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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3polN | Structured version Visualization version GIF version |
Description: Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2polss.a | β’ π΄ = (AtomsβπΎ) |
2polss.p | β’ β₯ = (β₯πβπΎ) |
Ref | Expression |
---|---|
3polN | β’ ((πΎ β HL β§ π β π΄) β ( β₯ β( β₯ β( β₯ βπ))) = ( β₯ βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlclat 38830 | . . . 4 β’ (πΎ β HL β πΎ β CLat) | |
2 | eqid 2728 | . . . . . 6 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | 2polss.a | . . . . . 6 β’ π΄ = (AtomsβπΎ) | |
4 | 2, 3 | atssbase 38762 | . . . . 5 β’ π΄ β (BaseβπΎ) |
5 | sstr 3988 | . . . . 5 β’ ((π β π΄ β§ π΄ β (BaseβπΎ)) β π β (BaseβπΎ)) | |
6 | 4, 5 | mpan2 690 | . . . 4 β’ (π β π΄ β π β (BaseβπΎ)) |
7 | eqid 2728 | . . . . 5 β’ (lubβπΎ) = (lubβπΎ) | |
8 | 2, 7 | clatlubcl 18494 | . . . 4 β’ ((πΎ β CLat β§ π β (BaseβπΎ)) β ((lubβπΎ)βπ) β (BaseβπΎ)) |
9 | 1, 6, 8 | syl2an 595 | . . 3 β’ ((πΎ β HL β§ π β π΄) β ((lubβπΎ)βπ) β (BaseβπΎ)) |
10 | eqid 2728 | . . . 4 β’ (ocβπΎ) = (ocβπΎ) | |
11 | eqid 2728 | . . . 4 β’ (pmapβπΎ) = (pmapβπΎ) | |
12 | 2polss.p | . . . 4 β’ β₯ = (β₯πβπΎ) | |
13 | 2, 10, 11, 12 | polpmapN 39385 | . . 3 β’ ((πΎ β HL β§ ((lubβπΎ)βπ) β (BaseβπΎ)) β ( β₯ β((pmapβπΎ)β((lubβπΎ)βπ))) = ((pmapβπΎ)β((ocβπΎ)β((lubβπΎ)βπ)))) |
14 | 9, 13 | syldan 590 | . 2 β’ ((πΎ β HL β§ π β π΄) β ( β₯ β((pmapβπΎ)β((lubβπΎ)βπ))) = ((pmapβπΎ)β((ocβπΎ)β((lubβπΎ)βπ)))) |
15 | 7, 3, 11, 12 | 2polvalN 39387 | . . 3 β’ ((πΎ β HL β§ π β π΄) β ( β₯ β( β₯ βπ)) = ((pmapβπΎ)β((lubβπΎ)βπ))) |
16 | 15 | fveq2d 6901 | . 2 β’ ((πΎ β HL β§ π β π΄) β ( β₯ β( β₯ β( β₯ βπ))) = ( β₯ β((pmapβπΎ)β((lubβπΎ)βπ)))) |
17 | 7, 10, 3, 11, 12 | polval2N 39379 | . 2 β’ ((πΎ β HL β§ π β π΄) β ( β₯ βπ) = ((pmapβπΎ)β((ocβπΎ)β((lubβπΎ)βπ)))) |
18 | 14, 16, 17 | 3eqtr4d 2778 | 1 β’ ((πΎ β HL β§ π β π΄) β ( β₯ β( β₯ β( β₯ βπ))) = ( β₯ βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wss 3947 βcfv 6548 Basecbs 17179 occoc 17240 lubclub 18300 CLatccla 18489 Atomscatm 38735 HLchlt 38822 pmapcpmap 38970 β₯πcpolN 39375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-proset 18286 df-poset 18304 df-plt 18321 df-lub 18337 df-glb 18338 df-join 18339 df-meet 18340 df-p0 18416 df-p1 18417 df-lat 18423 df-clat 18490 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-pmap 38977 df-polarityN 39376 |
This theorem is referenced by: 2polcon4bN 39391 2pmaplubN 39399 pmapocjN 39403 poml5N 39427 |
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