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Mirrors > Home > MPE Home > Th. List > fseqsupcl | Structured version Visualization version GIF version |
Description: The values of a finite real sequence have a supremum. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fseqsupcl | β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β sup(ran πΉ, β, < ) β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frn 6734 | . . 3 β’ (πΉ:(π...π)βΆβ β ran πΉ β β) | |
2 | 1 | adantl 480 | . 2 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β ran πΉ β β) |
3 | fzfi 13979 | . . . 4 β’ (π...π) β Fin | |
4 | ffn 6727 | . . . . . 6 β’ (πΉ:(π...π)βΆβ β πΉ Fn (π...π)) | |
5 | 4 | adantl 480 | . . . . 5 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β πΉ Fn (π...π)) |
6 | dffn4 6822 | . . . . 5 β’ (πΉ Fn (π...π) β πΉ:(π...π)βontoβran πΉ) | |
7 | 5, 6 | sylib 217 | . . . 4 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β πΉ:(π...π)βontoβran πΉ) |
8 | fofi 9372 | . . . 4 β’ (((π...π) β Fin β§ πΉ:(π...π)βontoβran πΉ) β ran πΉ β Fin) | |
9 | 3, 7, 8 | sylancr 585 | . . 3 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β ran πΉ β Fin) |
10 | fdm 6736 | . . . . . 6 β’ (πΉ:(π...π)βΆβ β dom πΉ = (π...π)) | |
11 | 10 | adantl 480 | . . . . 5 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β dom πΉ = (π...π)) |
12 | simpl 481 | . . . . . 6 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β π β (β€β₯βπ)) | |
13 | fzn0 13557 | . . . . . 6 β’ ((π...π) β β β π β (β€β₯βπ)) | |
14 | 12, 13 | sylibr 233 | . . . . 5 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β (π...π) β β ) |
15 | 11, 14 | eqnetrd 3005 | . . . 4 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β dom πΉ β β ) |
16 | dm0rn0 5931 | . . . . 5 β’ (dom πΉ = β β ran πΉ = β ) | |
17 | 16 | necon3bii 2990 | . . . 4 β’ (dom πΉ β β β ran πΉ β β ) |
18 | 15, 17 | sylib 217 | . . 3 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β ran πΉ β β ) |
19 | ltso 11334 | . . . 4 β’ < Or β | |
20 | fisupcl 9502 | . . . 4 β’ (( < Or β β§ (ran πΉ β Fin β§ ran πΉ β β β§ ran πΉ β β)) β sup(ran πΉ, β, < ) β ran πΉ) | |
21 | 19, 20 | mpan 688 | . . 3 β’ ((ran πΉ β Fin β§ ran πΉ β β β§ ran πΉ β β) β sup(ran πΉ, β, < ) β ran πΉ) |
22 | 9, 18, 2, 21 | syl3anc 1368 | . 2 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β sup(ran πΉ, β, < ) β ran πΉ) |
23 | 2, 22 | sseldd 3983 | 1 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β sup(ran πΉ, β, < ) β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 β wss 3949 β c0 4326 Or wor 5593 dom cdm 5682 ran crn 5683 Fn wfn 6548 βΆwf 6549 βontoβwfo 6551 βcfv 6553 (class class class)co 7426 Fincfn 8972 supcsup 9473 βcr 11147 < clt 11288 β€β₯cuz 12862 ...cfz 13526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-n0 12513 df-z 12599 df-uz 12863 df-fz 13527 |
This theorem is referenced by: (None) |
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