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Theorem polval2N 38372
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 38371 . 2 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ƒβ€˜π‘‹) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
6 hlop 37827 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
76ad2antrr 725 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ 𝐾 ∈ OP)
8 ssel2 3940 . . . . . . 7 ((𝑋 βŠ† 𝐴 ∧ 𝑝 ∈ 𝑋) β†’ 𝑝 ∈ 𝐴)
98adantll 713 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ 𝑝 ∈ 𝐴)
10 eqid 2737 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1110, 2atbase 37754 . . . . . 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 37659 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Baseβ€˜πΎ)) β†’ ( βŠ₯ β€˜π‘) ∈ (Baseβ€˜πΎ))
147, 12, 13syl2anc 585 . . . 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 2737 . . . 4 (glbβ€˜πΎ) = (glbβ€˜πΎ)
1710, 16, 2, 3pmapglb2xN 38238 . . 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 592 . 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 37844 . . . . 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 592 . . . 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 37660 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Baseβ€˜πΎ)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘)) = 𝑝)
237, 12, 22syl2anc 585 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) ∧ 𝑝 ∈ 𝑋) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘)) = 𝑝)
2423eqeq2d 2748 . . . . . . . . 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 2806 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ {π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜( βŠ₯ β€˜π‘))} = {π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = 𝑝})
27 df-rex 3075 . . . . . . . . . 10 (βˆƒπ‘ ∈ 𝑋 π‘₯ = 𝑝 ↔ βˆƒπ‘(𝑝 ∈ 𝑋 ∧ π‘₯ = 𝑝))
28 equcom 2022 . . . . . . . . . . . 12 (π‘₯ = 𝑝 ↔ 𝑝 = π‘₯)
2928anbi1ci 627 . . . . . . . . . . 11 ((𝑝 ∈ 𝑋 ∧ π‘₯ = 𝑝) ↔ (𝑝 = π‘₯ ∧ 𝑝 ∈ 𝑋))
3029exbii 1851 . . . . . . . . . 10 (βˆƒπ‘(𝑝 ∈ 𝑋 ∧ π‘₯ = 𝑝) ↔ βˆƒπ‘(𝑝 = π‘₯ ∧ 𝑝 ∈ 𝑋))
31 eleq1w 2821 . . . . . . . . . . 11 (𝑝 = π‘₯ β†’ (𝑝 ∈ 𝑋 ↔ π‘₯ ∈ 𝑋))
3231equsexvw 2009 . . . . . . . . . 10 (βˆƒπ‘(𝑝 = π‘₯ ∧ 𝑝 ∈ 𝑋) ↔ π‘₯ ∈ 𝑋)
3327, 30, 323bitri 297 . . . . . . . . 9 (βˆƒπ‘ ∈ 𝑋 π‘₯ = 𝑝 ↔ π‘₯ ∈ 𝑋)
3433abbii 2807 . . . . . . . 8 {π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = 𝑝} = {π‘₯ ∣ π‘₯ ∈ 𝑋}
35 abid2 2876 . . . . . . . 8 {π‘₯ ∣ π‘₯ ∈ 𝑋} = 𝑋
3634, 35eqtri 2765 . . . . . . 7 {π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = 𝑝} = 𝑋
3726, 36eqtrdi 2793 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ {π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜( βŠ₯ β€˜π‘))} = 𝑋)
3837fveq2d 6847 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜( βŠ₯ β€˜π‘))}) = (π‘ˆβ€˜π‘‹))
3938fveq2d 6847 . . . 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 2777 . . 3 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ((glbβ€˜πΎ)β€˜{π‘₯ ∣ βˆƒπ‘ ∈ 𝑋 π‘₯ = ( βŠ₯ β€˜π‘)}) = ( βŠ₯ β€˜(π‘ˆβ€˜π‘‹)))
4140fveq2d 6847 . 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 2783 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  βˆƒwrex 3074   ∩ cin 3910   βŠ† wss 3911  βˆ© ciin 4956  β€˜cfv 6497  Basecbs 17084  occoc 17142  lubclub 18199  glbcglb 18200  OPcops 37637  Atomscatm 37728  HLchlt 37815  pmapcpmap 37963  βŠ₯𝑃cpolN 38368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18185  df-poset 18203  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p1 18316  df-lat 18322  df-clat 18389  df-oposet 37641  df-ol 37643  df-oml 37644  df-ats 37732  df-hlat 37816  df-pmap 37970  df-polarityN 38369
This theorem is referenced by:  polsubN  38373  pol1N  38376  polpmapN  38378  2polvalN  38380  3polN  38382  poldmj1N  38394  pnonsingN  38399  ispsubcl2N  38413  polsubclN  38418  poml4N  38419
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