Theorem tgioo 9193

Description: The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals.

Hypotheses

Ref	Expression
remet.1	|- D = ((abs o. - ) |` (RR X. RR))
tgioo.2	|- J = (Open` D)

Assertion

Ref	Expression
tgioo	|- (topGen` ran (,)) = J

Proof of Theorem tgioo

Step	Hyp	Ref	Expression
1		remet.1	. . . . . 6 |- D = ((abs o. - ) |` (RR X. RR))
2	1	remet 9188	. . . . 5 |- D e. Met
3		eqid 1884	. . . . . 6 |- (Open` D) = (Open` D)
4	3	blbas 9149	. . . . 5 |- (D e. Met -> ran ( ball ` D) e. Bases)
5	2, 4	ax-mp 7	. . . 4 |- ran ( ball ` D) e. Bases
6		retopbas 8925	. . . 4 |- ran (,) e. Bases
7	5, 6	pm3.2i 307	. . 3 |- (ran ( ball ` D) e. Bases /\ ran (,) e. Bases)
8	1	blssioo 9191	. . . 4 |- ran ( ball ` D) C_ ran (,)
9	1	remetba 9187	. . . . . . . . 9 |- RR = dom dom D
10		tgioo.2	. . . . . . . . 9 |- J = (Open` D)
11	9, 10	isopn4 9139	. . . . . . . 8 |- (D e. Met -> (w e. J <-> (w C_ RR /\ A.v e. w E.u e. RR (0 < u /\ (v( ball ` D)u) C_ w))))
12	2, 11	ax-mp 7	. . . . . . 7 |- (w e. J <-> (w C_ RR /\ A.v e. w E.u e. RR (0 < u /\ (v( ball ` D)u) C_ w)))
13		elssuni 3206	. . . . . . . 8 |- (w e. ran (,) -> w C_ U.ran (,))
14		unirnioo 7571	. . . . . . . 8 |- U.ran (,) = RR
15	13, 14	syl6ss 2663	. . . . . . 7 |- (w e. ran (,) -> w C_ RR)
16		ioof 7569	. . . . . . . . . . 11 |- (,):(RR* X. RR*)-->~PRR
17		ffn 4562	. . . . . . . . . . 11 |- ((,):(RR* X. RR*)-->~PRR -> (,) Fn (RR* X. RR*))
18	16, 17	ax-mp 7	. . . . . . . . . 10 |- (,) Fn (RR* X. RR*)
19		oprvelrn 4969	. . . . . . . . . 10 |- ((,) Fn (RR* X. RR*) -> (w e. ran (,) <-> E.t e. RR* E.s e. RR* (t(,)s) = w))
20	18, 19	ax-mp 7	. . . . . . . . 9 |- (w e. ran (,) <-> E.t e. RR* E.s e. RR* (t(,)s) = w)
21	1	tgioolem 9192	. . . . . . . . . . . . 13 |- ((t e. RR* /\ s e. RR*) -> (v e. (t(,)s) -> E.u e. RR (0 < u /\ (v( ball ` D)u) C_ (t(,)s))))
22	21	adantr 425	. . . . . . . . . . . 12 |- (((t e. RR* /\ s e. RR*) /\ (t(,)s) = w) -> (v e. (t(,)s) -> E.u e. RR (0 < u /\ (v( ball ` D)u) C_ (t(,)s))))
23		eleq2 1958	. . . . . . . . . . . . 13 |- ((t(,)s) = w -> (v e. (t(,)s) <-> v e. w))
24	23	adantl 424	. . . . . . . . . . . 12 |- (((t e. RR* /\ s e. RR*) /\ (t(,)s) = w) -> (v e. (t(,)s) <-> v e. w))
25		sseq2 2639	. . . . . . . . . . . . . . 15 |- ((t(,)s) = w -> ((v( ball ` D)u) C_ (t(,)s) <-> (v( ball ` D)u) C_ w))
26	25	anbi2d 678	. . . . . . . . . . . . . 14 |- ((t(,)s) = w -> ((0 < u /\ (v( ball ` D)u) C_ (t(,)s)) <-> (0 < u /\ (v( ball ` D)u) C_ w)))
