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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 6718 | . . 3 β’ (πΉ:(π...π)βΆβ β ran πΉ β β) | |
2 | 1 | adantl 481 | . 2 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β ran πΉ β β) |
3 | fzfi 13943 | . . . 4 β’ (π...π) β Fin | |
4 | ffn 6711 | . . . . . 6 β’ (πΉ:(π...π)βΆβ β πΉ Fn (π...π)) | |
5 | 4 | adantl 481 | . . . . 5 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β πΉ Fn (π...π)) |
6 | dffn4 6805 | . . . . 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 9340 | . . . 4 β’ (((π...π) β Fin β§ πΉ:(π...π)βontoβran πΉ) β ran πΉ β Fin) | |
9 | 3, 7, 8 | sylancr 586 | . . 3 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β ran πΉ β Fin) |
10 | fdm 6720 | . . . . . 6 β’ (πΉ:(π...π)βΆβ β dom πΉ = (π...π)) | |
11 | 10 | adantl 481 | . . . . 5 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β dom πΉ = (π...π)) |
12 | simpl 482 | . . . . . 6 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β π β (β€β₯βπ)) | |
13 | fzn0 13521 | . . . . . 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 3002 | . . . 4 β’ ((π β (β€β₯βπ) β§ πΉ:(π...π)βΆβ) β dom πΉ β β ) |
16 | dm0rn0 5918 | . . . . 5 β’ (dom πΉ = β β ran πΉ = β ) | |
17 | 16 | necon3bii 2987 | . . . 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 11298 | . . . 4 β’ < Or β | |
20 | fisupcl 9466 | . . . 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 687 | . . 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 3978 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 β wss 3943 β c0 4317 Or wor 5580 dom cdm 5669 ran crn 5670 Fn wfn 6532 βΆwf 6533 βontoβwfo 6535 βcfv 6537 (class class class)co 7405 Fincfn 8941 supcsup 9437 βcr 11111 < clt 11252 β€β₯cuz 12826 ...cfz 13490 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 |
This theorem is referenced by: (None) |
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