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Theorem polvalN 39617
Description: Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
polfval.o = (oc‘𝐾)
polfval.a 𝐴 = (Atoms‘𝐾)
polfval.m 𝑀 = (pmap‘𝐾)
polfval.p 𝑃 = (⊥𝑃𝐾)
Assertion
Ref Expression
polvalN ((𝐾𝐵𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
Distinct variable groups:   𝐾,𝑝   𝑋,𝑝
Allowed substitution hints:   𝐴(𝑝)   𝐵(𝑝)   𝑃(𝑝)   𝑀(𝑝)   (𝑝)

Proof of Theorem polvalN
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 polfval.a . . . 4 𝐴 = (Atoms‘𝐾)
21fvexi 6907 . . 3 𝐴 ∈ V
32elpw2 5344 . 2 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
4 polfval.o . . . . 5 = (oc‘𝐾)
5 polfval.m . . . . 5 𝑀 = (pmap‘𝐾)
6 polfval.p . . . . 5 𝑃 = (⊥𝑃𝐾)
74, 1, 5, 6polfvalN 39616 . . . 4 (𝐾𝐵𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
87fveq1d 6895 . . 3 (𝐾𝐵 → (𝑃𝑋) = ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))‘𝑋))
9 iineq1 5010 . . . . 5 (𝑚 = 𝑋 𝑝𝑚 (𝑀‘( 𝑝)) = 𝑝𝑋 (𝑀‘( 𝑝)))
109ineq2d 4210 . . . 4 (𝑚 = 𝑋 → (𝐴 𝑝𝑚 (𝑀‘( 𝑝))) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
11 eqid 2726 . . . 4 (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))
122inex1 5314 . . . 4 (𝐴 𝑝𝑋 (𝑀‘( 𝑝))) ∈ V
1310, 11, 12fvmpt 7001 . . 3 (𝑋 ∈ 𝒫 𝐴 → ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))‘𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
148, 13sylan9eq 2786 . 2 ((𝐾𝐵𝑋 ∈ 𝒫 𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
153, 14sylan2br 593 1 ((𝐾𝐵𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  cin 3945  wss 3946  𝒫 cpw 4597   ciin 4994  cmpt 5228  cfv 6546  occoc 17269  Atomscatm 38974  pmapcpmap 39209  𝑃cpolN 39614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-iin 4996  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-polarityN 39615
This theorem is referenced by:  polval2N  39618  pol0N  39621  polcon3N  39629  polatN  39643
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