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Mirrors > Home > MPE Home > Th. List > Mathboxes > toslub | Structured version Visualization version GIF version |
Description: In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup(π΄, π΅, < ), defined for total orders. (these are the set.mm definitions: lowest upper bound and supremum are normally synonymous). Note that those two values are also equal if such a supremum does not exist: in that case, both are equal to the empty set. (Contributed by Thierry Arnoux, 15-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.) |
Ref | Expression |
---|---|
toslub.b | β’ π΅ = (BaseβπΎ) |
toslub.l | β’ < = (ltβπΎ) |
toslub.1 | β’ (π β πΎ β Toset) |
toslub.2 | β’ (π β π΄ β π΅) |
Ref | Expression |
---|---|
toslub | β’ (π β ((lubβπΎ)βπ΄) = sup(π΄, π΅, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toslub.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | toslub.l | . . . 4 β’ < = (ltβπΎ) | |
3 | toslub.1 | . . . 4 β’ (π β πΎ β Toset) | |
4 | toslub.2 | . . . 4 β’ (π β π΄ β π΅) | |
5 | eqid 2733 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
6 | 1, 2, 3, 4, 5 | toslublem 31881 | . . 3 β’ ((π β§ π β π΅) β ((βπ β π΄ π(leβπΎ)π β§ βπ β π΅ (βπ β π΄ π(leβπΎ)π β π(leβπΎ)π)) β (βπ β π΄ Β¬ π < π β§ βπ β π΅ (π < π β βπ β π΄ π < π)))) |
7 | 6 | riotabidva 7334 | . 2 β’ (π β (β©π β π΅ (βπ β π΄ π(leβπΎ)π β§ βπ β π΅ (βπ β π΄ π(leβπΎ)π β π(leβπΎ)π))) = (β©π β π΅ (βπ β π΄ Β¬ π < π β§ βπ β π΅ (π < π β βπ β π΄ π < π)))) |
8 | eqid 2733 | . . 3 β’ (lubβπΎ) = (lubβπΎ) | |
9 | biid 261 | . . 3 β’ ((βπ β π΄ π(leβπΎ)π β§ βπ β π΅ (βπ β π΄ π(leβπΎ)π β π(leβπΎ)π)) β (βπ β π΄ π(leβπΎ)π β§ βπ β π΅ (βπ β π΄ π(leβπΎ)π β π(leβπΎ)π))) | |
10 | 1, 5, 8, 9, 3, 4 | lubval 18250 | . 2 β’ (π β ((lubβπΎ)βπ΄) = (β©π β π΅ (βπ β π΄ π(leβπΎ)π β§ βπ β π΅ (βπ β π΄ π(leβπΎ)π β π(leβπΎ)π)))) |
11 | 1, 5, 2 | tosso 18313 | . . . . 5 β’ (πΎ β Toset β (πΎ β Toset β ( < Or π΅ β§ ( I βΎ π΅) β (leβπΎ)))) |
12 | 11 | ibi 267 | . . . 4 β’ (πΎ β Toset β ( < Or π΅ β§ ( I βΎ π΅) β (leβπΎ))) |
13 | 12 | simpld 496 | . . 3 β’ (πΎ β Toset β < Or π΅) |
14 | id 22 | . . . 4 β’ ( < Or π΅ β < Or π΅) | |
15 | 14 | supval2 9396 | . . 3 β’ ( < Or π΅ β sup(π΄, π΅, < ) = (β©π β π΅ (βπ β π΄ Β¬ π < π β§ βπ β π΅ (π < π β βπ β π΄ π < π)))) |
16 | 3, 13, 15 | 3syl 18 | . 2 β’ (π β sup(π΄, π΅, < ) = (β©π β π΅ (βπ β π΄ Β¬ π < π β§ βπ β π΅ (π < π β βπ β π΄ π < π)))) |
17 | 7, 10, 16 | 3eqtr4d 2783 | 1 β’ (π β ((lubβπΎ)βπ΄) = sup(π΄, π΅, < )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 βwrex 3070 β wss 3911 class class class wbr 5106 I cid 5531 Or wor 5545 βΎ cres 5636 βcfv 6497 β©crio 7313 supcsup 9381 Basecbs 17088 lecple 17145 ltcplt 18202 lubclub 18203 Tosetctos 18310 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
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-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-sup 9383 df-proset 18189 df-poset 18207 df-plt 18224 df-lub 18240 df-toset 18311 |
This theorem is referenced by: xrsp1 31922 |
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