )
37	35, 36	jca 554	. . . . . 6 |- ( ( 2 ^ C ) e. NN -> ( ( 2 ^ C ) e. RR /\ 0 < ( 2 ^ C ) ) )
38		lerec 10906	. . . . . 6 |- ( ( ( ( 2 ^ D ) e. RR /\ 0 < ( 2 ^ D ) ) /\ ( ( 2 ^ C ) e. RR /\ 0 < ( 2 ^ C ) ) ) -> ( ( 2 ^ D ) <_ ( 2 ^ C ) <-> ( 1 / ( 2 ^ C ) ) <_ ( 1 / ( 2 ^ D ) ) ) )
39	34, 37, 38	syl2an 494	. . . . 5 |- ( ( ( 2 ^ D ) e. NN /\ ( 2 ^ C ) e. NN ) -> ( ( 2 ^ D ) <_ ( 2 ^ C ) <-> ( 1 / ( 2 ^ C ) ) <_ ( 1 / ( 2 ^ D ) ) ) )
40	13, 31, 39	syl2anc 693	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( ( 2 ^ D ) <_ ( 2 ^ C ) <-> ( 1 / ( 2 ^ C ) ) <_ ( 1 / ( 2 ^ D ) ) ) )
41	29, 40	mpbird 247	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( 2 ^ D ) <_ ( 2 ^ C ) )
42		2re 11090	. . . . 5 |- 2 e. RR
43	42	a1i 11	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> 2 e. RR )
44	3	nn0zd 11480	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> D e. ZZ )
45	24	nn0zd 11480	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> C e. ZZ )
46		1lt2 11194	. . . . 5 |- 1 < 2
47	46	a1i 11	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> 1 < 2 )
48	43, 44, 45, 47	leexp2d 13039	. . 3 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> ( D <_ C <-> ( 2 ^ D ) <_ ( 2 ^ C ) ) )
49	41, 48	mpbird 247	. 2 |- ( ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) /\ ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) ) -> D <_ C )
50	49	ex 450	1 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( C e. NN0 /\ D e. NN0 ) ) -> ( ( [,] ` ( A F C ) ) C_ ( [,] ` ( B F D ) ) -> D <_ C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   [,]cicc 12178   ^cexp 12860   vol*covol 23231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233

This theorem is referenced by:  dyadmaxlem  23365