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Theorem polval2N 39908
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 39907 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
6 hlop 39363 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OP)
76ad2antrr 726 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝐾 ∈ OP)
8 ssel2 3978 . . . . . . 7 ((𝑋𝐴𝑝𝑋) → 𝑝𝐴)
98adantll 714 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝑝𝐴)
10 eqid 2737 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1110, 2atbase 39290 . . . . . 6 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
129, 11syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → 𝑝 ∈ (Base‘𝐾))
1310, 1opoccl 39195 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( 𝑝) ∈ (Base‘𝐾))
147, 12, 13syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → ( 𝑝) ∈ (Base‘𝐾))
1514ralrimiva 3146 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ∀𝑝𝑋 ( 𝑝) ∈ (Base‘𝐾))
16 eqid 2737 . . . 4 (glb‘𝐾) = (glb‘𝐾)
1710, 16, 2, 3pmapglb2xN 39774 . . 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 39380 . . . . 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 39196 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → ( ‘( 𝑝)) = 𝑝)
237, 12, 22syl2anc 584 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → ( ‘( 𝑝)) = 𝑝)
2423eqeq2d 2748 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ 𝑝𝑋) → (𝑥 = ( ‘( 𝑝)) ↔ 𝑥 = 𝑝))
2524rexbidva 3177 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (∃𝑝𝑋 𝑥 = ( ‘( 𝑝)) ↔ ∃𝑝𝑋 𝑥 = 𝑝))
2625abbidv 2808 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝐴) → {𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))} = {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝})
27 df-rex 3071 . . . . . . . . . 10 (∃𝑝𝑋 𝑥 = 𝑝 ↔ ∃𝑝(𝑝𝑋𝑥 = 𝑝))
28 equcom 2017 . . . . . . . . . . . 12 (𝑥 = 𝑝𝑝 = 𝑥)
2928anbi1ci 626 . . . . . . . . . . 11 ((𝑝𝑋𝑥 = 𝑝) ↔ (𝑝 = 𝑥𝑝𝑋))
3029exbii 1848 . . . . . . . . . 10 (∃𝑝(𝑝𝑋𝑥 = 𝑝) ↔ ∃𝑝(𝑝 = 𝑥𝑝𝑋))
31 eleq1w 2824 . . . . . . . . . . 11 (𝑝 = 𝑥 → (𝑝𝑋𝑥𝑋))
3231equsexvw 2004 . . . . . . . . . 10 (∃𝑝(𝑝 = 𝑥𝑝𝑋) ↔ 𝑥𝑋)
3327, 30, 323bitri 297 . . . . . . . . 9 (∃𝑝𝑋 𝑥 = 𝑝𝑥𝑋)
3433abbii 2809 . . . . . . . 8 {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝} = {𝑥𝑥𝑋}
35 abid2 2879 . . . . . . . 8 {𝑥𝑥𝑋} = 𝑋
3634, 35eqtri 2765 . . . . . . 7 {𝑥 ∣ ∃𝑝𝑋 𝑥 = 𝑝} = 𝑋
3726, 36eqtrdi 2793 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐴) → {𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))} = 𝑋)
3837fveq2d 6910 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))}) = (𝑈𝑋))
3938fveq2d 6910 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( ‘(𝑈‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( ‘( 𝑝))})) = ( ‘(𝑈𝑋)))
4021, 39eqtrd 2777 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐴) → ((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)}) = ( ‘(𝑈𝑋)))
4140fveq2d 6910 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑀‘((glb‘𝐾)‘{𝑥 ∣ ∃𝑝𝑋 𝑥 = ( 𝑝)})) = (𝑀‘( ‘(𝑈𝑋))))
425, 18, 413eqtr2d 2783 1 ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑃𝑋) = (𝑀‘( ‘(𝑈𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  cin 3950  wss 3951   ciin 4992  cfv 6561  Basecbs 17247  occoc 17305  lubclub 18355  glbcglb 18356  OPcops 39173  Atomscatm 39264  HLchlt 39351  pmapcpmap 39499  𝑃cpolN 39904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p1 18471  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-ats 39268  df-hlat 39352  df-pmap 39506  df-polarityN 39905
This theorem is referenced by:  polsubN  39909  pol1N  39912  polpmapN  39914  2polvalN  39916  3polN  39918  poldmj1N  39930  pnonsingN  39935  ispsubcl2N  39949  polsubclN  39954  poml4N  39955
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