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Theorem polvalN 38776
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 6906 . . 3 𝐴 ∈ V
32elpw2 5346 . 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 38775 . . . 4 (𝐾 ∈ 𝐡 β†’ 𝑃 = (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))))
87fveq1d 6894 . . 3 (𝐾 ∈ 𝐡 β†’ (π‘ƒβ€˜π‘‹) = ((π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘))))β€˜π‘‹))
9 iineq1 5015 . . . . 5 (π‘š = 𝑋 β†’ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)) = ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘)))
109ineq2d 4213 . . . 4 (π‘š = 𝑋 β†’ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
11 eqid 2733 . . . 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 5318 . . . 4 (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))) ∈ V
1310, 11, 12fvmpt 6999 . . 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 2793 . 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 3948   βŠ† wss 3949  π’« cpw 4603  βˆ© ciin 4999   ↦ cmpt 5232  β€˜cfv 6544  occoc 17205  Atomscatm 38133  pmapcpmap 38368  βŠ₯𝑃cpolN 38773
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-polarityN 38774
This theorem is referenced by:  polval2N  38777  pol0N  38780  polcon3N  38788  polatN  38802
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