Theorem ssblex 9133

Description: A nested ball exists whose radius is less than any desired amount.

Hypothesis

Ref	Expression
ssblex.1	|- X = dom dom D

Assertion

Ref	Expression
ssblex	|- (((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) C_ (P( ball ` D)S)))

Distinct variable groups:   x,D   x,P   x,R   x,S

Proof of Theorem ssblex

Step	Hyp	Ref	Expression
1		lelttric 6805	. . . 4 |- ((R e. RR /\ S e. RR) -> (R <_ S \/ S < R))
2	1	ad2ant2r 445	. . 3 |- (((R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) -> (R <_ S \/ S < R))
3	2	3adant1 894	. 2 |- (((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) -> (R <_ S \/ S < R))
4		rehalfcl 7220	. . . . . 6 |- (R e. RR -> (R / 2) e. RR)
5	4	ad2antrr 440	. . . . 5 |- (((R e. RR /\ 0 < R) /\ R <_ S) -> (R / 2) e. RR)
6	5	3ad2antl2 1039	. . . 4 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> (R / 2) e. RR)
7		halfpos2 7223	. . . . . . 7 |- (R e. RR -> (0 < R <-> 0 < (R / 2)))
8	7	biimpa 460	. . . . . 6 |- ((R e. RR /\ 0 < R) -> 0 < (R / 2))
9	8	adantr 425	. . . . 5 |- (((R e. RR /\ 0 < R) /\ R <_ S) -> 0 < (R / 2))
10	9	3ad2antl2 1039	. . . 4 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> 0 < (R / 2))
11		halfpos 7222	. . . . . . 7 |- (R e. RR -> (0 < R <-> (R / 2) < R))
12	11	biimpa 460	. . . . . 6 |- ((R e. RR /\ 0 < R) -> (R / 2) < R)
13	12	adantr 425	. . . . 5 |- (((R e. RR /\ 0 < R) /\ R <_ S) -> (R / 2) < R)
14	13	3ad2antl2 1039	. . . 4 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> (R / 2) < R)
15	4	adantr 425	. . . . . . . . . 10 |- ((R e. RR /\ 0 < R) -> (R / 2) e. RR)
16	15	ad2antrr 440	. . . . . . . . 9 |- ((((R e. RR /\ 0 < R) /\ S e. RR) /\ R <_ S) -> (R / 2) e. RR)
17		simplll 452	. . . . . . . . 9 |- ((((R e. RR /\ 0 < R) /\ S e. RR) /\ R <_ S) -> R e. RR)
18		simplr 449	. . . . . . . . 9 |- ((((R e. RR /\ 0 < R) /\ S e. RR) /\ R <_ S) -> S e. RR)
19	12	ad2antrr 440	. . . . . . . . 9 |- ((((R e. RR /\ 0 < R) /\ S e. RR) /\ R <_ S) -> (R / 2) < R)
20		simpr 350	. . . . . . . . 9 |- ((((R e. RR /\ 0 < R) /\ S e. RR) /\ R <_ S) -> R <_ S)
21	16, 17, 18, 19, 20	ltletrd 6698	. . . . . . . 8 |- ((((R e. RR /\ 0 < R) /\ S e. RR) /\ R <_ S) -> (R / 2) < S)
22		ltle 6690	. . . . . . . . . . 11 |- (((R / 2) e. RR /\ S e. RR) -> ((R / 2) < S -> (R / 2) <_ S))
23	22, 4	sylan 497	. . . . . . . . . 10 |- ((R e. RR /\ S e. RR) -> ((R / 2) < S -> (R / 2) <_ S))
24	23	adantlr 429	. . . . . . . . 9 |- (((R e. RR /\ 0 < R) /\ S e. RR) -> ((R / 2) < S -> (R / 2) <_ S))
25	24	adantr 425	. . . . . . . 8 |- ((((R e. RR /\ 0 < R) /\ S e. RR) /\ R <_ S) -> ((R / 2) < S -> (R / 2) <_ S))
26	21, 25	mpd 29	. . . . . . 7 |- ((((R e. RR /\ 0 < R) /\ S e. RR) /\ R <_ S) -> (R / 2) <_ S)
27	26	adantlrr 435	. . . . . 6 |- ((((R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> (R / 2) <_ S)
28	27	3adantl1 1032	. . . . 5 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> (R / 2) <_ S)
29		ssblex.1	. . . . . . 7 |- X = dom dom D
30	29	ssbl 9132	. . . . . 6 |- ((((D e. Met /\ P e. X) /\ ((R / 2) e. RR /\ 0 < (R / 2)) /\ (S e. RR /\ 0 < S)) /\ (R / 2) <_ S) -> (P( ball ` D)(R / 2)) C_ (P( ball ` D)S))
31	15, 8	jca 310	. . . . . 6 |- ((R e. RR /\ 0 < R) -> ((R / 2) e. RR /\ 0 < (R / 2)))
