Theorem iint 15012

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 15011	. . . . . . . 8 |- (y e. |^|_x e. RR+ ((A - x)(,)(A + x)) -> y e. RR)
2	1	adantl 424	. . . . . . 7 |- ((A e. RR /\ y e. |^|_x e. RR+ ((A - x)(,)(A + x))) -> y e. RR)
3		iintlem1 15010	. . . . . . 7 |- ((A e. RR /\ y e. |^|_x e. RR+ ((A - x)(,)(A + x))) -> (y e. RR -> y = A))
4	2, 3	jcai 313	. . . . . 6 |- ((A e. RR /\ y e. |^|_x e. RR+ ((A - x)(,)(A + x))) -> (y e. RR /\ y = A))
5	4	ex 402	. . . . 5 |- (A e. RR -> (y e. |^|_x e. RR+ ((A - x)(,)(A + x)) -> (y e. RR /\ y = A)))
6		eleq1 1957	. . . . . . . 8 |- (y = A -> (y e. |^|_x e. RR+ ((A - x)(,)(A + x)) <-> A e. |^|_x e. RR+ ((A - x)(,)(A + x))))
7		resubcl 6601	. . . . . . . . . . . . . . 15 |- ((A e. RR /\ x e. RR) -> (A - x) e. RR)
8		rpre 7236	. . . . . . . . . . . . . . 15 |- (x e. RR+ -> x e. RR)
9	7, 8	sylan2 500	. . . . . . . . . . . . . 14 |- ((A e. RR /\ x e. RR+) -> (A - x) e. RR)
10		rexr 6668	. . . . . . . . . . . . . 14 |- ((A - x) e. RR -> (A - x) e. RR*)
11	9, 10	syl 12	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. RR+) -> (A - x) e. RR*)
12		readdcl 6455	. . . . . . . . . . . . . . 15 |- ((A e. RR /\ x e. RR) -> (A + x) e. RR)
13	12, 8	sylan2 500	. . . . . . . . . . . . . 14 |- ((A e. RR /\ x e. RR+) -> (A + x) e. RR)
14		rexr 6668	. . . . . . . . . . . . . 14 |- ((A + x) e. RR -> (A + x) e. RR*)
15	13, 14	syl 12	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. RR+) -> (A + x) e. RR*)
16		rexr 6668	. . . . . . . . . . . . . 14 |- (A e. RR -> A e. RR*)
17	16	adantr 425	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. RR+) -> A e. RR*)
18	11, 15, 17	3jca 1050	. . . . . . . . . . . 12 |- ((A e. RR /\ x e. RR+) -> ((A - x) e. RR* /\ (A + x) e. RR* /\ A e. RR*))
19		ltsubpostb 15005	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. RR+) -> (A - x) < A)
20		ltaddpos2tb 15006	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. RR+) -> A < (A + x))
21	19, 20	jca 310	. . . . . . . . . . . 12 |- ((A e. RR /\ x e. RR+) -> ((A - x) < A /\ A < (A + x)))
22	18, 21	jca 310	. . . . . . . . . . 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 7547	. . . . . . . . . . . 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 12	. . . . . . . . . . 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 213	. . . . . . . . . 10 |- ((A e. RR /\ x e. RR+) -> A e. ((A - x)(,)(A + x)))
26	25	r19.21aiva 2176	. . . . . . . . 9 |- (A e. RR -> A.x e. RR+ A e. ((A - x)(,)(A + x)))
27		eliin 3260	. . . . . . . . 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 213	. . . . . . . 8 |- (A e. RR -> A e. |^|_x e. RR+ ((A - x)(,)(A + x)))
29	6, 28	syl5bir 227	. . . . . . 7 |- (y = A -> (A e. RR -> y e. |^|_x e. RR+ ((A - x)(,)(A + x))))
30	29	adantl 424	. . . . . 6 |- ((y e. RR /\ y = A) -> (A e. RR -> y e. |^|_x e. RR+ ((A - x)(,)(A + x))))
31	30	com12 14	. . . . 5 |- (A e. RR -> ((y e. RR /\ y = A) -> y e. |^|_x e. RR+ ((A - x)(,)(A + x))))
32	5, 31	impbid 574	. . . 4 |- (A e. RR -> (y e. |^|_x e. RR+ ((A - x)(,)(A + x)) <-> (y e. RR /\ y = A)))
33		eqeq1 1890	. . . . 5 |- (x = y -> (x = A <-> y = A))
34	33	elrab 2414	. . . 4 |- (y e. {x e. RR | x = A} <-> (y e. RR /\ y = A))
35	32, 34	syl6bbr 597	. . 3 |- (A e. RR -> (y e. |^|_x e. RR+ ((A - x)(,)(A + x)) <-> y e. {x e. RR | x = A}))
36	35	eqrdv 1882	. 2 |- (A e. RR -> |^|_x e. RR+ ((A - x)(,)(A + x)) = {x e. RR | x = A})
37		rabsn 3094	. 2 |- (A e. RR -> {x e. RR | x = A} = {A})
38	36, 37	eqtrd 1925	1 |- (A e. RR -> |^|_x e. RR+ ((A - x)(,)(A + x)) = {A})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  {csn 3044  |^|_ciin 3256   class class class wbr 3338  (class class class)co 4884  RRcr 6385   + caddc 6389   - cmin 6445  RR+crp 6453  RR*cxr 6652   < clt 6653  (,)cioo 7524

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-iin 3258  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-rp 7232  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

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