Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  polval2N Structured version   Visualization version   GIF version

Theorem polval2N 39080
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 π‘ˆ = (lubβ€˜πΎ)
polval2.o βŠ₯ = (ocβ€˜πΎ)
polval2.a 𝐴 = (Atomsβ€˜πΎ)
polval2.m 𝑀 = (pmapβ€˜πΎ)
polval2.p 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
polval2N ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ƒβ€˜π‘‹) = (π‘€β€˜( βŠ₯ β€˜(π‘ˆβ€˜π‘‹))))

Proof of Theorem polval2N
Dummy variables π‘₯ 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 polval2.o . . 3 βŠ₯ = (ocβ€˜πΎ)
2 polval2.a . . 3 𝐴 = (Atomsβ€˜πΎ)
3 polval2.m . . 3 𝑀 = (pmapβ€˜πΎ)
4 polval2.p . . 3 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
51, 2, 3, 4polvalN 39079 . 2 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ƒβ€˜π‘‹) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
6 hlop 38535 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
76ad2antrr 722 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ 𝐾 ∈ OP)
8 ssel2 3976 . . . . . . 7 ((𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝑋) β†’ 𝑝 ∈ 𝐴)
98adantll 710 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ 𝑝 ∈ 𝐴)
10 eqid 2730 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1110, 2atbase 38462 . . . . . 6 (𝑝 ∈ 𝐴 β†’ 𝑝 ∈ (Baseβ€˜πΎ))
129, 11syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
1310, 1opoccl 38367 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Baseβ€˜πΎ)) β†’ ( βŠ₯ β€˜π‘) ∈ (Baseβ€˜πΎ))
147, 12, 13syl2anc 582 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ ( βŠ₯ β€˜π‘) ∈ (Baseβ€˜πΎ))
1514ralrimiva 3144 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ βˆ€π‘ ∈ 𝑋 ( βŠ₯ β€˜π‘) ∈ (Baseβ€˜πΎ))
16 eqid 2730 . . . 4 (glbβ€˜πΎ) = (glbβ€˜πΎ)
1710, 16, 2, 3pmapglb2xN 38946 . . 3 ((𝐾 ∈ HL ∧ βˆ€π‘ ∈ 𝑋 ( βŠ₯ β€˜π‘) ∈ (Baseβ€˜πΎ)) β†’ (π‘€β€˜((glbβ€˜πΎ)β€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜π‘)})) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
1815, 17syldan 589 . 2 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘€β€˜((glbβ€˜πΎ)β€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜π‘)})) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
19 polval2.u . . . . . 6 π‘ˆ = (lubβ€˜πΎ)
2010, 19, 16, 1glbconxN 38552 . . . . 5 ((𝐾 ∈ HL ∧ βˆ€π‘ ∈ 𝑋 ( βŠ₯ β€˜π‘) ∈ (Baseβ€˜πΎ)) β†’ ((glbβ€˜πΎ)β€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜π‘)}) = ( βŠ₯ β€˜(π‘ˆβ€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜( βŠ₯ β€˜π‘))})))
2115, 20syldan 589 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((glbβ€˜πΎ)β€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜π‘)}) = ( βŠ₯ β€˜(π‘ˆβ€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜( βŠ₯ β€˜π‘))})))
2210, 1opococ 38368 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Baseβ€˜πΎ)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘)) = 𝑝)
237, 12, 22syl2anc 582 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘)) = 𝑝)
