Theorem polvalN 39864

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 = (⊥𝑃‘𝐾)

Assertion

Ref	Expression
polvalN	⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (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.

Step	Hyp	Ref	Expression
1		polfval.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2	1	fvexi 6936	. . 3 ⊢ 𝐴 ∈ V
3	2	elpw2 5352	. 2 ⊢ (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴)
4		polfval.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
5		polfval.m	. . . . 5 ⊢ 𝑀 = (pmap‘𝐾)
6		polfval.p	. . . . 5 ⊢ 𝑃 = (⊥𝑃‘𝐾)
7	4, 1, 5, 6	polfvalN 39863	. . . 4 ⊢ (𝐾 ∈ 𝐵 → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))))
8	7	fveq1d 6924	. . 3 ⊢ (𝐾 ∈ 𝐵 → (𝑃‘𝑋) = ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))‘𝑋))
9		iineq1 5032	. . . . 5 ⊢ (𝑚 = 𝑋 → ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)) = ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))
10	9	ineq2d 4241	. . . 4 ⊢ (𝑚 = 𝑋 → (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
11		eqid 2740	. . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))
12	2	inex1 5335	. . . 4 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))) ∈ V
13	10, 11, 12	fvmpt 7031	. . 3 ⊢ (𝑋 ∈ 𝒫 𝐴 → ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
14	8, 13	sylan9eq 2800	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
15	3, 14	sylan2br 594	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  ∩ ciin 5016   ↦ cmpt 5249  ‘cfv 6575  occoc 17321  Atomscatm 39221  pmapcpmap 39456  ⊥_𝑃cpolN 39861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-polarityN 39862

This theorem is referenced by:  polval2N  39865  pol0N  39868  polcon3N  39876  polatN  39890