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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 31236 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 4642 | . . . 4 β’ {π} = {π, π} | |
| 2 | 1 | fveq2i 6900 | . . 3 β’ (πβ{π}) = (πβ{π, π}) |
| 3 | lubsn.u | . . . 4 β’ π = (lubβπΎ) | |
| 4 | eqid 2728 | . . . 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 18369 | . . 3 β’ ((πΎ β Lat β§ π β π΅) β (π(joinβπΎ)π) = (πβ{π, π})) |
| 8 | 2, 7 | eqtr4id 2787 | . 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 18454 | . 2 β’ ((πΎ β Lat β§ π β π΅) β (π(joinβπΎ)π) = π) |
| 11 | 8, 10 | eqtrd 2768 | 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 1534 β wcel 2099 {csn 4629 {cpr 4631 βcfv 6548 (class class class)co 7420 Basecbs 17180 lubclub 18301 joincjn 18303 Latclat 18423 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-proset 18287 df-poset 18305 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-lat 18424 |
| This theorem is referenced by: lubel 18506 |
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