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Theorem polfvalN 35711
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 3352 . 2 (𝐾𝐵𝐾 ∈ V)
2 polfval.p . . 3 𝑃 = (⊥𝑃𝐾)
3 fveq2 6353 . . . . . . 7 ( = 𝐾 → (Atoms‘) = (Atoms‘𝐾))
4 polfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2812 . . . . . 6 ( = 𝐾 → (Atoms‘) = 𝐴)
65pweqd 4307 . . . . 5 ( = 𝐾 → 𝒫 (Atoms‘) = 𝒫 𝐴)
7 fveq2 6353 . . . . . . . . . 10 ( = 𝐾 → (pmap‘) = (pmap‘𝐾))
8 polfval.m . . . . . . . . . 10 𝑀 = (pmap‘𝐾)
97, 8syl6eqr 2812 . . . . . . . . 9 ( = 𝐾 → (pmap‘) = 𝑀)
10 fveq2 6353 . . . . . . . . . . 11 ( = 𝐾 → (oc‘) = (oc‘𝐾))
11 polfval.o . . . . . . . . . . 11 = (oc‘𝐾)
1210, 11syl6eqr 2812 . . . . . . . . . 10 ( = 𝐾 → (oc‘) = )
1312fveq1d 6355 . . . . . . . . 9 ( = 𝐾 → ((oc‘)‘𝑝) = ( 𝑝))
149, 13fveq12d 6359 . . . . . . . 8 ( = 𝐾 → ((pmap‘)‘((oc‘)‘𝑝)) = (𝑀‘( 𝑝)))
1514adantr 472 . . . . . . 7 (( = 𝐾𝑝𝑚) → ((pmap‘)‘((oc‘)‘𝑝)) = (𝑀‘( 𝑝)))
1615iineq2dv 4695 . . . . . 6 ( = 𝐾 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)) = 𝑝𝑚 (𝑀‘( 𝑝)))
175, 16ineq12d 3958 . . . . 5 ( = 𝐾 → ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝))) = (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))
186, 17mpteq12dv 4885 . . . 4 ( = 𝐾 → (𝑚 ∈ 𝒫 (Atoms‘) ↦ ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
19 df-polarityN 35710 . . . 4 𝑃 = ( ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘) ↦ ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)))))
20 fvex 6363 . . . . . . 7 (Atoms‘𝐾) ∈ V
214, 20eqeltri 2835 . . . . . 6 𝐴 ∈ V
2221pwex 4997 . . . . 5 𝒫 𝐴 ∈ V
2322mptex 6651 . . . 4 (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))) ∈ V
2418, 19, 23fvmpt 6445 . . 3 (𝐾 ∈ V → (⊥𝑃𝐾) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
252, 24syl5eq 2806 . 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 1632  wcel 2139  Vcvv 3340  cin 3714  𝒫 cpw 4302   ciin 4673  cmpt 4881  cfv 6049  occoc 16171  Atomscatm 35071  pmapcpmap 35304  𝑃cpolN 35709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-polarityN 35710
This theorem is referenced by:  polvalN  35712
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