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Theorem polfvalN 38370
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 3464 . 2 (𝐾 ∈ 𝐡 β†’ 𝐾 ∈ V)
2 polfval.p . . 3 𝑃 = (βŠ₯π‘ƒβ€˜πΎ)
3 fveq2 6843 . . . . . . 7 (β„Ž = 𝐾 β†’ (Atomsβ€˜β„Ž) = (Atomsβ€˜πΎ))
4 polfval.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2795 . . . . . 6 (β„Ž = 𝐾 β†’ (Atomsβ€˜β„Ž) = 𝐴)
65pweqd 4578 . . . . 5 (β„Ž = 𝐾 β†’ 𝒫 (Atomsβ€˜β„Ž) = 𝒫 𝐴)
7 fveq2 6843 . . . . . . . . . 10 (β„Ž = 𝐾 β†’ (pmapβ€˜β„Ž) = (pmapβ€˜πΎ))
8 polfval.m . . . . . . . . . 10 𝑀 = (pmapβ€˜πΎ)
97, 8eqtr4di 2795 . . . . . . . . 9 (β„Ž = 𝐾 β†’ (pmapβ€˜β„Ž) = 𝑀)
10 fveq2 6843 . . . . . . . . . . 11 (β„Ž = 𝐾 β†’ (ocβ€˜β„Ž) = (ocβ€˜πΎ))
11 polfval.o . . . . . . . . . . 11 βŠ₯ = (ocβ€˜πΎ)
1210, 11eqtr4di 2795 . . . . . . . . . 10 (β„Ž = 𝐾 β†’ (ocβ€˜β„Ž) = βŠ₯ )
1312fveq1d 6845 . . . . . . . . 9 (β„Ž = 𝐾 β†’ ((ocβ€˜β„Ž)β€˜π‘) = ( βŠ₯ β€˜π‘))
149, 13fveq12d 6850 . . . . . . . 8 (β„Ž = 𝐾 β†’ ((pmapβ€˜β„Ž)β€˜((ocβ€˜β„Ž)β€˜π‘)) = (π‘€β€˜( βŠ₯ β€˜π‘)))
1514adantr 482 . . . . . . 7 ((β„Ž = 𝐾 ∧ 𝑝 ∈ π‘š) β†’ ((pmapβ€˜β„Ž)β€˜((ocβ€˜β„Ž)β€˜π‘)) = (π‘€β€˜( βŠ₯ β€˜π‘)))
1615iineq2dv 4980 . . . . . 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 4174 . . . . 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 5197 . . . 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 38369 . . . 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 6857 . . . . . 6 𝐴 ∈ V
2120pwex 5336 . . . . 5 𝒫 𝐴 ∈ V
2221mptex 7174 . . . 4 (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))) ∈ V
2318, 19, 22fvmpt 6949 . . 3 (𝐾 ∈ V β†’ (βŠ₯π‘ƒβ€˜πΎ) = (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))))
242, 23eqtrid 2789 . 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 1542   ∈ wcel 2107  Vcvv 3446   ∩ cin 3910  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:  polvalN  38371
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