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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 6679 | . . 3 β’ (πΉ:(π...π)βΆβ β ran πΉ β β) | |
2 | 1 | adantl 483 | . 2 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β ran πΉ β β) |
3 | fzfi 13886 | . . . 4 β’ (π...π) β Fin | |
4 | ffn 6672 | . . . . . 6 β’ (πΉ:(π...π)βΆβ β πΉ Fn (π...π)) | |
5 | 4 | adantl 483 | . . . . 5 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β πΉ Fn (π...π)) |
6 | dffn4 6766 | . . . . 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 9288 | . . . 4 β’ (((π...π) β Fin β§ πΉ:(π...π)βontoβran πΉ) β ran πΉ β Fin) | |
9 | 3, 7, 8 | sylancr 588 | . . 3 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β ran πΉ β Fin) |
10 | fdm 6681 | . . . . . 6 β’ (πΉ:(π...π)βΆβ β dom πΉ = (π...π)) | |
11 | 10 | adantl 483 | . . . . 5 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β dom πΉ = (π...π)) |
12 | simpl 484 | . . . . . 6 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β π β (β€β₯βπ)) | |
13 | fzn0 13464 | . . . . . 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 3008 | . . . 4 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β dom πΉ β β ) |
16 | dm0rn0 5884 | . . . . 5 β’ (dom πΉ = β β ran πΉ = β ) | |
17 | 16 | necon3bii 2993 | . . . 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 11243 | . . . 4 β’ < Or β | |
20 | fisupcl 9413 | . . . 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 689 | . . 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 1372 | . 2 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β sup(ran πΉ, β, < ) β ran πΉ) |
23 | 2, 22 | sseldd 3949 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2940 β wss 3914 β c0 4286 Or wor 5548 dom cdm 5637 ran crn 5638 Fn wfn 6495 βΆwf 6496 βontoβwfo 6498 βcfv 6500 (class class class)co 7361 Fincfn 8889 supcsup 9384 βcr 11058 < clt 11197 β€β₯cuz 12771 ...cfz 13433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 |
This theorem is referenced by: (None) |
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