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Theorem pnonsingN 39462
Description: The intersection of a set of atoms and its polarity is empty. Definition of nonsingular in [Holland95] p. 214. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2polat.a 𝐴 = (Atomsβ€˜πΎ)
2polat.p 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
pnonsingN ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (𝑋 ∩ (π‘ƒβ€˜π‘‹)) = βˆ…)

Proof of Theorem pnonsingN
StepHypRef Expression
1 2polat.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
2 2polat.p . . . . 5 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
31, 22polssN 39444 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ 𝑋 βŠ† (π‘ƒβ€˜(π‘ƒβ€˜π‘‹)))
43ssrind 4230 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (𝑋 ∩ (π‘ƒβ€˜π‘‹)) βŠ† ((π‘ƒβ€˜(π‘ƒβ€˜π‘‹)) ∩ (π‘ƒβ€˜π‘‹)))
5 eqid 2725 . . . . . 6 (lubβ€˜πΎ) = (lubβ€˜πΎ)
6 eqid 2725 . . . . . 6 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
75, 1, 6, 22polvalN 39443 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ƒβ€˜(π‘ƒβ€˜π‘‹)) = ((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)))
8 eqid 2725 . . . . . 6 (ocβ€˜πΎ) = (ocβ€˜πΎ)
95, 8, 1, 6, 2polval2N 39435 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ƒβ€˜π‘‹) = ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹))))
107, 9ineq12d 4207 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((π‘ƒβ€˜(π‘ƒβ€˜π‘‹)) ∩ (π‘ƒβ€˜π‘‹)) = (((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) ∩ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)))))
11 hlop 38890 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
1211adantr 479 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ 𝐾 ∈ OP)
13 hlclat 38886 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ CLat)
14 eqid 2725 . . . . . . . . . 10 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1514, 1atssbase 38818 . . . . . . . . 9 𝐴 βŠ† (Baseβ€˜πΎ)
16 sstr 3981 . . . . . . . . 9 ((𝑋 βŠ† 𝐴 ∧ 𝐴 βŠ† (Baseβ€˜πΎ)) β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
1715, 16mpan2 689 . . . . . . . 8 (𝑋 βŠ† 𝐴 β†’ 𝑋 βŠ† (Baseβ€˜πΎ))
1814, 5clatlubcl 18494 . . . . . . . 8 ((𝐾 ∈ CLat ∧ 𝑋 βŠ† (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
1913, 17, 18syl2an 594 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ))
20 eqid 2725 . . . . . . . 8 (meetβ€˜πΎ) = (meetβ€˜πΎ)
21 eqid 2725 . . . . . . . 8 (0.β€˜πΎ) = (0.β€˜πΎ)
2214, 8, 20, 21opnoncon 38736 . . . . . . 7 ((𝐾 ∈ OP ∧ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ)) β†’ (((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹))) = (0.β€˜πΎ))
2312, 19, 22syl2anc 582 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹))) = (0.β€˜πΎ))
2423fveq2d 6896 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)))) = ((pmapβ€˜πΎ)β€˜(0.β€˜πΎ)))
25 simpl 481 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ 𝐾 ∈ HL)
2614, 8opoccl 38722 . . . . . . 7 ((𝐾 ∈ OP ∧ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) ∈ (Baseβ€˜πΎ))
2712, 19, 26syl2anc 582 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) ∈ (Baseβ€˜πΎ))
2814, 20, 1, 6pmapmeet 39302 . . . . . 6 ((𝐾 ∈ HL ∧ ((lubβ€˜πΎ)β€˜π‘‹) ∈ (Baseβ€˜πΎ) ∧ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) ∈ (Baseβ€˜πΎ)) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)))) = (((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) ∩ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)))))
2925, 19, 27, 28syl3anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((pmapβ€˜πΎ)β€˜(((lubβ€˜πΎ)β€˜π‘‹)(meetβ€˜πΎ)((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)))) = (((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) ∩ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)))))
30 hlatl 38888 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
3130adantr 479 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ 𝐾 ∈ AtLat)
3221, 6pmap0 39294 . . . . . 6 (𝐾 ∈ AtLat β†’ ((pmapβ€˜πΎ)β€˜(0.β€˜πΎ)) = βˆ…)
3331, 32syl 17 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((pmapβ€˜πΎ)β€˜(0.β€˜πΎ)) = βˆ…)
3424, 29, 333eqtr3d 2773 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (((pmapβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)) ∩ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π‘‹)))) = βˆ…)
3510, 34eqtrd 2765 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((π‘ƒβ€˜(π‘ƒβ€˜π‘‹)) ∩ (π‘ƒβ€˜π‘‹)) = βˆ…)
364, 35sseqtrd 4013 . 2 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (𝑋 ∩ (π‘ƒβ€˜π‘‹)) βŠ† βˆ…)
37 ss0b 4393 . 2 ((𝑋 ∩ (π‘ƒβ€˜π‘‹)) βŠ† βˆ… ↔ (𝑋 ∩ (π‘ƒβ€˜π‘‹)) = βˆ…)
3836, 37sylib 217 1 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (𝑋 ∩ (π‘ƒβ€˜π‘‹)) = βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ∩ cin 3938   βŠ† wss 3939  βˆ…c0 4318  β€˜cfv 6543  (class class class)co 7416  Basecbs 17179  occoc 17240  lubclub 18300  meetcmee 18303  0.cp0 18414  CLatccla 18489  OPcops 38700  Atomscatm 38791  AtLatcal 38792  HLchlt 38878  pmapcpmap 39026  βŠ₯𝑃cpolN 39431
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7372  df-ov 7419  df-oprab 7420  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 38704  df-ol 38706  df-oml 38707  df-covers 38794  df-ats 38795  df-atl 38826  df-cvlat 38850  df-hlat 38879  df-pmap 39033  df-polarityN 39432
This theorem is referenced by:  osumcllem4N  39488  pexmidN  39498  pexmidlem1N  39499
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