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Theorem polval2N 39889
Description: Alternate expression for value of the projective subspace polarity function. Equation for polarity in [Holland95] p. 223. (Contributed by NM, 22-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
polval2.u 𝑈 = (lub‘𝐾)
polval2.o = (oc‘𝐾)
polval2.a 𝐴 = (Atoms‘𝐾)
polval2.m 𝑀 = (pmap‘𝐾)
polval2.p 𝑃 = (⊥𝑃𝐾)
Assertion
Ref Expression
polval2N ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑃𝑋) = (𝑀‘( ‘(𝑈𝑋))))

Proof of Theorem polval2N
Dummy variables 𝑥 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 polval2.o . . 3 = (oc‘𝐾)
2 polval2.a . . 3 𝐴 = (Atoms‘𝐾)
3 polval2.m . . 3 𝑀 = (pmap‘𝐾)
4 polval2.p . . 3 𝑃 = (⊥𝑃𝐾)
51, 2, 3, 4polvalN 39888 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
6 hlop 39344 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OP)
76ad2antrr 726 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝐾 ∈ OP)
8 ssel2 3990 . . . . . . 7 ((𝑋𝐴𝑝𝑋) → 𝑝𝐴)
98adantll 714 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝑝𝐴)
10 eqid 2735 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1110, 2atbase 39271 . . . . . 6 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
129, 11syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝑝 ∈ (Base‘𝐾))
1310, 1opoccl 39176 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( 𝑝) ∈ (Base‘𝐾))
147, 12, 13syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → ( 𝑝) ∈ (Base‘𝐾))
1514ralrimiva 3144 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ∀𝑝𝑋 ( 𝑝) ∈ (Base‘𝐾))
16 eqid 2735 . . . 4 (glb‘𝐾) = (glb‘𝐾)
1710, 16, 2, 3pmapglb2xN 39755 . . 3 ((𝐾 ∈ HL ∧ ∀𝑝𝑋 ( 𝑝) ∈ (Base‘𝐾)) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)})) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
1815, 17syldan 591 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)})) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
19 polval2.u . . . . . 6 𝑈 = (lub‘𝐾)
2010, 19, 16, 1glbconxN 39361 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑝𝑋 ( 𝑝) ∈ (Base‘𝐾)) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)}) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))})))
2115, 20syldan 591 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)}) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))})))
2210, 1opococ 39177 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ‘( 𝑝)) = 𝑝)
237, 12, 22syl2anc 584 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → ( ‘( 𝑝)) = 𝑝)
2423eqeq2d 2746 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → (𝑥 = ( ‘( 𝑝)) ↔ 𝑥 = 𝑝))
2524rexbidva 3175 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (∃𝑝𝑋 𝑥 = ( ‘( 𝑝)) ↔ ∃𝑝𝑋 𝑥 = 𝑝))
2625abbidv 2806 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐴) → {𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))} = {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝})
27 df-rex 3069 . . . . . . . . . 10 (∃𝑝𝑋 𝑥 = 𝑝 ↔ ∃𝑝(𝑝𝑋𝑥 = 𝑝))
28 equcom 2015 . . . . . . . . . . . 12 (𝑥 = 𝑝𝑝 = 𝑥)
2928anbi1ci 626 . . . . . . . . . . 11 ((𝑝𝑋𝑥 = 𝑝) ↔ (𝑝 = 𝑥𝑝𝑋))
3029exbii 1845 . . . . . . . . . 10 (∃𝑝(𝑝𝑋𝑥 = 𝑝) ↔ ∃𝑝(𝑝 = 𝑥𝑝𝑋))
31 eleq1w 2822 . . . . . . . . . . 11 (𝑝 = 𝑥 → (𝑝𝑋𝑥𝑋))
3231equsexvw 2002 . . . . . . . . . 10 (∃𝑝(𝑝 = 𝑥𝑝𝑋) ↔ 𝑥𝑋)
3327, 30, 323bitri 297 . . . . . . . . 9 (∃𝑝𝑋 𝑥 = 𝑝𝑥𝑋)
3433abbii 2807 . . . . . . . 8 {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝} = {𝑥𝑥𝑋}
35 abid2 2877 . . . . . . . 8 {𝑥𝑥𝑋} = 𝑋
3634, 35eqtri 2763 . . . . . . 7 {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝} = 𝑋
3726, 36eqtrdi 2791 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐴) → {𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))} = 𝑋)
3837fveq2d 6911 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))}) = (𝑈𝑋))
3938fveq2d 6911 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( ‘(𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))})) = ( ‘(𝑈𝑋)))
4021, 39eqtrd 2775 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)}) = ( ‘(𝑈𝑋)))
4140fveq2d 6911 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)})) = (𝑀‘( ‘(𝑈𝑋))))
425, 18, 413eqtr2d 2781 1 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑃𝑋) = (𝑀‘( ‘(𝑈𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  cin 3962  wss 3963   ciin 4997  cfv 6563  Basecbs 17245  occoc 17306  lubclub 18367  glbcglb 18368  OPcops 39154  Atomscatm 39245  HLchlt 39332  pmapcpmap 39480  𝑃cpolN 39885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-ats 39249  df-hlat 39333  df-pmap 39487  df-polarityN 39886
This theorem is referenced by:  polsubN  39890  pol1N  39893  polpmapN  39895  2polvalN  39897  3polN  39899  poldmj1N  39911  pnonsingN  39916  ispsubcl2N  39930  polsubclN  39935  poml4N  39936
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