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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 37823 | . . . 4 β’ (πΎ β HL β πΎ β CLat) | |
2 | eqid 2737 | . . . . . 6 β’ (BaseβπΎ) = (BaseβπΎ) | |
3 | 2polss.a | . . . . . 6 β’ π΄ = (AtomsβπΎ) | |
4 | 2, 3 | atssbase 37755 | . . . . 5 β’ π΄ β (BaseβπΎ) |
5 | sstr 3953 | . . . . 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 2737 | . . . . 5 β’ (lubβπΎ) = (lubβπΎ) | |
8 | 2, 7 | clatlubcl 18393 | . . . 4 β’ ((πΎ β CLat β§ π β (BaseβπΎ)) β ((lubβπΎ)βπ) β (BaseβπΎ)) |
9 | 1, 6, 8 | syl2an 597 | . . 3 β’ ((πΎ β HL β§ π β π΄) β ((lubβπΎ)βπ) β (BaseβπΎ)) |
10 | eqid 2737 | . . . 4 β’ (ocβπΎ) = (ocβπΎ) | |
11 | eqid 2737 | . . . 4 β’ (pmapβπΎ) = (pmapβπΎ) | |
12 | 2polss.p | . . . 4 β’ β₯ = (β₯πβπΎ) | |
13 | 2, 10, 11, 12 | polpmapN 38378 | . . 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 592 | . 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 38380 | . . 3 β’ ((πΎ β HL β§ π β π΄) β ( β₯ β( β₯ βπ)) = ((pmapβπΎ)β((lubβπΎ)βπ))) |
16 | 15 | fveq2d 6847 | . 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 38372 | . 2 β’ ((πΎ β HL β§ π β π΄) β ( β₯ βπ) = ((pmapβπΎ)β((ocβπΎ)β((lubβπΎ)βπ)))) |
18 | 14, 16, 17 | 3eqtr4d 2787 | 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 397 = wceq 1542 β wcel 2107 β wss 3911 βcfv 6497 Basecbs 17084 occoc 17142 lubclub 18199 CLatccla 18388 Atomscatm 37728 HLchlt 37815 pmapcpmap 37963 β₯πcpolN 38368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-proset 18185 df-poset 18203 df-plt 18220 df-lub 18236 df-glb 18237 df-join 18238 df-meet 18239 df-p0 18315 df-p1 18316 df-lat 18322 df-clat 18389 df-oposet 37641 df-ol 37643 df-oml 37644 df-covers 37731 df-ats 37732 df-atl 37763 df-cvlat 37787 df-hlat 37816 df-pmap 37970 df-polarityN 38369 |
This theorem is referenced by: 2polcon4bN 38384 2pmaplubN 38392 pmapocjN 38396 poml5N 38420 |
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