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

Theorem polvalN 39079
Description: Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
polfval.o βŠ₯ = (ocβ€˜πΎ)
polfval.a 𝐴 = (Atomsβ€˜πΎ)
polfval.m 𝑀 = (pmapβ€˜πΎ)
polfval.p 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
Assertion
Ref Expression
polvalN ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ƒβ€˜π‘‹) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
Distinct variable groups:   𝐾,𝑝   𝑋,𝑝
Allowed substitution hints:   𝐴(𝑝)   𝐡(𝑝)   𝑃(𝑝)   𝑀(𝑝)   βŠ₯ (𝑝)

Proof of Theorem polvalN
Dummy variable π‘š is distinct from all other variables.
StepHypRef Expression
1 polfval.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
21fvexi 6904 . . 3 𝐴 ∈ V
32elpw2 5344 . 2 (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 βŠ† 𝐴)
4 polfval.o . . . . 5 βŠ₯ = (ocβ€˜πΎ)
5 polfval.m . . . . 5 𝑀 = (pmapβ€˜πΎ)
6 polfval.p . . . . 5 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
74, 1, 5, 6polfvalN 39078 . . . 4 (𝐾 ∈ 𝐡 β†’ 𝑃 = (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))))
87fveq1d 6892 . . 3 (𝐾 ∈ 𝐡 β†’ (π‘ƒβ€˜π‘‹) = ((π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘))))β€˜π‘‹))
9 iineq1 5013 . . . . 5 (π‘š = 𝑋 β†’ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)) = ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘)))
109ineq2d 4211 . . . 4 (π‘š = 𝑋 β†’ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
11 eqid 2730 . . . 4 (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))) = (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘))))
122inex1 5316 . . . 4 (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))) ∈ V
1310, 11, 12fvmpt 6997 . . 3 (𝑋 ∈ 𝒫 𝐴 β†’ ((π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘))))β€˜π‘‹) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
148, 13sylan9eq 2790 . 2 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ (π‘ƒβ€˜π‘‹) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
153, 14sylan2br 593 1 ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ƒβ€˜π‘‹) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  βˆ© ciin 4997   ↦ cmpt 5230  β€˜cfv 6542  occoc 17209  Atomscatm 38436  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
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-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-polarityN 39077
This theorem is referenced by:  polval2N  39080  pol0N  39083  polcon3N  39091  polatN  39105
  Copyright terms: Public domain W3C validator