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 38371
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 6857 . . 3 𝐴 ∈ V
32elpw2 5303 . 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 38370 . . . 4 (𝐾 ∈ 𝐡 β†’ 𝑃 = (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))))
87fveq1d 6845 . . 3 (𝐾 ∈ 𝐡 β†’ (π‘ƒβ€˜π‘‹) = ((π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘))))β€˜π‘‹))
9 iineq1 4972 . . . . 5 (π‘š = 𝑋 β†’ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)) = ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘)))
109ineq2d 4173 . . . 4 (π‘š = 𝑋 β†’ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
11 eqid 2737 . . . 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 5275 . . . 4 (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))) ∈ V
1310, 11, 12fvmpt 6949 . . 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 2797 . 2 ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝒫 𝐴) β†’ (π‘ƒβ€˜π‘‹) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
153, 14sylan2br 596 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 397   = wceq 1542   ∈ wcel 2107   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  βˆ© ciin 4956   ↦ cmpt 5189  β€˜cfv 6497  occoc 17142  Atomscatm 37728  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
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-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-polarityN 38369
This theorem is referenced by:  polval2N  38372  pol0N  38375  polcon3N  38383  polatN  38397
  Copyright terms: Public domain W3C validator