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Theorem polfvalN 39887
Description: 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
polfvalN (𝐾𝐵𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
Distinct variable groups:   𝐴,𝑚   𝑚,𝑝,𝐾
Allowed substitution hints:   𝐴(𝑝)   𝐵(𝑚,𝑝)   𝑃(𝑚,𝑝)   𝑀(𝑚,𝑝)   (𝑚,𝑝)

Proof of Theorem polfvalN
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3499 . 2 (𝐾𝐵𝐾 ∈ V)
2 polfval.p . . 3 𝑃 = (⊥𝑃𝐾)
3 fveq2 6907 . . . . . . 7 ( = 𝐾 → (Atoms‘) = (Atoms‘𝐾))
4 polfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2793 . . . . . 6 ( = 𝐾 → (Atoms‘) = 𝐴)
65pweqd 4622 . . . . 5 ( = 𝐾 → 𝒫 (Atoms‘) = 𝒫 𝐴)
7 fveq2 6907 . . . . . . . . . 10 ( = 𝐾 → (pmap‘) = (pmap‘𝐾))
8 polfval.m . . . . . . . . . 10 𝑀 = (pmap‘𝐾)
97, 8eqtr4di 2793 . . . . . . . . 9 ( = 𝐾 → (pmap‘) = 𝑀)
10 fveq2 6907 . . . . . . . . . . 11 ( = 𝐾 → (oc‘) = (oc‘𝐾))
11 polfval.o . . . . . . . . . . 11 = (oc‘𝐾)
1210, 11eqtr4di 2793 . . . . . . . . . 10 ( = 𝐾 → (oc‘) = )
1312fveq1d 6909 . . . . . . . . 9 ( = 𝐾 → ((oc‘)‘𝑝) = ( 𝑝))
149, 13fveq12d 6914 . . . . . . . 8 ( = 𝐾 → ((pmap‘)‘((oc‘)‘𝑝)) = (𝑀‘( 𝑝)))
1514adantr 480 . . . . . . 7 (( = 𝐾𝑝𝑚) → ((pmap‘)‘((oc‘)‘𝑝)) = (𝑀‘( 𝑝)))
1615iineq2dv 5022 . . . . . 6 ( = 𝐾 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)) = 𝑝𝑚 (𝑀‘( 𝑝)))
175, 16ineq12d 4229 . . . . 5 ( = 𝐾 → ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝))) = (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))
186, 17mpteq12dv 5239 . . . 4 ( = 𝐾 → (𝑚 ∈ 𝒫 (Atoms‘) ↦ ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
19 df-polarityN 39886 . . . 4 𝑃 = ( ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘) ↦ ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)))))
204fvexi 6921 . . . . . 6 𝐴 ∈ V
2120pwex 5386 . . . . 5 𝒫 𝐴 ∈ V
2221mptex 7243 . . . 4 (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))) ∈ V
2318, 19, 22fvmpt 7016 . . 3 (𝐾 ∈ V → (⊥𝑃𝐾) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
242, 23eqtrid 2787 . 2 (𝐾 ∈ V → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
251, 24syl 17 1 (𝐾𝐵𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  𝒫 cpw 4605   ciin 4997  cmpt 5231  cfv 6563  occoc 17306  Atomscatm 39245  pmapcpmap 39480  𝑃cpolN 39885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-polarityN 39886
This theorem is referenced by:  polvalN  39888
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