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Mirrors > Home > MPE Home > Th. List > fseqsupubi | Structured version Visualization version GIF version |
Description: The values of a finite real sequence are bounded by their supremum. (Contributed by NM, 20-Sep-2005.) |
Ref | Expression |
---|---|
fseqsupubi | β’ ((πΎ β (π...π) β§ πΉ:(π...π)βΆβ) β (πΉβπΎ) β€ sup(ran πΉ, β, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frn 6724 | . . 3 β’ (πΉ:(π...π)βΆβ β ran πΉ β β) | |
2 | 1 | adantl 480 | . 2 β’ ((πΎ β (π...π) β§ πΉ:(π...π)βΆβ) β ran πΉ β β) |
3 | fdm 6726 | . . 3 β’ (πΉ:(π...π)βΆβ β dom πΉ = (π...π)) | |
4 | ne0i 4330 | . . . 4 β’ (πΎ β (π...π) β (π...π) β β ) | |
5 | dm0rn0 5921 | . . . . . 6 β’ (dom πΉ = β β ran πΉ = β ) | |
6 | eqeq1 2729 | . . . . . . 7 β’ (dom πΉ = (π...π) β (dom πΉ = β β (π...π) = β )) | |
7 | 6 | biimpd 228 | . . . . . 6 β’ (dom πΉ = (π...π) β (dom πΉ = β β (π...π) = β )) |
8 | 5, 7 | biimtrrid 242 | . . . . 5 β’ (dom πΉ = (π...π) β (ran πΉ = β β (π...π) = β )) |
9 | 8 | necon3d 2951 | . . . 4 β’ (dom πΉ = (π...π) β ((π...π) β β β ran πΉ β β )) |
10 | 4, 9 | mpan9 505 | . . 3 β’ ((πΎ β (π...π) β§ dom πΉ = (π...π)) β ran πΉ β β ) |
11 | 3, 10 | sylan2 591 | . 2 β’ ((πΎ β (π...π) β§ πΉ:(π...π)βΆβ) β ran πΉ β β ) |
12 | fsequb2 13973 | . . 3 β’ (πΉ:(π...π)βΆβ β βπ₯ β β βπ¦ β ran πΉ π¦ β€ π₯) | |
13 | 12 | adantl 480 | . 2 β’ ((πΎ β (π...π) β§ πΉ:(π...π)βΆβ) β βπ₯ β β βπ¦ β ran πΉ π¦ β€ π₯) |
14 | ffn 6717 | . . 3 β’ (πΉ:(π...π)βΆβ β πΉ Fn (π...π)) | |
15 | fnfvelrn 7085 | . . . 4 β’ ((πΉ Fn (π...π) β§ πΎ β (π...π)) β (πΉβπΎ) β ran πΉ) | |
16 | 15 | ancoms 457 | . . 3 β’ ((πΎ β (π...π) β§ πΉ Fn (π...π)) β (πΉβπΎ) β ran πΉ) |
17 | 14, 16 | sylan2 591 | . 2 β’ ((πΎ β (π...π) β§ πΉ:(π...π)βΆβ) β (πΉβπΎ) β ran πΉ) |
18 | 2, 11, 13, 17 | suprubd 12206 | 1 β’ ((πΎ β (π...π) β§ πΉ:(π...π)βΆβ) β (πΉβπΎ) β€ sup(ran πΉ, β, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 βwral 3051 βwrex 3060 β wss 3939 β c0 4318 class class class wbr 5143 dom cdm 5672 ran crn 5673 Fn wfn 6538 βΆwf 6539 βcfv 6543 (class class class)co 7416 supcsup 9463 βcr 11137 < clt 11278 β€ cle 11279 ...cfz 13516 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 |
This theorem is referenced by: (None) |
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