Theorem bcthlem19 9295

Description: Lemma for bcth 9310. The distance between the center of a ball at m and any later ball in sequence g is less than half the radius of the ball at m.

Hypotheses

Ref	Expression
bcthlem18.1	|- D e. CMet
bcthlem18.3	|- X = dom dom D
bcthlem18.6	|- F = {<.<.j, y>., z>. | ((j e. NN /\ y e. A) /\ z = {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) C_ O))})}
bcthlem18.7	|- A = (X X. {x e. RR | 0 < x})
bcthlem18.8	|- O = ((X \ ((cls` J)` (M` j))) i^i ((1st` y)( ball ` D)((2nd` y) / 2)))

Assertion

Ref	Expression
bcthlem19	|- (((n e. NN /\ ((g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) /\ m e. NN)) /\ m < n) -> ((1st` (g` m))D(1st` (g` n))) < ((2nd` (g` m)) / 2))

Distinct variable groups:   j,m,n,y,z,A   j,p,r,D,m,n,y,z   k,m,n,F   j,J,p,r,y,z   j,M,p,r,y,z   z,O   j,X,m,n,p,r,y,z   j,k,x,g,p,r,y,z,m,n

Proof of Theorem bcthlem19

Step	Hyp	Ref	Expression
1		bcthlem18.1	. . . . 5 |- D e. CMet
2		bcthlem18.3	. . . . 5 |- X = dom dom D
3		bcthlem18.6	. . . . 5 |- F = {<.<.j, y>., z>. | ((j e. NN /\ y e. A) /\ z = {<.p, r>. | ((p e. X /\ (r e. RR /\ 0 < r)) /\ (r < ((2nd` y) / 2) /\ (p( ball ` D)r) C_ O))})}
4		bcthlem18.7	. . . . 5 |- A = (X X. {x e. RR | 0 < x})
5		bcthlem18.8	. . . . 5 |- O = ((X \ ((cls` J)` (M` j))) i^i ((1st` y)( ball ` D)((2nd` y) / 2)))
6	1, 2, 3, 4, 5	bcthlem18 9294	. . . 4 |- (n e. NN -> (((g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) /\ m e. NN) -> (m < n -> ((1st` (g` n))( ball ` D)(2nd` (g` n))) C_ ((1st` (g` m))( ball ` D)((2nd` (g` m)) / 2)))))
7	6	imp31 389	. . 3 |- (((n e. NN /\ ((g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) /\ m e. NN)) /\ m < n) -> ((1st` (g` n))( ball ` D)(2nd` (g` n))) C_ ((1st` (g` m))( ball ` D)((2nd` (g` m)) / 2)))
8	4	bcthlem4 9280	. . . . . . . 8 |- ((g:NN-->A /\ n e. NN) -> ((1st` (g` n)) e. X /\ ((2nd` (g` n)) e. RR /\ 0 < (2nd` (g` n)))))
9	8	ancoms 484	. . . . . . 7 |- ((n e. NN /\ g:NN-->A) -> ((1st` (g` n)) e. X /\ ((2nd` (g` n)) e. RR /\ 0 < (2nd` (g` n)))))
10	1	cmsmeti 9240	. . . . . . . 8 |- D e. Met
11	2	blcntr 9122	. . . . . . . 8 |- (((D e. Met /\ (1st` (g` n)) e. X) /\ ((2nd` (g` n)) e. RR /\ 0 < (2nd` (g` n)))) -> (1st` (g` n)) e. ((1st` (g` n))( ball ` D)(2nd` (g` n))))
12	10, 11	mpanl1 770	. . . . . . 7 |- (((1st` (g` n)) e. X /\ ((2nd` (g` n)) e. RR /\ 0 < (2nd` (g` n)))) -> (1st` (g` n)) e. ((1st` (g` n))( ball ` D)(2nd` (g` n))))
13	9, 12	syl 12	. . . . . 6 |- ((n e. NN /\ g:NN-->A) -> (1st` (g` n)) e. ((1st` (g` n))( ball ` D)(2nd` (g` n))))
14	13	adantrr 431	. . . . 5 |- ((n e. NN /\ (g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k)))) -> (1st` (g` n)) e. ((1st` (g` n))( ball ` D)(2nd` (g` n))))
15	14	adantrr 431	. . . 4 |- ((n e. NN /\ ((g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) /\ m e. NN)) -> (1st` (g` n)) e. ((1st` (g` n))( ball ` D)(2nd` (g` n))))
16	15	adantr 425	. . 3 |- (((n e. NN /\ ((g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) /\ m e. NN)) /\ m < n) -> (1st` (g` n)) e. ((1st` (g` n))( ball ` D)(2nd` (g` n))))
17	7, 16	sseldd 2620	. 2 |- (((n e. NN /\ ((g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) /\ m e. NN)) /\ m < n) -> (1st` (g` n)) e. ((1st` (g` m))( ball ` D)((2nd` (g` m)) / 2)))
18	4	bcthlem4 9280	. . . . . . . 8 |- ((g:NN-->A /\ m e. NN) -> ((1st` (g` m)) e. X /\ ((2nd` (g` m)) e. RR /\ 0 < (2nd` (g` m)))))
