Theorem opni3 7866

Description: An open set of a metric space includes an arbitrarily small ball around each of its points.

Hypothesis

Ref	Expression
opni.1	|- J = (Open` D)

Assertion

Ref	Expression
opni3	|- (((D e. Met /\ A e. J /\ P e. A) /\ (R e. RR /\ 0 < R)) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ A))

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

Proof of Theorem opni3

Step	Hyp	Ref	Expression
1		opni.1	. . . 4 |- J = (Open` D)
2	1	opni2 7865	. . 3 |- ((D e. Met /\ A e. J /\ P e. A) -> E.y e. RR (0 < y /\ (P( ball ` D)y) (_ A))
3	2	adantr 389	. 2 |- (((D e. Met /\ A e. J /\ P e. A) /\ (R e. RR /\ 0 < R)) -> E.y e. RR (0 < y /\ (P( ball ` D)y) (_ A))
4		eqid 1475	. . . . . . . . 9 |- dom dom D = dom dom D
5	4	ssblex 7856	. . . . . . . 8 |- (((D e. Met /\ P e. dom dom D) /\ (R e. RR /\ 0 < R) /\ (y e. RR /\ 0 < y)) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ (P( ball ` D)y)))
6	5	3expa 833	. . . . . . 7 |- ((((D e. Met /\ P e. dom dom D) /\ (R e. RR /\ 0 < R)) /\ (y e. RR /\ 0 < y)) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ (P( ball ` D)y)))
7	6	adantrrr 403	. . . . . 6 |- ((((D e. Met /\ P e. dom dom D) /\ (R e. RR /\ 0 < R)) /\ (y e. RR /\ (0 < y /\ (P( ball ` D)y) (_ A))) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ (P( ball ` D)y)))
8		idd 61	. . . . . . . . . 10 |- ((P( ball ` D)y) (_ A -> (0 < x -> 0 < x))
9		idd 61	. . . . . . . . . 10 |- ((P( ball ` D)y) (_ A -> (x < R -> x < R))
10		sstr2 2071	. . . . . . . . . . 11 |- ((P( ball ` D)x) (_ (P( ball ` D)y) -> ((P( ball ` D)y) (_ A -> (P( ball ` D)x) (_ A))
11	10	com12 11	. . . . . . . . . 10 |- ((P( ball ` D)y) (_ A -> ((P( ball ` D)x) (_ (P( ball ` D)y) -> (P( ball ` D)x) (_ A))
12	8, 9, 11	3anim123d 900	. . . . . . . . 9 |- ((P( ball ` D)y) (_ A -> ((0 < x /\ x < R /\ (P( ball ` D)x) (_ (P( ball ` D)y)) -> (0 < x /\ x < R /\ (P( ball ` D)x) (_ A)))
13	12	r19.22sdv 1738	. . . . . . . 8 |- ((P( ball ` D)y) (_ A -> (E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ (P( ball ` D)y)) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ A)))
14	13	adantl 388	. . . . . . 7 |- ((0 < y /\ (P( ball ` D)y) (_ A) -> (E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ (P( ball ` D)y)) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ A)))
15	14	ad2antll 407	. . . . . 6 |- ((((D e. Met /\ P e. dom dom D) /\ (R e. RR /\ 0 < R)) /\ (y e. RR /\ (0 < y /\ (P( ball ` D)y) (_ A))) -> (E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ (P( ball ` D)y)) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ A)))
16	7, 15	mpd 26	. . . . 5 |- ((((D e. Met /\ P e. dom dom D) /\ (R e. RR /\ 0 < R)) /\ (y e. RR /\ (0 < y /\ (P( ball ` D)y) (_ A))) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ A))
17	16	exp32 377	. . . 4 |- (((D e. Met /\ P e. dom dom D) /\ (R e. RR /\ 0 < R)) -> (y e. RR -> ((0 < y /\ (P( ball ` D)y) (_ A) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ A))))
18	17	r19.23adv 1746	. . 3 |- (((D e. Met /\ P e. dom dom D) /\ (R e. RR /\ 0 < R)) -> (E.y e. RR (0 < y /\ (P( ball ` D)y) (_ A) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ A)))
19		3simp1 788	. . . 4 |- ((D e. Met /\ A e. J /\ P e. A) -> D e. Met)
20		ssel2 2064	. . . . . 6 |- ((A (_ dom dom D /\ P e. A) -> P e. dom dom D)
21	4, 1	opnss 7863	. . . . . 6 |- ((D e. Met /\ A e. J) -> A (_ dom dom D)
22	20, 21	sylan 448	. . . . 5 |- (((D e. Met /\ A e. J) /\ P e. A) -> P e. dom dom D)
23	22	3impa 828	. . . 4 |- ((D e. Met /\ A e. J /\ P e. A) -> P e. dom dom D)
24	19, 23	jca 288	. . 3 |- ((D e. Met /\ A e. J /\ P e. A) -> (D e. Met /\ P e. dom dom D))
25	18, 24	sylan 448	. 2 |- (((D e. Met /\ A e. J /\ P e. A) /\ (R e. RR /\ 0 < R)) -> (E.y e. RR (0 < y /\ (P( ball ` D)y) (_ A) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ A)))
26	3, 25	mpd 26	1 |- (((D e. Met /\ A e. J /\ P e. A) /\ (R e. RR /\ 0 < R)) -> E.x e. RR (0 < x /\ x < R /\ (P( ball ` D)x) (_ A))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wrex 1646   (_ wss 2047   class class class wbr 2619  dom cdm 3170  ` cfv 3182  (class class class)co 3963  RRcr 5233  0cc0 5234   < clt 5486  Metcme 7789   ball cbl 7791  Opencopn 7792

This theorem is referenced by:  bcthlem8 8006  bcthlem14 8012

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-2 5970  df-met 7793  df-bl 7795  df-opn 7796

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