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Theorem polval2N 35927
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 35926 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
6 hlop 35383 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OP)
76ad2antrr 718 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝐾 ∈ OP)
8 ssel2 3793 . . . . . . 7 ((𝑋𝐴𝑝𝑋) → 𝑝𝐴)
98adantll 706 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝑝𝐴)
10 eqid 2799 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1110, 2atbase 35310 . . . . . 6 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
129, 11syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝑝 ∈ (Base‘𝐾))
1310, 1opoccl 35215 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( 𝑝) ∈ (Base‘𝐾))
147, 12, 13syl2anc 580 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → ( 𝑝) ∈ (Base‘𝐾))
1514ralrimiva 3147 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ∀𝑝𝑋 ( 𝑝) ∈ (Base‘𝐾))
16 eqid 2799 . . . 4 (glb‘𝐾) = (glb‘𝐾)
1710, 16, 2, 3pmapglb2xN 35793 . . 3 ((𝐾 ∈ HL ∧ ∀𝑝𝑋 ( 𝑝) ∈ (Base‘𝐾)) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)})) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
1815, 17syldan 586 . 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 35399 . . . . 5 ((𝐾 ∈ HL ∧ ∀𝑝𝑋 ( 𝑝) ∈ (Base‘𝐾)) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)}) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))})))
2115, 20syldan 586 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)}) = ( ‘(𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))})))
2210, 1opococ 35216 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ‘( 𝑝)) = 𝑝)
237, 12, 22syl2anc 580 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → ( ‘( 𝑝)) = 𝑝)
2423eqeq2d 2809 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → (𝑥 = ( ‘( 𝑝)) ↔ 𝑥 = 𝑝))
2524rexbidva 3230 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (∃𝑝𝑋 𝑥 = ( ‘( 𝑝)) ↔ ∃𝑝𝑋 𝑥 = 𝑝))
2625abbidv 2918 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐴) → {𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))} = {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝})
27 df-rex 3095 . . . . . . . . . 10 (∃𝑝𝑋 𝑥 = 𝑝 ↔ ∃𝑝(𝑝𝑋𝑥 = 𝑝))
28 equcom 2117 . . . . . . . . . . . . 13 (𝑥 = 𝑝𝑝 = 𝑥)
2928anbi2i 617 . . . . . . . . . . . 12 ((𝑝𝑋𝑥 = 𝑝) ↔ (𝑝𝑋𝑝 = 𝑥))
30 ancom 453 . . . . . . . . . . . 12 ((𝑝𝑋𝑝 = 𝑥) ↔ (𝑝 = 𝑥𝑝𝑋))
3129, 30bitri 267 . . . . . . . . . . 11 ((𝑝𝑋𝑥 = 𝑝) ↔ (𝑝 = 𝑥𝑝𝑋))
3231exbii 1944 . . . . . . . . . 10 (∃𝑝(𝑝𝑋𝑥 = 𝑝) ↔ ∃𝑝(𝑝 = 𝑥𝑝𝑋))
33 vex 3388 . . . . . . . . . . 11 𝑥 ∈ V
34 eleq1w 2861 . . . . . . . . . . 11 (𝑝 = 𝑥 → (𝑝𝑋𝑥𝑋))
3533, 34ceqsexv 3430 . . . . . . . . . 10 (∃𝑝(𝑝 = 𝑥𝑝𝑋) ↔ 𝑥𝑋)
3627, 32, 353bitri 289 . . . . . . . . 9 (∃𝑝𝑋 𝑥 = 𝑝𝑥𝑋)
3736abbii 2916 . . . . . . . 8 {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝} = {𝑥𝑥𝑋}
38 abid2 2922 . . . . . . . 8 {𝑥𝑥𝑋} = 𝑋
3937, 38eqtri 2821 . . . . . . 7 {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝} = 𝑋
4026, 39syl6eq 2849 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐴) → {𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))} = 𝑋)
4140fveq2d 6415 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))}) = (𝑈𝑋))
4241fveq2d 6415 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( ‘(𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))})) = ( ‘(𝑈𝑋)))
4321, 42eqtrd 2833 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)}) = ( ‘(𝑈𝑋)))
4443fveq2d 6415 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)})) = (𝑀‘( ‘(𝑈𝑋))))
455, 18, 443eqtr2d 2839 1 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑃𝑋) = (𝑀‘( ‘(𝑈𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wex 1875  wcel 2157  {cab 2785  wral 3089  wrex 3090  cin 3768  wss 3769   ciin 4711  cfv 6101  Basecbs 16184  occoc 16275  lubclub 17257  glbcglb 17258  OPcops 35193  Atomscatm 35284  HLchlt 35371  pmapcpmap 35518  𝑃cpolN 35923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-riotaBAD 34974
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-iin 4713  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-undef 7637  df-proset 17243  df-poset 17261  df-lub 17289  df-glb 17290  df-join 17291  df-meet 17292  df-p1 17355  df-lat 17361  df-clat 17423  df-oposet 35197  df-ol 35199  df-oml 35200  df-ats 35288  df-hlat 35372  df-pmap 35525  df-polarityN 35924
This theorem is referenced by:  polsubN  35928  pol1N  35931  polpmapN  35933  2polvalN  35935  3polN  35937  poldmj1N  35949  pnonsingN  35954  ispsubcl2N  35968  polsubclN  35973  poml4N  35974
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