Theorem tgioo 7915

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 7910	. . . . 5 |- D e. Met
3		eqid 1475	. . . . . 6 |- (Open` D) = (Open` D)
4	3	blbas 7872	. . . . 5 |- (D e. Met -> ran ( ball ` D) e. Bases)
5	2, 4	ax-mp 7	. . . 4 |- ran ( ball ` D) e. Bases
6		retopbas 7655	. . . 4 |- ran (,) e. Bases
7	5, 6	pm3.2i 285	. . 3 |- (ran ( ball ` D) e. Bases /\ ran (,) e. Bases)
8	1	blssioo 7913	. . . 4 |- ran ( ball ` D) (_ ran (,)
9		elssuni 2526	. . . . . . . . 9 |- (w e. ran (,) -> w (_ U.ran (,))
10		unirnioo 6402	. . . . . . . . 9 |- U.ran (,) = RR
11	9, 10	syl6ss 2107	. . . . . . . 8 |- (w e. ran (,) -> w (_ RR)
12		ioof 6400	. . . . . . . . . . . 12 |- (,):(RR* X. RR*)-->P~RR
13		ffn 3627	. . . . . . . . . . . 12 |- ((,):(RR* X. RR*)-->P~RR -> (,) Fn (RR* X. RR*))
14	12, 13	ax-mp 7	. . . . . . . . . . 11 |- (,) Fn (RR* X. RR*)
15		oprvalelrn 4039	. . . . . . . . . . 11 |- ((,) Fn (RR* X. RR*) -> (w e. ran (,) <-> E.t e. RR* E.s e. RR* (t(,)s) = w))
16	14, 15	ax-mp 7	. . . . . . . . . 10 |- (w e. ran (,) <-> E.t e. RR* E.s e. RR* (t(,)s) = w)
17	1	tgioolem 7914	. . . . . . . . . . . . . 14 |- ((t e. RR* /\ s e. RR*) -> (v e. (t(,)s) -> E.u e. RR (0 < u /\ (v( ball ` D)u) (_ (t(,)s))))
18	17	adantr 389	. . . . . . . . . . . . 13 |- (((t e. RR* /\ s e. RR*) /\ (t(,)s) = w) -> (v e. (t(,)s) -> E.u e. RR (0 < u /\ (v( ball ` D)u) (_ (t(,)s))))
19		eleq2 1535	. . . . . . . . . . . . . 14 |- ((t(,)s) = w -> (v e. (t(,)s) <-> v e. w))
20	19	adantl 388	. . . . . . . . . . . . 13 |- (((t e. RR* /\ s e. RR*) /\ (t(,)s) = w) -> (v e. (t(,)s) <-> v e. w))
21		sseq2 2083	. . . . . . . . . . . . . . . 16 |- ((t(,)s) = w -> ((v( ball ` D)u) (_ (t(,)s) <-> (v( ball ` D)u) (_ w))
22	21	anbi2d 616	. . . . . . . . . . . . . . 15 |- ((t(,)s) = w -> ((0 < u /\ (v( ball ` D)u) (_ (t(,)s)) <-> (0 < u /\ (v( ball ` D)u) (_ w)))
23	22	rexbidv 1664	. . . . . . . . . . . . . 14 |- ((t(,)s) = w -> (E.u e. RR (0 < u /\ (v( ball ` D)u) (_ (t(,)s)) <-> E.u e. RR (0 < u /\ (v( ball ` D)u) (_ w)))
24	23	adantl 388	. . . . . . . . . . . . 13 |- (((t e. RR* /\ s e. RR*) /\ (t(,)s) = w) -> (E.u e. RR (0 < u /\ (v( ball ` D)u) (_ (t(,)s)) <-> E.u e. RR (0 < u /\ (v( ball ` D)u) (_ w)))
25	18, 20, 24	3imtr3d 542	. . . . . . . . . . . 12 |- (((t e. RR* /\ s e. RR*) /\ (t(,)s) = w) -> (v e. w -> E.u e. RR (0 < u /\ (v( ball ` D)u) (_ w)))
26	25	ex 373	. . . . . . . . . . 11 |- ((t e. RR* /\ s e. RR*) -> ((t(,)s) = w -> (v e. w -> E.u e. RR (0 < u /\ (v( ball ` D)u) (_ w))))
