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Mirrors > Home > MPE Home > Th. List > joincom | Structured version Visualization version GIF version |
Description: The join of a poset is commutative. (The antecedent β¨π, πβ© β dom β¨ β§ β¨π, πβ© β dom β¨ i.e., "the joins exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.) |
Ref | Expression |
---|---|
joincom.b | β’ π΅ = (BaseβπΎ) |
joincom.j | β’ β¨ = (joinβπΎ) |
Ref | Expression |
---|---|
joincom | 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 | joincom.b | . . 3 β’ π΅ = (BaseβπΎ) | |
2 | joincom.j | . . 3 β’ β¨ = (joinβπΎ) | |
3 | 1, 2 | joincomALT 18398 | . 2 β’ ((πΎ β Poset β§ π β π΅ β§ π β π΅) β (π β¨ π) = (π β¨ π)) |
4 | 3 | adantr 479 | 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 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β¨cop 4636 dom cdm 5680 βcfv 6551 (class class class)co 7424 Basecbs 17185 Posetcpo 18304 joincjn 18308 |
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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-lub 18343 df-join 18345 |
This theorem is referenced by: latjcom 18444 |
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