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Theorem polvalN 36585
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 6555 . . 3 𝐴 ∈ V
32elpw2 5142 . 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 36584 . . . 4 (𝐾𝐵𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
87fveq1d 6543 . . 3 (𝐾𝐵 → (𝑃𝑋) = ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))‘𝑋))
9 iineq1 4843 . . . . 5 (𝑚 = 𝑋 𝑝𝑚 (𝑀‘( 𝑝)) = 𝑝𝑋 (𝑀‘( 𝑝)))
109ineq2d 4111 . . . 4 (𝑚 = 𝑋 → (𝐴 𝑝𝑚 (𝑀‘( 𝑝))) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
11 eqid 2794 . . . 4 (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))
122inex1 5115 . . . 4 (𝐴 𝑝𝑋 (𝑀‘( 𝑝))) ∈ V
1310, 11, 12fvmpt 6638 . . 3 (𝑋 ∈ 𝒫 𝐴 → ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝))))‘𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
148, 13sylan9eq 2850 . 2 ((𝐾𝐵𝑋 ∈ 𝒫 𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
153, 14sylan2br 594 1 ((𝐾𝐵𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2080  cin 3860  wss 3861  𝒫 cpw 4455   ciin 4828  cmpt 5043  cfv 6228  occoc 16402  Atomscatm 35943  pmapcpmap 36177  𝑃cpolN 36582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-reu 3111  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-iun 4829  df-iin 4830  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-polarityN 36583
This theorem is referenced by:  polval2N  36586  pol0N  36589  polcon3N  36597  polatN  36611
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