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Theorem polfvalN 39078
Description: 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
polfvalN (𝐾 ∈ 𝐡 β†’ 𝑃 = (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))))
Distinct variable groups:   𝐴,π‘š   π‘š,𝑝,𝐾
Allowed substitution hints:   𝐴(𝑝)   𝐡(π‘š,𝑝)   𝑃(π‘š,𝑝)   𝑀(π‘š,𝑝)   βŠ₯ (π‘š,𝑝)

Proof of Theorem polfvalN
Dummy variable β„Ž is distinct from all other variables.
StepHypRef Expression
1 elex 3491 . 2 (𝐾 ∈ 𝐡 β†’ 𝐾 ∈ V)
2 polfval.p . . 3 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
3 fveq2 6890 . . . . . . 7 (β„Ž = 𝐾 β†’ (Atomsβ€˜β„Ž) = (Atomsβ€˜πΎ))
4 polfval.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2788 . . . . . 6 (β„Ž = 𝐾 β†’ (Atomsβ€˜β„Ž) = 𝐴)
65pweqd 4618 . . . . 5 (β„Ž = 𝐾 β†’ 𝒫 (Atomsβ€˜β„Ž) = 𝒫 𝐴)
7 fveq2 6890 . . . . . . . . . 10 (β„Ž = 𝐾 β†’ (pmapβ€˜β„Ž) = (pmapβ€˜πΎ))
8 polfval.m . . . . . . . . . 10 𝑀 = (pmapβ€˜πΎ)
97, 8eqtr4di 2788 . . . . . . . . 9 (β„Ž = 𝐾 β†’ (pmapβ€˜β„Ž) = 𝑀)
10 fveq2 6890 . . . . . . . . . . 11 (β„Ž = 𝐾 β†’ (ocβ€˜β„Ž) = (ocβ€˜πΎ))
11 polfval.o . . . . . . . . . . 11 βŠ₯ = (ocβ€˜πΎ)
1210, 11eqtr4di 2788 . . . . . . . . . 10 (β„Ž = 𝐾 β†’ (ocβ€˜β„Ž) = βŠ₯ )
1312fveq1d 6892 . . . . . . . . 9 (β„Ž = 𝐾 β†’ ((ocβ€˜β„Ž)β€˜π‘) = ( βŠ₯ β€˜π‘))
149, 13fveq12d 6897 . . . . . . . 8 (β„Ž = 𝐾 β†’ ((pmapβ€˜β„Ž)β€˜((ocβ€˜β„Ž)β€˜π‘)) = (π‘€β€˜( βŠ₯ β€˜π‘)))
1514adantr 479 . . . . . . 7 ((β„Ž = 𝐾 ∧ 𝑝 ∈ π‘š) β†’ ((pmapβ€˜β„Ž)β€˜((ocβ€˜β„Ž)β€˜π‘)) = (π‘€β€˜( βŠ₯ β€˜π‘)))
1615iineq2dv 5021 . . . . . 6 (β„Ž = 𝐾 β†’ ∩ 𝑝 ∈ π‘š ((pmapβ€˜β„Ž)β€˜((ocβ€˜β„Ž)β€˜π‘)) = ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))
175, 16ineq12d 4212 . . . . 5 (β„Ž = 𝐾 β†’ ((Atomsβ€˜β„Ž) ∩ ∩ 𝑝 ∈ π‘š ((pmapβ€˜β„Ž)β€˜((ocβ€˜β„Ž)β€˜π‘))) = (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘))))
186, 17mpteq12dv 5238 . . . 4 (β„Ž = 𝐾 β†’ (π‘š ∈ 𝒫 (Atomsβ€˜β„Ž) ↦ ((Atomsβ€˜β„Ž) ∩ ∩ 𝑝 ∈ π‘š ((pmapβ€˜β„Ž)β€˜((ocβ€˜β„Ž)β€˜π‘)))) = (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))))
19 df-polarityN 39077 . . . 4 βŠ₯𝑃 = (β„Ž ∈ V ↦ (π‘š ∈ 𝒫 (Atomsβ€˜β„Ž) ↦ ((Atomsβ€˜β„Ž) ∩ ∩ 𝑝 ∈ π‘š ((pmapβ€˜β„Ž)β€˜((ocβ€˜β„Ž)β€˜π‘)))))
204fvexi 6904 . . . . . 6 𝐴 ∈ V
2120pwex 5377 . . . . 5 𝒫 𝐴 ∈ V
2221mptex 7226 . . . 4 (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))) ∈ V
2318, 19, 22fvmpt 6997 . . 3 (𝐾 ∈ V β†’ (βŠ₯π‘ƒβ€˜πΎ) = (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))))
242, 23eqtrid 2782 . 2 (𝐾 ∈ V β†’ 𝑃 = (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))))
251, 24syl 17 1 (𝐾 ∈ 𝐡 β†’ 𝑃 = (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ∩ cin 3946  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:  polvalN  39079
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