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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 38735, (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 38737. (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 2935 | . . . 4 β’ (π β π β Β¬ π = π) | |
2 | 1 | imbi1i 349 | . . 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 7412 | . . . 4 β’ (π = π β (π β¨ π ) = (π β¨ π )) | |
4 | 3 | biantrur 530 | . . 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 1021 | . . 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 298 | . 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 38726 | . . . 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 273 | . 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 283 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6537 (class class class)co 7405 lecple 17213 joincjn 18276 Atomscatm 38646 CvLatclc 38648 |
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 7722 |
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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-lat 18397 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 |
This theorem is referenced by: ishlat3N 38737 hlsupr2 38771 |
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