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Theorem polvalN 34005
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 6098 . . . 4 (Atoms‘𝐾) ∈ V
31, 2eqeltri 2683 . . 3 𝐴 ∈ V
43elpw2 4750 . 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 34004 . . . 4 (𝐾𝐵𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
98fveq1d 6090 . . 3 (𝐾𝐵 → (𝑃𝑋) = ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))‘𝑋))
10 iineq1 4465 . . . . 5 (𝑚 = 𝑋 𝑝𝑚 (𝑀‘( 𝑝)) = 𝑝𝑋 (𝑀‘( 𝑝)))
1110ineq2d 3775 . . . 4 (𝑚 = 𝑋 → (𝐴 𝑝𝑚 (𝑀‘( 𝑝))) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
12 eqid 2609 . . . 4 (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))
133inex1 4722 . . . 4 (𝐴 𝑝𝑋 (𝑀‘( 𝑝))) ∈ V
1411, 12, 13fvmpt 6176 . . 3 (𝑋 ∈ 𝒫 𝐴 → ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))‘𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
159, 14sylan9eq 2663 . 2 ((𝐾𝐵𝑋 ∈ 𝒫 𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
164, 15sylan2br 491 1 ((𝐾𝐵𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cin 3538  wss 3539  𝒫 cpw 4107   ciin 4450  cmpt 4637  cfv 5790  occoc 15722  Atomscatm 33364  pmapcpmap 33597  𝑃cpolN 34002
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 4712  ax-pow 4764  ax-pr 4828
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 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-polarityN 34003
This theorem is referenced by:  polval2N  34006  pol0N  34009  polcon3N  34017  polatN  34031
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