Theorem limsupval 7772

Description: The superior limit of an infinite sequence F of extended real numbers, which is the infimum (indicated by `' <) of the set of suprema of all upper infinite subsequences of F. Definition 12-4.1 of [Gleason] p. 175.

Assertion

Ref	Expression
limsupval	|- (F e. A -> (limsup` F) = sup({x | E.k e. ZZ x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < ))

Distinct variable group:   x,k,F

Proof of Theorem limsupval

Step	Hyp	Ref	Expression
1		xrltso 6729	. . . . 5 |- < Or RR*
2		cnvso 4428	. . . . 5 |- ( < Or RR* <-> `' < Or RR*)
3	1, 2	mpbi 206	. . . 4 |- `' < Or RR*
4	3	supex 5667	. . 3 |- sup({x | E.k e. ZZ x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < ) e. _V
5		imaeq1 4259	. . . . . . . . . 10 |- (z = F -> (z"(ZZ>=` k)) = (F"(ZZ>=` k)))
6	5	ineq1d 2795	. . . . . . . . 9 |- (z = F -> ((z"(ZZ>=` k)) i^i RR*) = ((F"(ZZ>=` k)) i^i RR*))
7		supeq1 5665	. . . . . . . . 9 |- (((z"(ZZ>=` k)) i^i RR*) = ((F"(ZZ>=` k)) i^i RR*) -> sup(((z"(ZZ>=` k)) i^i RR*), RR*, < ) = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < ))
8	6, 7	syl 12	. . . . . . . 8 |- (z = F -> sup(((z"(ZZ>=` k)) i^i RR*), RR*, < ) = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < ))
9	8	eqeq2d 1895	. . . . . . 7 |- (z = F -> (x = sup(((z"(ZZ>=` k)) i^i RR*), RR*, < ) <-> x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )))
10	9	rexbidv 2124	. . . . . 6 |- (z = F -> (E.k e. ZZ x = sup(((z"(ZZ>=` k)) i^i RR*), RR*, < ) <-> E.k e. ZZ x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )))
11	10	abbidv 2008	. . . . 5 |- (z = F -> {x | E.k e. ZZ x = sup(((z"(ZZ>=` k)) i^i RR*), RR*, < )} = {x | E.k e. ZZ x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )})
12		supeq1 5665	. . . . 5 |- ({x | E.k e. ZZ x = sup(((z"(ZZ>=` k)) i^i RR*), RR*, < )} = {x | E.k e. ZZ x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )} -> sup({x | E.k e. ZZ x = sup(((z"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < ) = sup({x | E.k e. ZZ x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < ))
13	11, 12	syl 12	. . . 4 |- (z = F -> sup({x | E.k e. ZZ x = sup(((z"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < ) = sup({x | E.k e. ZZ x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < ))
14	13	fvopabg 4748	. . 3 |- ((F e. A /\ sup({x | E.k e. ZZ x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < ) e. _V) -> ({<.z, y>. | y = sup({x | E.k e. ZZ x = sup(((z"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < )}` F) = sup({x | E.k e. ZZ x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < ))
15	4, 14	mpan2 760	. 2 |- (F e. A -> ({<.z, y>. | y = sup({x | E.k e. ZZ x = sup(((z"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < )}` F) = sup({x | E.k e. ZZ x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < ))
16		df-limsup 7771	. . 3 |- limsup = {<.z, y>. | y = sup({x | E.k e. ZZ x = sup(((z"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < )}
17	16	fveq1i 4682	. 2 |- (limsup` F) = ({<.z, y>. | y = sup({x | E.k e. ZZ x = sup(((z"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < )}` F)
18	15, 17	syl5eq 1940	1 |- (F e. A -> (limsup` F) = sup({x | E.k e. ZZ x = sup(((F"(ZZ>=` k)) i^i RR*), RR*, < )}, RR*, `' < ))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  _Vcvv 2292   i^i cin 2592  {copab 3395   Or wor 3590  `'ccnv 3985  "cima 3989  ` cfv 3998  supcsup 5663  ZZcz 6451  RR*cxr 6652   < clt 6653  ZZ>=cuz 7586  limsupclsp 7770

This theorem is referenced by:  limsupcl 7773

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-1st 5020  df-2nd 5021  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-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-ltp 6242  df-enr 6318  df-nr 6319  df-ltr 6322  df-0r 6323  df-c 6392  df-r 6396  df-lt 6399  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-limsup 7771

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