2423eqeq2d 2741 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ (π‘₯ = ( βŠ₯ β€˜( βŠ₯ β€˜π‘)) ↔ π‘₯ = 𝑝))
2524rexbidva 3174 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜( βŠ₯ β€˜π‘)) ↔ βˆƒπ‘ ∈ 𝑋 π‘₯ = 𝑝))
2625abbidv 2799 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ {π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜( βŠ₯ β€˜π‘))} = {π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = 𝑝})
27 df-rex 3069 . . . . . . . . . 10 (βˆƒπ‘ ∈ 𝑋 π‘₯ = 𝑝 ↔ βˆƒπ‘(𝑝 ∈ 𝑋 ∧ π‘₯ = 𝑝))
28 equcom 2019 . . . . . . . . . . . 12 (π‘₯ = 𝑝 ↔ 𝑝 = π‘₯)
2928anbi1ci 624 . . . . . . . . . . 11 ((𝑝 ∈ 𝑋 ∧ π‘₯ = 𝑝) ↔ (𝑝 = π‘₯ ∧ 𝑝 ∈ 𝑋))
3029exbii 1848 . . . . . . . . . 10 (βˆƒπ‘(𝑝 ∈ 𝑋 ∧ π‘₯ = 𝑝) ↔ βˆƒπ‘(𝑝 = π‘₯ ∧ 𝑝 ∈ 𝑋))
31 eleq1w 2814 . . . . . . . . . . 11 (𝑝 = π‘₯ β†’ (𝑝 ∈ 𝑋 ↔ π‘₯ ∈ 𝑋))
3231equsexvw 2006 . . . . . . . . . 10 (βˆƒπ‘(𝑝 = π‘₯ ∧ 𝑝 ∈ 𝑋) ↔ π‘₯ ∈ 𝑋)
3327, 30, 323bitri 296 . . . . . . . . 9 (βˆƒπ‘ ∈ 𝑋 π‘₯ = 𝑝 ↔ π‘₯ ∈ 𝑋)
3433abbii 2800 . . . . . . . 8 {π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = 𝑝} = {π‘₯ ∣ π‘₯ ∈ 𝑋}
35 abid2 2869 . . . . . . . 8 {π‘₯ ∣ π‘₯ ∈ 𝑋} = 𝑋
3634, 35eqtri 2758 . . . . . . 7 {π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = 𝑝} = 𝑋
3726, 36eqtrdi 2786 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ {π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜( βŠ₯ β€˜π‘))} = 𝑋)
3837fveq2d 6894 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜( βŠ₯ β€˜π‘))}) = (π‘ˆβ€˜π‘‹))
3938fveq2d 6894 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜(π‘ˆβ€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜( βŠ₯ β€˜π‘))})) = ( βŠ₯ β€˜(π‘ˆβ€˜π‘‹)))
4021, 39eqtrd 2770 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((glbβ€˜πΎ)β€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜π‘)}) = ( βŠ₯ β€˜(π‘ˆβ€˜π‘‹)))
4140fveq2d 6894 . 2 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘€β€˜((glbβ€˜πΎ)β€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜π‘)})) = (π‘€β€˜( βŠ₯ β€˜(π‘ˆβ€˜π‘‹))))
425, 18, 413eqtr2d 2776 1 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ƒβ€˜π‘‹) = (π‘€β€˜( βŠ₯ β€˜(π‘ˆβ€˜π‘‹))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  {cab 2707  βˆ€wral 3059  βˆƒwrex 3068   ∩ cin 3946   βŠ† wss 3947  βˆ© ciin 4997  β€˜cfv 6542  Basecbs 17148  occoc 17209  lubclub 18266  glbcglb 18267  OPcops 38345  Atomscatm 38436  HLchlt 38523  pmapcpmap 38671  βŠ₯𝑃cpolN 39076
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-proset 18252  df-poset 18270  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p1 18383  df-lat 18389  df-clat 18456  df-oposet 38349  df-ol 38351  df-oml 38352  df-ats 38440  df-hlat 38524  df-pmap 38678  df-polarityN 39077
This theorem is referenced by:  polsubN  39081  pol1N  39084  polpmapN  39086  2polvalN  39088  3polN  39090  poldmj1N  39102  pnonsingN  39107  ispsubcl2N  39121  polsubclN  39126  poml4N  39127
  Copyright terms: Public domain W3C validator