Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  polfvalN Structured version   Visualization version   GIF version

Theorem polfvalN 36491
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 3433 . 2 (𝐾𝐵𝐾 ∈ V)
2 polfval.p . . 3 𝑃 = (⊥𝑃𝐾)
3 fveq2 6499 . . . . . . 7 ( = 𝐾 → (Atoms‘) = (Atoms‘𝐾))
4 polfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2832 . . . . . 6 ( = 𝐾 → (Atoms‘) = 𝐴)
65pweqd 4427 . . . . 5 ( = 𝐾 → 𝒫 (Atoms‘) = 𝒫 𝐴)
7 fveq2 6499 . . . . . . . . . 10 ( = 𝐾 → (pmap‘) = (pmap‘𝐾))
8 polfval.m . . . . . . . . . 10 𝑀 = (pmap‘𝐾)
97, 8syl6eqr 2832 . . . . . . . . 9 ( = 𝐾 → (pmap‘) = 𝑀)
10 fveq2 6499 . . . . . . . . . . 11 ( = 𝐾 → (oc‘) = (oc‘𝐾))
11 polfval.o . . . . . . . . . . 11 = (oc‘𝐾)
1210, 11syl6eqr 2832 . . . . . . . . . 10 ( = 𝐾 → (oc‘) = )
1312fveq1d 6501 . . . . . . . . 9 ( = 𝐾 → ((oc‘)‘𝑝) = ( 𝑝))
149, 13fveq12d 6506 . . . . . . . 8 ( = 𝐾 → ((pmap‘)‘((oc‘)‘𝑝)) = (𝑀‘( 𝑝)))
1514adantr 473 . . . . . . 7 (( = 𝐾𝑝𝑚) → ((pmap‘)‘((oc‘)‘𝑝)) = (𝑀‘( 𝑝)))
1615iineq2dv 4816 . . . . . 6 ( = 𝐾 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)) = 𝑝𝑚 (𝑀‘( 𝑝)))
175, 16ineq12d 4077 . . . . 5 ( = 𝐾 → ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝))) = (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))
186, 17mpteq12dv 5012 . . . 4 ( = 𝐾 → (𝑚 ∈ 𝒫 (Atoms‘) ↦ ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
19 df-polarityN 36490 . . . 4 𝑃 = ( ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘) ↦ ((Atoms‘) ∩ 𝑝𝑚 ((pmap‘)‘((oc‘)‘𝑝)))))
204fvexi 6513 . . . . . 6 𝐴 ∈ V
2120pwex 5134 . . . . 5 𝒫 𝐴 ∈ V
2221mptex 6812 . . . 4 (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))) ∈ V
2318, 19, 22fvmpt 6595 . . 3 (𝐾 ∈ V → (⊥𝑃𝐾) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
242, 23syl5eq 2826 . 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 1507  wcel 2050  Vcvv 3415  cin 3828  𝒫 cpw 4422   ciin 4793  cmpt 5008  cfv 6188  occoc 16429  Atomscatm 35850  pmapcpmap 36084  𝑃cpolN 36489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-polarityN 36490
This theorem is referenced by:  polvalN  36492
  Copyright terms: Public domain W3C validator