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Theorem polfvalN 33991
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 3184 . 2 (𝐾𝐵𝐾 ∈ V)
2 polfval.p . . 3 𝑃 = (⊥𝑃𝐾)
3 fveq2 6087 . . . . . . 7 ( = 𝐾 → (Atoms‘) = (Atoms‘𝐾))
4 polfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2661 . . . . . 6 ( = 𝐾 → (Atoms‘) = 𝐴)
65pweqd 4112 . . . . 5 ( = 𝐾 → 𝒫 (Atoms‘) = 𝒫 𝐴)
7 fveq2 6087 . . . . . . . . . 10 ( = 𝐾 → (pmap‘) = (pmap‘𝐾))
8 polfval.m . . . . . . . . . 10 𝑀 = (pmap‘𝐾)
97, 8syl6eqr 2661 . . . . . . . . 9 ( = 𝐾 → (pmap‘) = 𝑀)
10 fveq2 6087 . . . . . . . . . . 11 ( = 𝐾 → (oc‘) = (oc‘𝐾))
11 polfval.o . . . . . . . . . . 11 = (oc‘𝐾)
1210, 11syl6eqr 2661 . . . . . . . . . 10 ( = 𝐾 → (oc‘) = )
1312fveq1d 6089 . . . . . . . . 9 ( = 𝐾 → ((oc‘)‘𝑝) = ( 𝑝))
149, 13fveq12d 6093 . . . . . . . 8 ( = 𝐾 → ((pmap‘)‘((oc‘)‘𝑝)) = (𝑀‘( 𝑝)))
1514adantr 479 . . . . . . 7 (( = 𝐾𝑝𝑚) → ((pmap‘)‘((oc‘)‘𝑝)) = (𝑀‘( 𝑝)))
1615iineq2dv 4473 . . . . . 6 ( = 𝐾 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)) = 𝑝𝑚 (𝑀‘( 𝑝)))
175, 16ineq12d 3776 . . . . 5 ( = 𝐾 → ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝))) = (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))
186, 17mpteq12dv 4657 . . . 4 ( = 𝐾 → (𝑚 ∈ 𝒫 (Atoms‘) ↦ ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
19 df-polarityN 33990 . . . 4 𝑃 = ( ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘) ↦ ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)))))
20 fvex 6097 . . . . . . 7 (Atoms‘𝐾) ∈ V
214, 20eqeltri 2683 . . . . . 6 𝐴 ∈ V
2221pwex 4768 . . . . 5 𝒫 𝐴 ∈ V
2322mptex 6367 . . . 4 (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))) ∈ V
2418, 19, 23fvmpt 6175 . . 3 (𝐾 ∈ V → (⊥𝑃𝐾) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
252, 24syl5eq 2655 . 2 (𝐾 ∈ V → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
261, 25syl 17 1 (𝐾𝐵𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3172  cin 3538  𝒫 cpw 4107   ciin 4450  cmpt 4637  cfv 5789  occoc 15724  Atomscatm 33351  pmapcpmap 33584  𝑃cpolN 33989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-polarityN 33990
This theorem is referenced by:  polvalN  33992
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