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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 2730 | . . . 4 β’ (leβπΎ) = (leβπΎ) | |
6 | 1, 2, 3, 4, 5 | toslublem 32407 | . . 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 7389 | . 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 2730 | . . 3 β’ (lubβπΎ) = (lubβπΎ) | |
9 | biid 260 | . . 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 18315 | . 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 18378 | . . . . 5 β’ (πΎ β Toset β (πΎ β Toset β ( < Or π΅ β§ ( I βΎ π΅) β (leβπΎ)))) |
12 | 11 | ibi 266 | . . . 4 β’ (πΎ β Toset β ( < Or π΅ β§ ( I βΎ π΅) β (leβπΎ))) |
13 | 12 | simpld 493 | . . 3 β’ (πΎ β Toset β < Or π΅) |
14 | id 22 | . . . 4 β’ ( < Or π΅ β < Or π΅) | |
15 | 14 | supval2 9454 | . . 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 2780 | 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 394 = wceq 1539 β wcel 2104 βwral 3059 βwrex 3068 β wss 3949 class class class wbr 5149 I cid 5574 Or wor 5588 βΎ cres 5679 βcfv 6544 β©crio 7368 supcsup 9439 Basecbs 17150 lecple 17210 ltcplt 18267 lubclub 18268 Tosetctos 18375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-sup 9441 df-proset 18254 df-poset 18272 df-plt 18289 df-lub 18305 df-toset 18376 |
This theorem is referenced by: xrsp1 32448 |
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