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Theorem pol1N 38376
Description: The polarity of the whole projective subspace is the empty space. Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
polssat.a 𝐴 = (Atomsβ€˜πΎ)
polssat.p βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
pol1N (𝐾 ∈ HL β†’ ( βŠ₯ β€˜π΄) = βˆ…)

Proof of Theorem pol1N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 ssid 3967 . . 3 𝐴 βŠ† 𝐴
2 eqid 2737 . . . 4 (lubβ€˜πΎ) = (lubβ€˜πΎ)
3 eqid 2737 . . . 4 (ocβ€˜πΎ) = (ocβ€˜πΎ)
4 polssat.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
5 eqid 2737 . . . 4 (pmapβ€˜πΎ) = (pmapβ€˜πΎ)
6 polssat.p . . . 4 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
72, 3, 4, 5, 6polval2N 38372 . . 3 ((𝐾 ∈ HL ∧ 𝐴 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π΄) = ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π΄))))
81, 7mpan2 690 . 2 (𝐾 ∈ HL β†’ ( βŠ₯ β€˜π΄) = ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π΄))))
9 hlop 37827 . . . . . . . . . 10 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
10 eqid 2737 . . . . . . . . . . 11 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1110, 4atbase 37754 . . . . . . . . . 10 (𝑝 ∈ 𝐴 β†’ 𝑝 ∈ (Baseβ€˜πΎ))
12 eqid 2737 . . . . . . . . . . 11 (leβ€˜πΎ) = (leβ€˜πΎ)
13 eqid 2737 . . . . . . . . . . 11 (1.β€˜πΎ) = (1.β€˜πΎ)
1410, 12, 13ople1 37656 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Baseβ€˜πΎ)) β†’ 𝑝(leβ€˜πΎ)(1.β€˜πΎ))
159, 11, 14syl2an 597 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐴) β†’ 𝑝(leβ€˜πΎ)(1.β€˜πΎ))
1615ralrimiva 3144 . . . . . . . 8 (𝐾 ∈ HL β†’ βˆ€π‘ ∈ 𝐴 𝑝(leβ€˜πΎ)(1.β€˜πΎ))
17 rabid2 3437 . . . . . . . 8 (𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ)(1.β€˜πΎ)} ↔ βˆ€π‘ ∈ 𝐴 𝑝(leβ€˜πΎ)(1.β€˜πΎ))
1816, 17sylibr 233 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ)(1.β€˜πΎ)})
1918fveq2d 6847 . . . . . 6 (𝐾 ∈ HL β†’ ((lubβ€˜πΎ)β€˜π΄) = ((lubβ€˜πΎ)β€˜{𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ)(1.β€˜πΎ)}))
20 hlomcmat 37830 . . . . . . 7 (𝐾 ∈ HL β†’ (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
2110, 13op1cl 37650 . . . . . . . 8 (𝐾 ∈ OP β†’ (1.β€˜πΎ) ∈ (Baseβ€˜πΎ))
229, 21syl 17 . . . . . . 7 (𝐾 ∈ HL β†’ (1.β€˜πΎ) ∈ (Baseβ€˜πΎ))
2310, 12, 2, 4atlatmstc 37784 . . . . . . 7 (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ (1.β€˜πΎ) ∈ (Baseβ€˜πΎ)) β†’ ((lubβ€˜πΎ)β€˜{𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ)(1.β€˜πΎ)}) = (1.β€˜πΎ))
2420, 22, 23syl2anc 585 . . . . . 6 (𝐾 ∈ HL β†’ ((lubβ€˜πΎ)β€˜{𝑝 ∈ 𝐴 ∣ 𝑝(leβ€˜πΎ)(1.β€˜πΎ)}) = (1.β€˜πΎ))
2519, 24eqtr2d 2778 . . . . 5 (𝐾 ∈ HL β†’ (1.β€˜πΎ) = ((lubβ€˜πΎ)β€˜π΄))
2625fveq2d 6847 . . . 4 (𝐾 ∈ HL β†’ ((ocβ€˜πΎ)β€˜(1.β€˜πΎ)) = ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π΄)))
27 eqid 2737 . . . . . 6 (0.β€˜πΎ) = (0.β€˜πΎ)
2827, 13, 3opoc1 37667 . . . . 5 (𝐾 ∈ OP β†’ ((ocβ€˜πΎ)β€˜(1.β€˜πΎ)) = (0.β€˜πΎ))
299, 28syl 17 . . . 4 (𝐾 ∈ HL β†’ ((ocβ€˜πΎ)β€˜(1.β€˜πΎ)) = (0.β€˜πΎ))
3026, 29eqtr3d 2779 . . 3 (𝐾 ∈ HL β†’ ((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π΄)) = (0.β€˜πΎ))
3130fveq2d 6847 . 2 (𝐾 ∈ HL β†’ ((pmapβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜((lubβ€˜πΎ)β€˜π΄))) = ((pmapβ€˜πΎ)β€˜(0.β€˜πΎ)))
32 hlatl 37825 . . 3 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
3327, 5pmap0 38231 . . 3 (𝐾 ∈ AtLat β†’ ((pmapβ€˜πΎ)β€˜(0.β€˜πΎ)) = βˆ…)
3432, 33syl 17 . 2 (𝐾 ∈ HL β†’ ((pmapβ€˜πΎ)β€˜(0.β€˜πΎ)) = βˆ…)
358, 31, 343eqtrd 2781 1 (𝐾 ∈ HL β†’ ( βŠ₯ β€˜π΄) = βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3408   βŠ† wss 3911  βˆ…c0 4283   class class class wbr 5106  β€˜cfv 6497  Basecbs 17084  lecple 17141  occoc 17142  lubclub 18199  0.cp0 18313  1.cp1 18314  CLatccla 18388  OPcops 37637  OMLcoml 37640  Atomscatm 37728  AtLatcal 37729  HLchlt 37815  pmapcpmap 37963  βŠ₯𝑃cpolN 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:  2pol0N  38377  1psubclN  38410
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