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Theorem polfvalN 39906
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 3501 . 2 (𝐾𝐵𝐾 ∈ V)
2 polfval.p . . 3 𝑃 = (⊥𝑃𝐾)
3 fveq2 6906 . . . . . . 7 ( = 𝐾 → (Atoms‘) = (Atoms‘𝐾))
4 polfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2795 . . . . . 6 ( = 𝐾 → (Atoms‘) = 𝐴)
65pweqd 4617 . . . . 5 ( = 𝐾 → 𝒫 (Atoms‘) = 𝒫 𝐴)
7 fveq2 6906 . . . . . . . . . 10 ( = 𝐾 → (pmap‘) = (pmap‘𝐾))
8 polfval.m . . . . . . . . . 10 𝑀 = (pmap‘𝐾)
97, 8eqtr4di 2795 . . . . . . . . 9 ( = 𝐾 → (pmap‘) = 𝑀)
10 fveq2 6906 . . . . . . . . . . 11 ( = 𝐾 → (oc‘) = (oc‘𝐾))
11 polfval.o . . . . . . . . . . 11 = (oc‘𝐾)
1210, 11eqtr4di 2795 . . . . . . . . . 10 ( = 𝐾 → (oc‘) = )
1312fveq1d 6908 . . . . . . . . 9 ( = 𝐾 → ((oc‘)‘𝑝) = ( 𝑝))
149, 13fveq12d 6913 . . . . . . . 8 ( = 𝐾 → ((pmap‘)‘((oc‘)‘𝑝)) = (𝑀‘( 𝑝)))
1514adantr 480 . . . . . . 7 (( = 𝐾𝑝𝑚) → ((pmap‘)‘((oc‘)‘𝑝)) = (𝑀‘( 𝑝)))
1615iineq2dv 5017 . . . . . 6 ( = 𝐾 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)) = 𝑝𝑚 (𝑀‘( 𝑝)))
175, 16ineq12d 4221 . . . . 5 ( = 𝐾 → ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝))) = (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))
186, 17mpteq12dv 5233 . . . 4 ( = 𝐾 → (𝑚 ∈ 𝒫 (Atoms‘) ↦ ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
19 df-polarityN 39905 . . . 4 𝑃 = ( ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘) ↦ ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)))))
204fvexi 6920 . . . . . 6 𝐴 ∈ V
2120pwex 5380 . . . . 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 2789 . 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 1540  wcel 2108  Vcvv 3480  cin 3950  𝒫 cpw 4600   ciin 4992  cmpt 5225  cfv 6561  occoc 17305  Atomscatm 39264  pmapcpmap 39499  𝑃cpolN 39904
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-polarityN 39905
This theorem is referenced by:  polvalN  39907
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