Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > lubsn | Structured version Visualization version GIF version | ||
| Description: The least upper bound of a singleton. (chsupsn 31161 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 4634 | . . . 4 β’ {π} = {π, π} | |
| 2 | 1 | fveq2i 6885 | . . 3 β’ (πβ{π}) = (πβ{π, π}) |
| 3 | lubsn.u | . . . 4 β’ π = (lubβπΎ) | |
| 4 | eqid 2724 | . . . 4 β’ (joinβπΎ) = (joinβπΎ) | |
| 5 | simpl 482 | . . . 4 β’ ((πΎ β Lat β§ π β π΅) β πΎ β Lat) | |
| 6 | simpr 484 | . . . 4 β’ ((πΎ β Lat β§ π β π΅) β π β π΅) | |
| 7 | 3, 4, 5, 6, 6 | joinval 18338 | . . 3 β’ ((πΎ β Lat β§ π β π΅) β (π(joinβπΎ)π) = (πβ{π, π})) |
| 8 | 2, 7 | eqtr4id 2783 | . 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 18423 | . 2 β’ ((πΎ β Lat β§ π β π΅) β (π(joinβπΎ)π) = π) |
| 11 | 8, 10 | eqtrd 2764 | 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 395 = wceq 1533 β wcel 2098 {csn 4621 {cpr 4623 βcfv 6534 (class class class)co 7402 Basecbs 17149 lubclub 18270 joincjn 18272 Latclat 18392 |
| 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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-proset 18256 df-poset 18274 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-lat 18393 |
| This theorem is referenced by: lubel 18475 |
| Copyright terms: Public domain | W3C validator |