Theorem fseqsupubi 7705

Description: The values of a finite real sequence are bounded by their supremum.

Hypothesis

Ref	Expression
fseqsupub.1	|- N e. _V

Assertion

Ref	Expression
fseqsupubi	|- ((K e. (M...N) /\ F:(M...N)-->RR) -> (F` K) <_ sup(ran F, RR, < ))

Proof of Theorem fseqsupubi

Step	Hyp	Ref	Expression
1		frn 4569	. . 3 |- (F:(M...N)-->RR -> ran F C_ RR)
2	1	adantl 424	. 2 |- ((K e. (M...N) /\ F:(M...N)-->RR) -> ran F C_ RR)
3		eqeq1 1890	. . . . . . . 8 |- (dom F = (M...N) -> (dom F = (/) <-> (M...N) = (/)))
4	3	biimpd 170	. . . . . . 7 |- (dom F = (M...N) -> (dom F = (/) -> (M...N) = (/)))
5		dm0rn0 4175	. . . . . . 7 |- (dom F = (/) <-> ran F = (/))
6	4, 5	syl5ibr 224	. . . . . 6 |- (dom F = (M...N) -> (ran F = (/) -> (M...N) = (/)))
7	6	necon3d 2041	. . . . 5 |- (dom F = (M...N) -> ((M...N) =/= (/) -> ran F =/= (/)))
8		ne0i 2881	. . . . 5 |- (K e. (M...N) -> (M...N) =/= (/))
9	7, 8	syl5com 63	. . . 4 |- (K e. (M...N) -> (dom F = (M...N) -> ran F =/= (/)))
10	9	imp 377	. . 3 |- ((K e. (M...N) /\ dom F = (M...N)) -> ran F =/= (/))
11		fdm 4567	. . 3 |- (F:(M...N)-->RR -> dom F = (M...N))
12	10, 11	sylan2 500	. 2 |- ((K e. (M...N) /\ F:(M...N)-->RR) -> ran F =/= (/))
13		fsequb2 7703	. . 3 |- ((N e. (ZZ>=` M) /\ F:(M...N)-->RR) -> E.x e. RR A.y e. ran F y <_ x)
14		fseqsupub.1	. . . 4 |- N e. _V
15		elfzuz2 7656	. . . 4 |- ((N e. _V /\ K e. (M...N)) -> N e. (ZZ>=` M))
16	14, 15	mpan 759	. . 3 |- (K e. (M...N) -> N e. (ZZ>=` M))
17	13, 16	sylan 497	. 2 |- ((K e. (M...N) /\ F:(M...N)-->RR) -> E.x e. RR A.y e. ran F y <_ x)
18		fnfvelrn 4786	. . . 4 |- ((F Fn (M...N) /\ K e. (M...N)) -> (F` K) e. ran F)
19	18	ancoms 484	. . 3 |- ((K e. (M...N) /\ F Fn (M...N)) -> (F` K) e. ran F)
20		ffn 4562	. . 3 |- (F:(M...N)-->RR -> F Fn (M...N))
21	19, 20	sylan2 500	. 2 |- ((K e. (M...N) /\ F:(M...N)-->RR) -> (F` K) e. ran F)
22		suprub 7265	. 2 |- (((ran F C_ RR /\ ran F =/= (/) /\ E.x e. RR A.y e. ran F y <_ x) /\ (F` K) e. ran F) -> (F` K) <_ sup(ran F, RR, < ))
23	2, 12, 17, 21, 22	syl31anc 1103	1 |- ((K e. (M...N) /\ F:(M...N)-->RR) -> (F` K) <_ sup(ran F, RR, < ))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  (/)c0 2875   class class class wbr 3338  dom cdm 3986  ran crn 3987   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  supcsup 5663  RRcr 6385   <_ cle 6448   < clt 6653  ZZ>=cuz 7586  ...cfz 7637

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-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-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638

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