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

Theorem polvalN 37482
 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)
21fvexi 6673 . . 3 𝐴 ∈ V
32elpw2 5216 . 2 (𝑋 ∈ 𝒫 𝐴𝑋𝐴)
4 polfval.o . . . . 5 = (oc‘𝐾)
5 polfval.m . . . . 5 𝑀 = (pmap‘𝐾)
6 polfval.p . . . . 5 𝑃 = (⊥𝑃𝐾)
74, 1, 5, 6polfvalN 37481 . . . 4 (𝐾𝐵𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
87fveq1d 6661 . . 3 (𝐾𝐵 → (𝑃𝑋) = ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))‘𝑋))
9 iineq1 4901 . . . . 5 (𝑚 = 𝑋 𝑝𝑚 (𝑀‘( 𝑝)) = 𝑝𝑋 (𝑀‘( 𝑝)))
109ineq2d 4118 . . . 4 (𝑚 = 𝑋 → (𝐴 𝑝𝑚 (𝑀‘( 𝑝))) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
11 eqid 2759 . . . 4 (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))
122inex1 5188 . . . 4 (𝐴 𝑝𝑋 (𝑀‘( 𝑝))) ∈ V
1310, 11, 12fvmpt 6760 . . 3 (𝑋 ∈ 𝒫 𝐴 → ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))‘𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
148, 13sylan9eq 2814 . 2 ((𝐾𝐵𝑋 ∈ 𝒫 𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
153, 14sylan2br 598 1 ((𝐾𝐵𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4495  ∩ ciin 4885   ↦ cmpt 5113  ‘cfv 6336  occoc 16632  Atomscatm 36840  pmapcpmap 37074  ⊥𝑃cpolN 37479 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-polarityN 37480 This theorem is referenced by:  polval2N  37483  pol0N  37486  polcon3N  37494  polatN  37508
 Copyright terms: Public domain W3C validator