27	26	rexbidv 2124	. . . . . . . . . . . . 13 |- ((t(,)s) = w -> (E.u e. RR (0 < u /\ (v( ball ` D)u) C_ (t(,)s)) <-> E.u e. RR (0 < u /\ (v( ball ` D)u) C_ w)))
28	27	adantl 424	. . . . . . . . . . . 12 |- (((t e. RR* /\ s e. RR*) /\ (t(,)s) = w) -> (E.u e. RR (0 < u /\ (v( ball ` D)u) C_ (t(,)s)) <-> E.u e. RR (0 < u /\ (v( ball ` D)u) C_ w)))
29	22, 24, 28	3imtr3d 601	. . . . . . . . . . 11 |- (((t e. RR* /\ s e. RR*) /\ (t(,)s) = w) -> (v e. w -> E.u e. RR (0 < u /\ (v( ball ` D)u) C_ w)))
30	29	ex 402	. . . . . . . . . 10 |- ((t e. RR* /\ s e. RR*) -> ((t(,)s) = w -> (v e. w -> E.u e. RR (0 < u /\ (v( ball ` D)u) C_ w))))
31	30	r19.23aivv 2217	. . . . . . . . 9 |- (E.t e. RR* E.s e. RR* (t(,)s) = w -> (v e. w -> E.u e. RR (0 < u /\ (v( ball ` D)u) C_ w)))
32	20, 31	sylbi 216	. . . . . . . 8 |- (w e. ran (,) -> (v e. w -> E.u e. RR (0 < u /\ (v( ball ` D)u) C_ w)))
33	32	r19.21aiv 2175	. . . . . . 7 |- (w e. ran (,) -> A.v e. w E.u e. RR (0 < u /\ (v( ball ` D)u) C_ w))
34	12, 15, 33	sylanbrc 527	. . . . . 6 |- (w e. ran (,) -> w e. J)
35	34	ssriv 2621	. . . . 5 |- ran (,) C_ J
36	10	tgbl 9148	. . . . . 6 |- (D e. Met -> (topGen` ran ( ball ` D)) = J)
37	2, 36	ax-mp 7	. . . . 5 |- (topGen` ran ( ball ` D)) = J
38	35, 37	sseqtr4i 2650	. . . 4 |- ran (,) C_ (topGen` ran ( ball ` D))
39	8, 38	pm3.2i 307	. . 3 |- (ran ( ball ` D) C_ ran (,) /\ ran (,) C_ (topGen` ran ( ball ` D)))
40		2basgen 8911	. . 3 |- (((ran ( ball ` D) e. Bases /\ ran (,) e. Bases) /\ (ran ( ball ` D) C_ ran (,) /\ ran (,) C_ (topGen` ran ( ball ` D)))) -> (topGen` ran ( ball ` D)) = (topGen` ran (,)))
41	7, 39, 40	mp2an 761	. 2 |- (topGen` ran ( ball ` D)) = (topGen` ran (,))
42	41, 37	eqtr3i 1910	1 |- (topGen` ran (,)) = J

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593  ~Pcpw 3032  U.cuni 3177   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   - cmin 6445  RR*cxr 6652   < clt 6653  (,)cioo 7524  abscabs 8000  Basesctb 8859  topGenctg 8860  Metcme 9066   ball cbl 9068  Opencopn 9069

This theorem is referenced by:  qdensere2 9194  rehaus 9195  nmcn2 9679  reconnlem4 15449  reconnlem5 15450  ivthALT 15454  stioo 15845  iccst 15875  icccmp 16027  pcocn 16076  pcopt 16084

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-top 8861  df-bases 8863  df-topgen 8864  df-met 9070  df-bl 9072  df-opn 9073

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