19	18	simplld 348	. . . . . . 7 |- ((g:NN-->A /\ m e. NN) -> (1st` (g` m)) e. X)
20	19	adantr 425	. . . . . 6 |- (((g:NN-->A /\ m e. NN) /\ n e. NN) -> (1st` (g` m)) e. X)
21	8	simplld 348	. . . . . . 7 |- ((g:NN-->A /\ n e. NN) -> (1st` (g` n)) e. X)
22	21	adantlr 429	. . . . . 6 |- (((g:NN-->A /\ m e. NN) /\ n e. NN) -> (1st` (g` n)) e. X)
23	18	simprd 352	. . . . . . . 8 |- ((g:NN-->A /\ m e. NN) -> ((2nd` (g` m)) e. RR /\ 0 < (2nd` (g` m))))
24		rehalfcl 7220	. . . . . . . . . 10 |- ((2nd` (g` m)) e. RR -> ((2nd` (g` m)) / 2) e. RR)
25	24	adantr 425	. . . . . . . . 9 |- (((2nd` (g` m)) e. RR /\ 0 < (2nd` (g` m))) -> ((2nd` (g` m)) / 2) e. RR)
26		halfpos2 7223	. . . . . . . . . 10 |- ((2nd` (g` m)) e. RR -> (0 < (2nd` (g` m)) <-> 0 < ((2nd` (g` m)) / 2)))
27	26	biimpa 460	. . . . . . . . 9 |- (((2nd` (g` m)) e. RR /\ 0 < (2nd` (g` m))) -> 0 < ((2nd` (g` m)) / 2))
28	25, 27	jca 310	. . . . . . . 8 |- (((2nd` (g` m)) e. RR /\ 0 < (2nd` (g` m))) -> (((2nd` (g` m)) / 2) e. RR /\ 0 < ((2nd` (g` m)) / 2)))
29	23, 28	syl 12	. . . . . . 7 |- ((g:NN-->A /\ m e. NN) -> (((2nd` (g` m)) / 2) e. RR /\ 0 < ((2nd` (g` m)) / 2)))
30	29	adantr 425	. . . . . 6 |- (((g:NN-->A /\ m e. NN) /\ n e. NN) -> (((2nd` (g` m)) / 2) e. RR /\ 0 < ((2nd` (g` m)) / 2)))
31	2	elbl2 9116	. . . . . . 7 |- (((D e. Met /\ (1st` (g` m)) e. X /\ (1st` (g` n)) e. X) /\ (((2nd` (g` m)) / 2) e. RR /\ 0 < ((2nd` (g` m)) / 2))) -> ((1st` (g` n)) e. ((1st` (g` m))( ball ` D)((2nd` (g` m)) / 2)) <-> ((1st` (g` m))D(1st` (g` n))) < ((2nd` (g` m)) / 2)))
32	10, 31	mp3anl1 1185	. . . . . 6 |- ((((1st` (g` m)) e. X /\ (1st` (g` n)) e. X) /\ (((2nd` (g` m)) / 2) e. RR /\ 0 < ((2nd` (g` m)) / 2))) -> ((1st` (g` n)) e. ((1st` (g` m))( ball ` D)((2nd` (g` m)) / 2)) <-> ((1st` (g` m))D(1st` (g` n))) < ((2nd` (g` m)) / 2)))
33	20, 22, 30, 32	syl21anc 1099	. . . . 5 |- (((g:NN-->A /\ m e. NN) /\ n e. NN) -> ((1st` (g` n)) e. ((1st` (g` m))( ball ` D)((2nd` (g` m)) / 2)) <-> ((1st` (g` m))D(1st` (g` n))) < ((2nd` (g` m)) / 2)))
34	33	ancoms 484	. . . 4 |- ((n e. NN /\ (g:NN-->A /\ m e. NN)) -> ((1st` (g` n)) e. ((1st` (g` m))( ball ` D)((2nd` (g` m)) / 2)) <-> ((1st` (g` m))D(1st` (g` n))) < ((2nd` (g` m)) / 2)))
35	34	adantrlr 437	. . 3 |- ((n e. NN /\ ((g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) /\ m e. NN)) -> ((1st` (g` n)) e. ((1st` (g` m))( ball ` D)((2nd` (g` m)) / 2)) <-> ((1st` (g` m))D(1st` (g` n))) < ((2nd` (g` m)) / 2)))
36	35	adantr 425	. 2 |- (((n e. NN /\ ((g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) /\ m e. NN)) /\ m < n) -> ((1st` (g` n)) e. ((1st` (g` m))( ball ` D)((2nd` (g` m)) / 2)) <-> ((1st` (g` m))D(1st` (g` n))) < ((2nd` (g` m)) / 2)))
37	17, 36	mpbid 212	1 |- (((n e. NN /\ ((g:NN-->A /\ A.k e. NN (g` (k + 1)) e. ((k + 1)F(g` k))) /\ m e. NN)) /\ m < n) -> ((1st` (g` m))D(1st` (g` n))) < ((2nd` (g` m)) / 2))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108   \ cdif 2590   i^i cin 2592   C_ wss 2593   class class class wbr 3338  {copab 3395   X. cxp 3984  dom cdm 3986  -->wf 3994  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   / cdiv 6447  NNcn 6449   < clt 6653  2c2 7145  clsccl 8938  Metcme 9066   ball cbl 9068  CMetcms 9199

This theorem is referenced by:  bcthlem20 9296  bcthlem24 9300

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-met 9070  df-bl 9072  df-cmet 9202

Copyright terms: Public domain