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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvlsupr3 | Structured version Visualization version GIF version |
Description: Two equivalent ways of expressing that π is a superposition of π and π, which can replace the superposition part of ishlat1 38856, (π₯ β π¦ β βπ§ β π΄(π§ β π₯ β§ π§ β π¦ β§ π§ β€ (π₯ β¨ π¦)) ), with the simpler βπ§ β π΄(π₯ β¨ π§) = (π¦ β¨ π§) as shown in ishlat3N 38858. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
cvlsupr2.a | β’ π΄ = (AtomsβπΎ) |
cvlsupr2.l | β’ β€ = (leβπΎ) |
cvlsupr2.j | β’ β¨ = (joinβπΎ) |
Ref | Expression |
---|---|
cvlsupr3 | β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β ((π β¨ π ) = (π β¨ π ) β (π β π β (π β π β§ π β π β§ π β€ (π β¨ π))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2938 | . . . 4 β’ (π β π β Β¬ π = π) | |
2 | 1 | imbi1i 348 | . . 3 β’ ((π β π β (π β¨ π ) = (π β¨ π )) β (Β¬ π = π β (π β¨ π ) = (π β¨ π ))) |
3 | oveq1 7433 | . . . 4 β’ (π = π β (π β¨ π ) = (π β¨ π )) | |
4 | 3 | biantrur 529 | . . 3 β’ ((Β¬ π = π β (π β¨ π ) = (π β¨ π )) β ((π = π β (π β¨ π ) = (π β¨ π )) β§ (Β¬ π = π β (π β¨ π ) = (π β¨ π )))) |
5 | pm4.83 1022 | . . 3 β’ (((π = π β (π β¨ π ) = (π β¨ π )) β§ (Β¬ π = π β (π β¨ π ) = (π β¨ π ))) β (π β¨ π ) = (π β¨ π )) | |
6 | 2, 4, 5 | 3bitrri 297 | . 2 β’ ((π β¨ π ) = (π β¨ π ) β (π β π β (π β¨ π ) = (π β¨ π ))) |
7 | cvlsupr2.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
8 | cvlsupr2.l | . . . . 5 β’ β€ = (leβπΎ) | |
9 | cvlsupr2.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
10 | 7, 8, 9 | cvlsupr2 38847 | . . . 4 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΄) β§ π β π) β ((π β¨ π ) = (π β¨ π ) β (π β π β§ π β π β§ π β€ (π β¨ π)))) |
11 | 10 | 3expia 1118 | . . 3 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β (π β π β ((π β¨ π ) = (π β¨ π ) β (π β π β§ π β π β§ π β€ (π β¨ π))))) |
12 | 11 | pm5.74d 272 | . 2 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β ((π β π β (π β¨ π ) = (π β¨ π )) β (π β π β (π β π β§ π β π β§ π β€ (π β¨ π))))) |
13 | 6, 12 | bitrid 282 | 1 β’ ((πΎ β CvLat β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β ((π β¨ π ) = (π β¨ π ) β (π β π β (π β π β§ π β π β§ π β€ (π β¨ π))))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 class class class wbr 5152 βcfv 6553 (class class class)co 7426 lecple 17247 joincjn 18310 Atomscatm 38767 CvLatclc 38769 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-proset 18294 df-poset 18312 df-plt 18329 df-lub 18345 df-glb 18346 df-join 18347 df-meet 18348 df-p0 18424 df-lat 18431 df-covers 38770 df-ats 38771 df-atl 38802 df-cvlat 38826 |
This theorem is referenced by: ishlat3N 38858 hlsupr2 38892 |
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