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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 18364 | . 2 β’ ((πΎ β Poset β§ π β π΅ β§ π β π΅) β (π β¨ π) = (π β¨ π)) |
4 | 3 | adantr 480 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β¨cop 4629 dom cdm 5669 βcfv 6536 (class class class)co 7404 Basecbs 17151 Posetcpo 18270 joincjn 18274 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-lub 18309 df-join 18311 |
This theorem is referenced by: latjcom 18410 |
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