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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 4736 | . . . 4 β’ {π, π} = {π, π} | |
| 2 | 1 | fveq2i 6894 | . . 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 2732 | . . 3 β’ (lubβπΎ) = (lubβπΎ) | |
| 5 | joincom.j | . . 3 β’ β¨ = (joinβπΎ) | |
| 6 | simp1 1136 | . . 3 β’ ((πΎ β π β§ π β π΅ β§ π β π΅) β πΎ β π) | |
| 7 | simp3 1138 | . . 3 β’ ((πΎ β π β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 8 | simp2 1137 | . . 3 β’ ((πΎ β π β§ π β π΅ β§ π β π΅) β π β π΅) | |
| 9 | 4, 5, 6, 7, 8 | joinval 18329 | . 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 18329 | . 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 2783 | 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 1087 = wceq 1541 β wcel 2106 {cpr 4630 βcfv 6543 (class class class)co 7408 Basecbs 17143 lubclub 18261 joincjn 18263 |
| 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-lub 18298 df-join 18300 |
| This theorem is referenced by: joincom 18354 |
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