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Mirrors > Home > MPE Home > Th. List > latcl2 | Structured version Visualization version GIF version |
Description: The join and meet of any two elements exist. (Contributed by NM, 14-Sep-2018.) |
Ref | Expression |
---|---|
latcl2.b | β’ π΅ = (BaseβπΎ) |
latcl2.j | β’ β¨ = (joinβπΎ) |
latcl2.m | β’ β§ = (meetβπΎ) |
latcl2.k | β’ (π β πΎ β Lat) |
latcl2.x | β’ (π β π β π΅) |
latcl2.y | β’ (π β π β π΅) |
Ref | Expression |
---|---|
latcl2 | β’ (π β (β¨π, πβ© β dom β¨ β§ β¨π, πβ© β dom β§ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latcl2.x | . . . 4 β’ (π β π β π΅) | |
2 | latcl2.y | . . . 4 β’ (π β π β π΅) | |
3 | 1, 2 | opelxpd 5717 | . . 3 β’ (π β β¨π, πβ© β (π΅ Γ π΅)) |
4 | latcl2.k | . . . . 5 β’ (π β πΎ β Lat) | |
5 | latcl2.b | . . . . . 6 β’ π΅ = (BaseβπΎ) | |
6 | latcl2.j | . . . . . 6 β’ β¨ = (joinβπΎ) | |
7 | latcl2.m | . . . . . 6 β’ β§ = (meetβπΎ) | |
8 | 5, 6, 7 | islat 18425 | . . . . 5 β’ (πΎ β Lat β (πΎ β Poset β§ (dom β¨ = (π΅ Γ π΅) β§ dom β§ = (π΅ Γ π΅)))) |
9 | 4, 8 | sylib 217 | . . . 4 β’ (π β (πΎ β Poset β§ (dom β¨ = (π΅ Γ π΅) β§ dom β§ = (π΅ Γ π΅)))) |
10 | 9 | simprld 771 | . . 3 β’ (π β dom β¨ = (π΅ Γ π΅)) |
11 | 3, 10 | eleqtrrd 2832 | . 2 β’ (π β β¨π, πβ© β dom β¨ ) |
12 | 9 | simprrd 773 | . . 3 β’ (π β dom β§ = (π΅ Γ π΅)) |
13 | 3, 12 | eleqtrrd 2832 | . 2 β’ (π β β¨π, πβ© β dom β§ ) |
14 | 11, 13 | jca 511 | 1 β’ (π β (β¨π, πβ© β dom β¨ β§ β¨π, πβ© β dom β§ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β¨cop 4635 Γ cxp 5676 dom cdm 5678 βcfv 6548 Basecbs 17180 Posetcpo 18299 joincjn 18303 meetcmee 18304 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-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
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-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5684 df-dm 5688 df-iota 6500 df-fv 6556 df-lat 18424 |
This theorem is referenced by: latlej1 18440 latlej2 18441 latjle12 18442 latmle1 18456 latmle2 18457 latlem12 18458 |
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