Theorem iint 10585

Description: Indexed intersection of a set of open intervals centered on A. This theorem is a rough justification for taking finite intersections in the definition of a topology. If we consider we are in the standard topology of RR this theorem means a non finite intersection of open sets can result in a closed set.

Assertion

Ref	Expression
iint	|- (A e. RR -> |^|_x e. RR+ ((A - x)(,)(A + x)) = {A})

Distinct variable group:   x,A

Proof of Theorem iint

Step	Hyp	Ref	Expression
1		iintlem2 10584	. . . . . . . 8 |- (y e. |^|_x e. RR+ ((A - x)(,)(A + x)) -> y e. RR)
2	1	adantl 388	. . . . . . 7 |- ((A e. RR /\ y e. |^|_x e. RR+ ((A - x)(,)(A + x))) -> y e. RR)
3		iintlem1 10583	. . . . . . 7 |- ((A e. RR /\ y e. |^|_x e. RR+ ((A - x)(,)(A + x))) -> (y e. RR -> y = A))
4	2, 3	jcai 289	. . . . . 6 |- ((A e. RR /\ y e. |^|_x e. RR+ ((A - x)(,)(A + x))) -> (y e. RR /\ y = A))
5	4	ex 373	. . . . 5 |- (A e. RR -> (y e. |^|_x e. RR+ ((A - x)(,)(A + x)) -> (y e. RR /\ y = A)))
6		eleq1 1533	. . . . . . . 8 |- (y = A -> (y e. |^|_x e. RR+ ((A - x)(,)(A + x)) <-> A e. |^|_x e. RR+ ((A - x)(,)(A + x))))
7		resubclt 5425	. . . . . . . . . . . . . . 15 |- ((A e. RR /\ x e. RR) -> (A - x) e. RR)
8		rpret 6239	. . . . . . . . . . . . . . 15 |- (x e. RR+ -> x e. RR)
9	7, 8	sylan2 451	. . . . . . . . . . . . . 14 |- ((A e. RR /\ x e. RR+) -> (A - x) e. RR)
10		rexrt 5486	. . . . . . . . . . . . . 14 |- ((A - x) e. RR -> (A - x) e. RR*)
11	9, 10	syl 10	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. RR+) -> (A - x) e. RR*)
12		axaddrcl 5259	. . . . . . . . . . . . . . 15 |- ((A e. RR /\ x e. RR) -> (A + x) e. RR)
13	12, 8	sylan2 451	. . . . . . . . . . . . . 14 |- ((A e. RR /\ x e. RR+) -> (A + x) e. RR)
14		rexrt 5486	. . . . . . . . . . . . . 14 |- ((A + x) e. RR -> (A + x) e. RR*)
15	13, 14	syl 10	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. RR+) -> (A + x) e. RR*)
16		rexrt 5486	. . . . . . . . . . . . . 14 |- (A e. RR -> A e. RR*)
17	16	adantr 389	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. RR+) -> A e. RR*)
18	11, 15, 17	3jca 818	. . . . . . . . . . . 12 |- ((A e. RR /\ x e. RR+) -> ((A - x) e. RR* /\ (A + x) e. RR* /\ A e. RR*))
19		ltsubpostb 10578	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. RR+) -> (A - x) < A)
20		ltaddpos2tb 10579	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. RR+) -> A < (A + x))
21	19, 20	jca 288	. . . . . . . . . . . 12 |- ((A e. RR /\ x e. RR+) -> ((A - x) < A /\ A < (A + x)))
22	18, 21	jca 288	. . . . . . . . . . 11 |- ((A e. RR /\ x e. RR+) -> (((A - x) e. RR* /\ (A + x) e. RR* /\ A e. RR*) /\ ((A - x) < A /\ A < (A + x))))
23		elioo3g 6335	. . . . . . . . . . . 12 |- ((A + x) e. RR -> (A e. ((A - x)(,)(A + x)) <-> (((A - x) e. RR* /\ (A + x) e. RR* /\ A e. RR*) /\ ((A - x) < A /\ A < (A + x)))))
24	13, 23	syl 10	. . . . . . . . . . 11 |- ((A e. RR /\ x e. RR+) -> (A e. ((A - x)(,)(A + x)) <-> (((A - x) e. RR* /\ (A + x) e. RR* /\ A e. RR*) /\ ((A - x) < A /\ A < (A + x)))))
25	22, 24	mpbird 196	. . . . . . . . . 10 |- ((A e. RR /\ x e. RR+) -> A e. ((A - x)(,)(A + x)))
26	25	r19.21aiva 1713	. . . . . . . . 9 |- (A e. RR -> A.x e. RR+ A e. ((A - x)(,)(A + x)))
27		eliin 2568	. . . . . . . . 9 |- (A e. RR -> (A e. |^|_x e. RR+ ((A - x)(,)(A + x)) <-> A.x e. RR+ A e. ((A - x)(,)(A + x))))
28	26, 27	mpbird 196	. . . . . . . 8 |- (A e. RR -> A e. |^|_x e. RR+ ((A - x)(,)(A + x)))
29	6, 28	syl5bir 210	. . . . . . 7 |- (y = A -> (A e. RR -> y e. |^|_x e. RR+ ((A - x)(,)(A + x))))
30	29	adantl 388	. . . . . 6 |- ((y e. RR /\ y = A) -> (A e. RR -> y e. |^|_x e. RR+ ((A - x)(,)(A + x))))
31	30	com12 11	. . . . 5 |- (A e. RR -> ((y e. RR /\ y = A) -> y e. |^|_x e. RR+ ((A - x)(,)(A + x))))
32	5, 31	impbid 515	. . . 4 |- (A e. RR -> (y e. |^|_x e. RR+ ((A - x)(,)(A + x)) <-> (y e. RR /\ y = A)))
33		eqeq1 1480	. . . . 5 |- (x = y -> (x = A <-> y = A))
34	33	elrab 1903	. . . 4 |- (y e. {x e. RR | x = A} <-> (y e. RR /\ y = A))
35	32, 34	syl6bbr 537	. . 3 |- (A e. RR -> (y e. |^|_x e. RR+ ((A - x)(,)(A + x)) <-> y e. {x e. RR | x = A}))
36	35	eqrdv 1473	. 2 |- (A e. RR -> |^|_x e. RR+ ((A - x)(,)(A + x)) = {x e. RR | x = A})
37		rabsn 2443	. 2 |- (A e. RR -> {x e. RR | x = A} = {A})
38	36, 37	eqtrd 1506	1 |- (A e. RR -> |^|_x e. RR+ ((A - x)(,)(A + x)) = {A})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  A.wral 1644  {crab 1647  {csn 2407  |^|_ciin 2564   class class class wbr 2616  (class class class)co 3960  RRcr 5220   + caddc 5224   - cmin 5279  RR+crp 5287  RR*cxr 5472   < clt 5473  (,)cioo 6312

