Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > meetcom | Structured version Visualization version GIF version | ||
| Description: The meet of a poset is commutative. (The antecedent β¨π, πβ© β dom β§ β§ β¨π, πβ© β dom β§ i.e., "the meets exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 17-Sep-2011.) (Revised by NM, 12-Sep-2018.) |
| Ref | Expression |
|---|---|
| meetcom.b | β’ π΅ = (BaseβπΎ) |
| meetcom.m | β’ β§ = (meetβπΎ) |
| Ref | Expression |
|---|---|
| meetcom | β’ (((πΎ β Poset β§ π β π΅ β§ π β π΅) β§ (β¨π, πβ© β dom β§ β§ β¨π, πβ© β dom β§ )) β (π β§ π) = (π β§ π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | meetcom.b | . . 3 β’ π΅ = (BaseβπΎ) | |
| 2 | meetcom.m | . . 3 β’ β§ = (meetβπΎ) | |
| 3 | 1, 2 | meetcomALT 18355 | . 2 β’ ((πΎ β Poset β§ π β π΅ β§ π β π΅) β (π β§ π) = (π β§ π)) |
| 4 | 3 | adantr 481 | 1 β’ (((πΎ β Poset β§ π β π΅ β§ π β π΅) β§ (β¨π, πβ© β dom β§ β§ β¨π, πβ© β dom β§ )) β (π β§ π) = (π β§ π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β¨cop 4634 dom cdm 5676 βcfv 6543 (class class class)co 7408 Basecbs 17143 Posetcpo 18259 meetcmee 18264 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-glb 18299 df-meet 18301 |
| This theorem is referenced by: latmcom 18415 |
| Copyright terms: Public domain | W3C validator |