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Theorem polval2N 34210
 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 34209 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
6 hlop 33667 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OP)
76ad2antrr 758 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝐾 ∈ OP)
8 ssel2 3563 . . . . . . 7 ((𝑋𝐴𝑝𝑋) → 𝑝𝐴)
98adantll 746 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝑝𝐴)
10 eqid 2610 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1110, 2atbase 33594 . . . . . 6 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
129, 11syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝑝 ∈ (Base‘𝐾))
1310, 1opoccl 33499 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( 𝑝) ∈ (Base‘𝐾))
147, 12, 13syl2anc 691 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → ( 𝑝) ∈ (Base‘𝐾))
1514ralrimiva 2949 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ∀𝑝𝑋 ( 𝑝) ∈ (Base‘𝐾))
16 eqid 2610 . . . 4 (glb‘𝐾) = (glb‘𝐾)
1710, 16, 2, 3pmapglb2xN 34076 . . 3 ((𝐾 ∈ HL ∧ ∀𝑝𝑋 ( 𝑝) ∈ (Base‘𝐾)) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)})) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
1815, 17syldan 486 . 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 33682 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑝𝑋 ( 𝑝) ∈ (Base‘𝐾)) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)}) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))})))
2115, 20syldan 486 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)}) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))})))
2210, 1opococ 33500 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ‘( 𝑝)) = 𝑝)
237, 12, 22syl2anc 691 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → ( ‘( 𝑝)) = 𝑝)
2423eqeq2d 2620 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → (𝑥 = ( ‘( 𝑝)) ↔ 𝑥 = 𝑝))
2524rexbidva 3031 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (∃𝑝𝑋 𝑥 = ( ‘( 𝑝)) ↔ ∃𝑝𝑋 𝑥 = 𝑝))
2625abbidv 2728 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐴) → {𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))} = {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝})
27 df-rex 2902 . . . . . . . . . 10 (∃𝑝𝑋 𝑥 = 𝑝 ↔ ∃𝑝(𝑝𝑋𝑥 = 𝑝))
28 equcom 1932 . . . . . . . . . . . . 13 (𝑥 = 𝑝𝑝 = 𝑥)
2928anbi2i 726 . . . . . . . . . . . 12 ((𝑝𝑋𝑥 = 𝑝) ↔ (𝑝𝑋𝑝 = 𝑥))
30 ancom 465 . . . . . . . . . . . 12 ((𝑝𝑋𝑝 = 𝑥) ↔ (𝑝 = 𝑥𝑝𝑋))
3129, 30bitri 263 . . . . . . . . . . 11 ((𝑝𝑋𝑥 = 𝑝) ↔ (𝑝 = 𝑥𝑝𝑋))
3231exbii 1764 . . . . . . . . . 10 (∃𝑝(𝑝𝑋𝑥 = 𝑝) ↔ ∃𝑝(𝑝 = 𝑥𝑝𝑋))
33 vex 3176 . . . . . . . . . . 11 𝑥 ∈ V
34 eleq1 2676 . . . . . . . . . . 11 (𝑝 = 𝑥 → (𝑝𝑋𝑥𝑋))
3533, 34ceqsexv 3215 . . . . . . . . . 10 (∃𝑝(𝑝 = 𝑥𝑝𝑋) ↔ 𝑥𝑋)
3627, 32, 353bitri 285 . . . . . . . . 9 (∃𝑝𝑋 𝑥 = 𝑝𝑥𝑋)
3736abbii 2726 . . . . . . . 8 {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝} = {𝑥𝑥𝑋}
38 abid2 2732 . . . . . . . 8 {𝑥𝑥𝑋} = 𝑋
3937, 38eqtri 2632 . . . . . . 7 {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝} = 𝑋
4026, 39syl6eq 2660 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐴) → {𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))} = 𝑋)
4140fveq2d 6107 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))}) = (𝑈𝑋))
4241fveq2d 6107 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( ‘(𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))})) = ( ‘(𝑈𝑋)))
4321, 42eqtrd 2644 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)}) = ( ‘(𝑈𝑋)))
4443fveq2d 6107 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)})) = (𝑀‘( ‘(𝑈𝑋))))
455, 18, 443eqtr2d 2650 1 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑃𝑋) = (𝑀‘( ‘(𝑈𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  ∩ ciin 4456  ‘cfv 5804  Basecbs 15695  occoc 15776  lubclub 16765  glbcglb 16766  OPcops 33477  Atomscatm 33568  HLchlt 33655  pmapcpmap 33801  ⊥𝑃cpolN 34206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-undef 7286  df-preset 16751  df-poset 16769  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-ats 33572  df-hlat 33656  df-pmap 33808  df-polarityN 34207 This theorem is referenced by:  polsubN  34211  pol1N  34214  polpmapN  34216  2polvalN  34218  3polN  34220  poldmj1N  34232  pnonsingN  34237  ispsubcl2N  34251  polsubclN  34256  poml4N  34257
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