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-nel 1587  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-iin 2566  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-1o 4130  df-oadd 4132  df-omul 4133  df-er 4258  df-ec 4260  df-qs 4263  df-en 4364  df-dom 4365  df-sdom 4366  df-sup 4561  df-ni 4987  df-pli 4988  df-mi 4989  df-lti 4990  df-plpq 5022  df-mpq 5023  df-enq 5024  df-nq 5025  df-plq 5026  df-mq 5027  df-rq 5028  df-ltq 5029  df-1q 5030  df-np 5073  df-1p 5074  df-plp 5075  df-mp 5076  df-ltp 5077  df-plpr 5151  df-mpr 5152  df-enr 5153  df-nr 5154  df-plr 5155  df-mr 5156  df-ltr 5157  df-0r 5158  df-1r 5159  df-m1r 5160  df-c 5227  df-0 5228  df-1 5229  df-i 5230  df-r 5231  df-plus 5232  df-mul 5233  df-lt 5234  df-sub 5343  df-neg 5345  df-pnf 5474  df-mnf 5475  df-xr 5476  df-ltxr 5477  df-le 5478  df-div 5686  df-n 5887  df-2 5931  df-n0 6061  df-z 6097  df-q 6211  df-rp 6236  df-seq1 6263  df-ioo 6316  df-exp 6519  df-sqr 6621  df-re 6703  df-im 6704  df-cj 6705  df-abs 6706

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