27	26	r19.23aivv 1748	. . . . . . . . . 10 |- (E.t e. RR* E.s e. RR* (t(,)s) = w -> (v e. w -> E.u e. RR (0 < u /\ (v( ball ` D)u) (_ w)))
28	16, 27	sylbi 199	. . . . . . . . 9 |- (w e. ran (,) -> (v e. w -> E.u e. RR (0 < u /\ (v( ball ` D)u) (_ w)))
29	28	r19.21aiv 1713	. . . . . . . 8 |- (w e. ran (,) -> A.v e. w E.u e. RR (0 < u /\ (v( ball ` D)u) (_ w))
30	11, 29	jca 288	. . . . . . 7 |- (w e. ran (,) -> (w (_ RR /\ A.v e. w E.u e. RR (0 < u /\ (v( ball ` D)u) (_ w)))
31	1	remetba 7909	. . . . . . . . 9 |- RR = dom dom D
32		tgioo.2	. . . . . . . . 9 |- J = (Open` D)
33	31, 32	isopn4 7862	. . . . . . . 8 |- (D e. Met -> (w e. J <-> (w (_ RR /\ A.v e. w E.u e. RR (0 < u /\ (v( ball ` D)u) (_ w))))
34	2, 33	ax-mp 7	. . . . . . 7 |- (w e. J <-> (w (_ RR /\ A.v e. w E.u e. RR (0 < u /\ (v( ball ` D)u) (_ w)))
35	30, 34	sylibr 200	. . . . . 6 |- (w e. ran (,) -> w e. J)
36	35	ssriv 2069	. . . . 5 |- ran (,) (_ J
37	32	tgbl 7871	. . . . . 6 |- (D e. Met -> (topGen` ran ( ball ` D)) = J)
38	2, 37	ax-mp 7	. . . . 5 |- (topGen` ran ( ball ` D)) = J
39	36, 38	sseqtr4 2094	. . . 4 |- ran (,) (_ (topGen` ran ( ball ` D))
40	8, 39	pm3.2i 285	. . 3 |- (ran ( ball ` D) (_ ran (,) /\ ran (,) (_ (topGen` ran ( ball ` D)))
41		2basgent 7641	. . 3 |- (((ran ( ball ` D) e. Bases /\ ran (,) e. Bases) /\ (ran ( ball ` D) (_ ran (,) /\ ran (,) (_ (topGen` ran ( ball ` D)))) -> (topGen` ran ( ball ` D)) = (topGen` ran (,)))
42	7, 40, 41	mp2an 697	. 2 |- (topGen` ran ( ball ` D)) = (topGen` ran (,))
43	42, 38	eqtr3 1497	1 |- (topGen` ran (,)) = J

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646   (_ wss 2047  P~cpw 2401  U.cuni 2503   class class class wbr 2619   X. cxp 3168  ran crn 3171   |` cres 3172   o. ccom 3174   Fn wfn 3177  -->wf 3178  ` cfv 3182  (class class class)co 3963  RRcr 5233  0cc0 5234   - cmin 5292  RR*cxr 5485   < clt 5486  (,)cioo 6357  abscabs 6750  Basesctb 7590  topGenctg 7591  Metcme 7789   ball cbl 7791  Opencopn 7792

This theorem is referenced by:  qdensere2 7916  rehaus 7917  nmcn2 8340

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-sup 4574  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-n0 6100  df-z 6136  df-q 6256  df-seq1 6308  df-ioo 6361  df-exp 6569  df-sqr 6670  df-re 6751  df-im 6752  df-cj 6753  df-abs 6754  df-top 7592  df-bases 7594  df-topgen 7595  df-met 7793  df-bl 7795  df-opn 7796

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