32	30, 31	syl3anl2 1146	. . . . 5 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ (R / 2) <_ S) -> (P( ball ` D)(R / 2)) C_ (P( ball ` D)S))
33	28, 32	syldan 516	. . . 4 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> (P( ball ` D)(R / 2)) C_ (P( ball ` D)S))
34		breq2 3342	. . . . . 6 |- (x = (R / 2) -> (0 < x <-> 0 < (R / 2)))
35		breq1 3341	. . . . . 6 |- (x = (R / 2) -> (x < R <-> (R / 2) < R))
36		opreq2 4890	. . . . . . 7 |- (x = (R / 2) -> (P( ball ` D)x) = (P( ball ` D)(R / 2)))
37	36	sseq1d 2644	. . . . . 6 |- (x = (R / 2) -> ((P( ball ` D)x) C_ (P( ball ` D)S) <-> (P( ball ` D)(R / 2)) C_ (P( ball ` D)S)))
38	34, 35, 37	3anbi123d 1168	. . . . 5 |- (x = (R / 2) -> ((0 < x /\ x < R /\ (P( ball ` D)x) C_ (P( ball ` D)S)) <-> (0 < (R / 2) /\ (R / 2) < R /\ (P( ball ` D)(R / 2)) C_ (P( ball ` D)S))))
39	38	rcla4ev 2381	. . . 4 |- (((R / 2) e. RR /\ (0 < (R / 2) /\ (R / 2) < R /\ (P( ball ` D)(R / 2)) C_ (P( ball ` D)S))) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) C_ (P( ball ` D)S)))
40	6, 10, 14, 33, 39	syl13anc 1102	. . 3 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) C_ (P( ball ` D)S)))
41		simpl3l 931	. . . 4 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ S < R) -> S e. RR)
42		simpl3r 932	. . . 4 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ S < R) -> 0 < S)
43		simpr 350	. . . 4 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ S < R) -> S < R)
44		ssid 2634	. . . . 5 |- (P( ball ` D)S) C_ (P( ball ` D)S)
45	44	a1i 8	. . . 4 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ S < R) -> (P( ball ` D)S) C_ (P( ball ` D)S))
46		breq2 3342	. . . . . 6 |- (x = S -> (0 < x <-> 0 < S))
47		breq1 3341	. . . . . 6 |- (x = S -> (x < R <-> S < R))
48		opreq2 4890	. . . . . . 7 |- (x = S -> (P( ball ` D)x) = (P( ball ` D)S))
49	48	sseq1d 2644	. . . . . 6 |- (x = S -> ((P( ball ` D)x) C_ (P( ball ` D)S) <-> (P( ball ` D)S) C_ (P( ball ` D)S)))
50	46, 47, 49	3anbi123d 1168	. . . . 5 |- (x = S -> ((0 < x /\ x < R /\ (P( ball ` D)x) C_ (P( ball ` D)S)) <-> (0 < S /\ S < R /\ (P( ball ` D)S) C_ (P( ball ` D)S))))
51	50	rcla4ev 2381	. . . 4 |- ((S e. RR /\ (0 < S /\ S < R /\ (P( ball ` D)S) C_ (P( ball ` D)S))) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) C_ (P( ball ` D)S)))
52	41, 42, 43, 45, 51	syl13anc 1102	. . 3 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ S < R) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) C_ (P( ball ` D)S)))
53	40, 52	jaodan 471	. 2 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ (R <_ S \/ S < R)) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) C_ (P( ball ` D)S)))
54	3, 53	mpdan 768	1 |- (((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) C_ (P( ball ` D)S)))

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106   C_ wss 2593   class class class wbr 3338  dom cdm 3986  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386   / cdiv 6447   <_ cle 6448   < clt 6653  2c2 7145  Metcme 9066   ball cbl 9068

This theorem is referenced by:  opni3 9143

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-2 7154  df-met 9070  df-bl 9072

Copyright terms: Public domain