Theorem bcthlem13 8011

Description: Lemma for bcth 8032. In the sequence g of balls (expressed as ordered pairs), for any m, there is a larger n whose ball's center distance from limit p is less than half of the ball radius at m.

Hypotheses

Ref	Expression
bcthlem12.1	|- D e. CMet
bcthlem12.3	|- X = dom dom D
bcthlem12.7	|- A = (X X. {x e. RR | 0 < x})

Assertion

Ref	Expression
bcthlem13	|- (((g:NN-->A /\ m e. NN) /\ (1st o. g)(~~>m` D)q) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < ((2nd` (g` m)) / 2)))

Distinct variable groups:   D,n   x,n,g   m,n,x   n,q

Proof of Theorem bcthlem13

Step	Hyp	Ref	Expression
1		visset 1813	. . . 4 |- q e. V
2		bcthlem12.1	. . . . . 6 |- D e. CMet
3	2	cmsmeti 7962	. . . . 5 |- D e. Met
4		bcthlem12.3	. . . . . 6 |- X = dom dom D
5		1z 6159	. . . . . 6 |- 1 e. ZZ
6		nnuz 6439	. . . . . 6 |- NN = (ZZ>` 1)
7	4, 5, 6	lmcvg2 7933	. . . . 5 |- (((D e. Met /\ q e. V /\ (1st o. g)(~~>m` D)q) /\ (((2nd` (g` m)) / 2) e. RR /\ 0 < ((2nd` (g` m)) / 2))) -> E.j e. NN A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < ((2nd` (g` m)) / 2)))
8	3, 7	mp3anl1 910	. . . 4 |- (((q e. V /\ (1st o. g)(~~>m` D)q) /\ (((2nd` (g` m)) / 2) e. RR /\ 0 < ((2nd` (g` m)) / 2))) -> E.j e. NN A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < ((2nd` (g` m)) / 2)))
9	1, 8	mpanl1 706	. . 3 |- (((1st o. g)(~~>m` D)q /\ (((2nd` (g` m)) / 2) e. RR /\ 0 < ((2nd` (g` m)) / 2))) -> E.j e. NN A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < ((2nd` (g` m)) / 2)))
10		pm3.27 323	. . 3 |- (((g:NN-->A /\ m e. NN) /\ (1st o. g)(~~>m` D)q) -> (1st o. g)(~~>m` D)q)
11		bcthlem12.7	. . . . . . 7 |- A = (X X. {x e. RR | 0 < x})
12	11	bcthlem4 8002	. . . . . 6 |- ((g:NN-->A /\ m e. NN) -> ((1st` (g` m)) e. X /\ ((2nd` (g` m)) e. RR /\ 0 < (2nd` (g` m)))))
13	12	pm3.27d 325	. . . . 5 |- ((g:NN-->A /\ m e. NN) -> ((2nd` (g` m)) e. RR /\ 0 < (2nd` (g` m))))
14		rehalfclt 6034	. . . . . . 7 |- ((2nd` (g` m)) e. RR -> ((2nd` (g` m)) / 2) e. RR)
15	14	adantr 389	. . . . . 6 |- (((2nd` (g` m)) e. RR /\ 0 < (2nd` (g` m))) -> ((2nd` (g` m)) / 2) e. RR)
16		halfpos2t 6037	. . . . . . 7 |- ((2nd` (g` m)) e. RR -> (0 < (2nd` (g` m)) <-> 0 < ((2nd` (g` m)) / 2)))
17	16	biimpa 416	. . . . . 6 |- (((2nd` (g` m)) e. RR /\ 0 < (2nd` (g` m))) -> 0 < ((2nd` (g` m)) / 2))
18	15, 17	jca 288	. . . . 5 |- (((2nd` (g` m)) e. RR /\ 0 < (2nd` (g` m))) -> (((2nd` (g` m)) / 2) e. RR /\ 0 < ((2nd` (g` m)) / 2)))
19	13, 18	syl 10	. . . 4 |- ((g:NN-->A /\ m e. NN) -> (((2nd` (g` m)) / 2) e. RR /\ 0 < ((2nd` (g` m)) / 2)))
20	19	adantr 389	. . 3 |- (((g:NN-->A /\ m e. NN) /\ (1st o. g)(~~>m` D)q) -> (((2nd` (g` m)) / 2) e. RR /\ 0 < ((2nd` (g` m)) / 2)))
21	9, 10, 20	sylanc 471	. 2 |- (((g:NN-->A /\ m e. NN) /\ (1st o. g)(~~>m` D)q) -> E.j e. NN A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < ((2nd` (g` m)) / 2)))
22		bcthlem2 8000	. . . . 5 |- (((m e. NN /\ j e. NN) /\ A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < ((2nd` (g` m)) / 2))) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < ((2nd` (g` m)) / 2)))
23	22	ex 373	. . . 4 |- ((m e. NN /\ j e. NN) -> (A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < ((2nd` (g` m)) / 2)) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < ((2nd` (g` m)) / 2))))
24	23	r19.23adva 1747	. . 3 |- (m e. NN -> (E.j e. NN A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < ((2nd` (g` m)) / 2)) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < ((2nd` (g` m)) / 2))))
25	24	ad2antlr 405	. 2 |- (((g:NN-->A /\ m e. NN) /\ (1st o. g)(~~>m` D)q) -> (E.j e. NN A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < ((2nd` (g` m)) / 2)) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < ((2nd` (g` m)) / 2))))
26	21, 25	mpd 26	1 |- (((g:NN-->A /\ m e. NN) /\ (1st o. g)(~~>m` D)q) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < ((2nd` (g` m)) / 2)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  {crab 1648  Vcvv 1811   class class class wbr 2619   X. cxp 3168  dom cdm 3170   o. ccom 3174  -->wf 3178  ` cfv 3182  (class class class)co 3963  1stc1st 4077  2ndc2nd 4078  RRcr 5233  0cc0 5234  1c1 5235   / cdiv 5294   <_ cle 5295  NNcn 5296   < clt 5486  2c2 5961  Metcme 7789  ~~>mclm 7919  CMetcms 7921

This theorem is referenced by:  bcthlem26 8024

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-2 5970  df-z 6136  df-uz 6418  df-lm 7922  df-cmet 7924

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