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Mirrors > Home > MPE Home > Th. List > lubsn | Structured version Visualization version GIF version |
Description: The least upper bound of a singleton. (chsupsn 29820 analog.) (Contributed by NM, 20-Oct-2011.) |
Ref | Expression |
---|---|
lubsn.b | β’ π΅ = (BaseβπΎ) |
lubsn.u | β’ π = (lubβπΎ) |
Ref | Expression |
---|---|
lubsn | β’ ((πΎ β Lat β§ π β π΅) β (πβ{π}) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 4578 | . . . 4 β’ {π} = {π, π} | |
2 | 1 | fveq2i 6807 | . . 3 β’ (πβ{π}) = (πβ{π, π}) |
3 | lubsn.u | . . . 4 β’ π = (lubβπΎ) | |
4 | eqid 2736 | . . . 4 β’ (joinβπΎ) = (joinβπΎ) | |
5 | simpl 484 | . . . 4 β’ ((πΎ β Lat β§ π β π΅) β πΎ β Lat) | |
6 | simpr 486 | . . . 4 β’ ((πΎ β Lat β§ π β π΅) β π β π΅) | |
7 | 3, 4, 5, 6, 6 | joinval 18140 | . . 3 β’ ((πΎ β Lat β§ π β π΅) β (π(joinβπΎ)π) = (πβ{π, π})) |
8 | 2, 7 | eqtr4id 2795 | . 2 β’ ((πΎ β Lat β§ π β π΅) β (πβ{π}) = (π(joinβπΎ)π)) |
9 | lubsn.b | . . 3 β’ π΅ = (BaseβπΎ) | |
10 | 9, 4 | latjidm 18225 | . 2 β’ ((πΎ β Lat β§ π β π΅) β (π(joinβπΎ)π) = π) |
11 | 8, 10 | eqtrd 2776 | 1 β’ ((πΎ β Lat β§ π β π΅) β (πβ{π}) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1539 β wcel 2104 {csn 4565 {cpr 4567 βcfv 6458 (class class class)co 7307 Basecbs 16957 lubclub 18072 joincjn 18074 Latclat 18194 |
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 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-proset 18058 df-poset 18076 df-lub 18109 df-glb 18110 df-join 18111 df-meet 18112 df-lat 18195 |
This theorem is referenced by: lubel 18277 |
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