Theorem blssioo 9191

Description: The balls of the standard real metric space are included in the open real intervals.

Hypothesis

Ref	Expression
remet.1	|- D = ((abs o. - ) |` (RR X. RR))

Assertion

Ref	Expression
blssioo	|- ran ( ball ` D) C_ ran (,)

Proof of Theorem blssioo

Step	Hyp	Ref	Expression
1		remet.1	. . . . . . . . 9 |- D = ((abs o. - ) |` (RR X. RR))
2	1	bl2ioo 9189	. . . . . . . 8 |- ((v e. RR /\ u e. RR /\ 0 < u) -> (v( ball ` D)u) = ((v - u)(,)(v + u)))
3	2	eqeq1d 1892	. . . . . . 7 |- ((v e. RR /\ u e. RR /\ 0 < u) -> ((v( ball ` D)u) = w <-> ((v - u)(,)(v + u)) = w))
4		resubcl 6601	. . . . . . . . 9 |- ((v e. RR /\ u e. RR) -> (v - u) e. RR)
5		readdcl 6455	. . . . . . . . 9 |- ((v e. RR /\ u e. RR) -> (v + u) e. RR)
6		opreq1 4889	. . . . . . . . . . . . 13 |- (t = (v - u) -> (t(,)s) = ((v - u)(,)s))
7	6	eqeq1d 1892	. . . . . . . . . . . 12 |- (t = (v - u) -> ((t(,)s) = w <-> ((v - u)(,)s) = w))
8		opreq2 4890	. . . . . . . . . . . . 13 |- (s = (v + u) -> ((v - u)(,)s) = ((v - u)(,)(v + u)))
9	8	eqeq1d 1892	. . . . . . . . . . . 12 |- (s = (v + u) -> (((v - u)(,)s) = w <-> ((v - u)(,)(v + u)) = w))
10	7, 9	rcla42ev 2385	. . . . . . . . . . 11 |- (((v - u) e. RR* /\ (v + u) e. RR* /\ ((v - u)(,)(v + u)) = w) -> E.t e. RR* E.s e. RR* (t(,)s) = w)
11	10	3expia 1069	. . . . . . . . . 10 |- (((v - u) e. RR* /\ (v + u) e. RR*) -> (((v - u)(,)(v + u)) = w -> E.t e. RR* E.s e. RR* (t(,)s) = w))
12		rexr 6668	. . . . . . . . . 10 |- ((v - u) e. RR -> (v - u) e. RR*)
13		rexr 6668	. . . . . . . . . 10 |- ((v + u) e. RR -> (v + u) e. RR*)
14	11, 12, 13	syl2an 503	. . . . . . . . 9 |- (((v - u) e. RR /\ (v + u) e. RR) -> (((v - u)(,)(v + u)) = w -> E.t e. RR* E.s e. RR* (t(,)s) = w))
15	4, 5, 14	syl11anc 524	. . . . . . . 8 |- ((v e. RR /\ u e. RR) -> (((v - u)(,)(v + u)) = w -> E.t e. RR* E.s e. RR* (t(,)s) = w))
16	15	3adant3 896	. . . . . . 7 |- ((v e. RR /\ u e. RR /\ 0 < u) -> (((v - u)(,)(v + u)) = w -> E.t e. RR* E.s e. RR* (t(,)s) = w))
17	3, 16	sylbid 220	. . . . . 6 |- ((v e. RR /\ u e. RR /\ 0 < u) -> ((v( ball ` D)u) = w -> E.t e. RR* E.s e. RR* (t(,)s) = w))
18	17	3expb 1068	. . . . 5 |- ((v e. RR /\ (u e. RR /\ 0 < u)) -> ((v( ball ` D)u) = w -> E.t e. RR* E.s e. RR* (t(,)s) = w))
19		repos 7568	. . . . 5 |- (u e. (0(,) +oo) <-> (u e. RR /\ 0 < u))
20	18, 19	sylan2b 501	. . . 4 |- ((v e. RR /\ u e. (0(,) +oo)) -> ((v( ball ` D)u) = w -> E.t e. RR* E.s e. RR* (t(,)s) = w))
21	20	r19.23aivv 2217	. . 3 |- (E.v e. RR E.u e. (0(,) +oo)(v( ball ` D)u) = w -> E.t e. RR* E.s e. RR* (t(,)s) = w)
22	1	remet 9188	. . . . . 6 |- D e. Met
23	1	remetba 9187	. . . . . . 7 |- RR = dom dom D
24	23	blf 9121	. . . . . 6 |- (D e. Met -> ( ball ` D):(RR X. (0(,) +oo))-->~PRR)
25	22, 24	ax-mp 7	. . . . 5 |- ( ball ` D):(RR X. (0(,) +oo))-->~PRR
26		ffn 4562	. . . . 5 |- (( ball ` D):(RR X. (0(,) +oo))-->~PRR -> ( ball ` D) Fn (RR X. (0(,) +oo)))
27	25, 26	ax-mp 7	. . . 4 |- ( ball ` D) Fn (RR X. (0(,) +oo))
28		oprvelrn 4969	. . . 4 |- (( ball ` D) Fn (RR X. (0(,) +oo)) -> (w e. ran ( ball ` D) <-> E.v e. RR E.u e. (0(,) +oo)(v( ball ` D)u) = w))
29	27, 28	ax-mp 7	. . 3 |- (w e. ran ( ball ` D) <-> E.v e. RR E.u e. (0(,) +oo)(v( ball ` D)u) = w)
30		ioof 7569	. . . . 5 |- (,):(RR* X. RR*)-->~PRR
31		ffn 4562	. . . . 5 |- ((,):(RR* X. RR*)-->~PRR -> (,) Fn (RR* X. RR*))
32	30, 31	ax-mp 7	. . . 4 |- (,) Fn (RR* X. RR*)
33		oprvelrn 4969	. . . 4 |- ((,) Fn (RR* X. RR*) -> (w e. ran (,) <-> E.t e. RR* E.s e. RR* (t(,)s) = w))
34	32, 33	ax-mp 7	. . 3 |- (w e. ran (,) <-> E.t e. RR* E.s e. RR* (t(,)s) = w)
35	21, 29, 34	3imtr4i 236	. 2 |- (w e. ran ( ball ` D) -> w e. ran (,))
36	35	ssriv 2621	1 |- ran ( ball ` D) C_ ran (,)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106   C_ wss 2593  ~Pcpw 3032   class class class wbr 3338   X. cxp 3984  ran crn 3987   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386   + caddc 6389   - cmin 6445   +oocpnf 6650  RR*cxr 6652   < clt 6653  (,)cioo 7524  abscabs 8000  Metcme 9066   ball cbl 9068

This theorem is referenced by:  tgioo 9193

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-sup 5664  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-n0 7309  df-z 7345  df-q 7436  df-ioo 7528  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-met 9070  df-bl 9072

Copyright terms: Public domain