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| Mirrors > Home > MPE Home > Th. List > joincomALT | Structured version Visualization version GIF version | ||
| Description: The join of a poset is commutative. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 16-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| joincom.b | β’ π΅ = (BaseβπΎ) |
| joincom.j | β’ β¨ = (joinβπΎ) |
| Ref | Expression |
|---|---|
| joincomALT | β’ ((πΎ β π β§ π β π΅ β§ π β π΅) β (π β¨ π) = (π β¨ π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prcom 4732 | . . . 4 β’ {π, π} = {π, π} | |
| 2 | 1 | fveq2i 6895 | . . 3 β’ ((lubβπΎ)β{π, π}) = ((lubβπΎ)β{π, π}) |
| 3 | 2 | a1i 11 | . 2 β’ ((πΎ β π β§ π β π΅ β§ π β π΅) β ((lubβπΎ)β{π, π}) = ((lubβπΎ)β{π, π})) |
| 4 | eqid 2725 | . . 3 β’ (lubβπΎ) = (lubβπΎ) | |
| 5 | joincom.j | . . 3 β’ β¨ = (joinβπΎ) | |
| 6 | simp1 1133 | . . 3 β’ ((πΎ β π β§ π β π΅ β§ π β π΅) β πΎ β π) | |
| 7 | simp3 1135 | . . 3 β’ ((πΎ β π β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 8 | simp2 1134 | . . 3 β’ ((πΎ β π β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 9 | 4, 5, 6, 7, 8 | joinval 18368 | . 2 β’ ((πΎ β π β§ π β π΅ β§ π β π΅) β (π β¨ π) = ((lubβπΎ)β{π, π})) |
| 10 | 4, 5, 6, 8, 7 | joinval 18368 | . 2 β’ ((πΎ β π β§ π β π΅ β§ π β π΅) β (π β¨ π) = ((lubβπΎ)β{π, π})) |
| 11 | 3, 9, 10 | 3eqtr4rd 2776 | 1 β’ ((πΎ β π β§ π β π΅ β§ π β π΅) β (π β¨ π) = (π β¨ π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 {cpr 4626 βcfv 6543 (class class class)co 7416 Basecbs 17179 lubclub 18300 joincjn 18302 |
| 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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
| 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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-lub 18337 df-join 18339 |
| This theorem is referenced by: joincom 18393 |
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