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Theorem polvalN 35712
Description: Value of 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
polvalN ((𝐾𝐵𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
Distinct variable groups:   𝐾,𝑝   𝑋,𝑝
Allowed substitution hints:   𝐴(𝑝)   𝐵(𝑝)   𝑃(𝑝)   𝑀(𝑝)   (𝑝)

Proof of Theorem polvalN
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 polfval.a . . . 4 𝐴 = (Atoms‘𝐾)
2 fvex 6363 . . . 4 (Atoms‘𝐾) ∈ V
31, 2eqeltri 2835 . . 3 𝐴 ∈ V
43elpw2 4977 . 2 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
5 polfval.o . . . . 5 = (oc‘𝐾)
6 polfval.m . . . . 5 𝑀 = (pmap‘𝐾)
7 polfval.p . . . . 5 𝑃 = (⊥𝑃𝐾)
85, 1, 6, 7polfvalN 35711 . . . 4 (𝐾𝐵𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
98fveq1d 6355 . . 3 (𝐾𝐵 → (𝑃𝑋) = ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))‘𝑋))
10 iineq1 4687 . . . . 5 (𝑚 = 𝑋 𝑝𝑚 (𝑀‘( 𝑝)) = 𝑝𝑋 (𝑀‘( 𝑝)))
1110ineq2d 3957 . . . 4 (𝑚 = 𝑋 → (𝐴 𝑝𝑚 (𝑀‘( 𝑝))) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
12 eqid 2760 . . . 4 (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))
133inex1 4951 . . . 4 (𝐴 𝑝𝑋 (𝑀‘( 𝑝))) ∈ V
1411, 12, 13fvmpt 6445 . . 3 (𝑋 ∈ 𝒫 𝐴 → ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))‘𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
159, 14sylan9eq 2814 . 2 ((𝐾𝐵𝑋 ∈ 𝒫 𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
164, 15sylan2br 494 1 ((𝐾𝐵𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  wss 3715  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:  polval2N  35713  pol0N  35716  polcon3N  35724  polatN  35738
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