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 = (⊥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		fvex 6363	. . . 4 ⊢ (Atoms‘𝐾) ∈ V
3	1, 2	eqeltri 2835	. . 3 ⊢ 𝐴 ∈ V
4	3	elpw2 4977	. 2 ⊢ (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴)
5		polfval.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
6		polfval.m	. . . . 5 ⊢ 𝑀 = (pmap‘𝐾)
7		polfval.p	. . . . 5 ⊢ 𝑃 = (⊥𝑃‘𝐾)
8	5, 1, 6, 7	polfvalN 35711	. . . 4 ⊢ (𝐾 ∈ 𝐵 → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))))
9	8	fveq1d 6355	. . 3 ⊢ (𝐾 ∈ 𝐵 → (𝑃‘𝑋) = ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))‘𝑋))
10		iineq1 4687	. . . . 5 ⊢ (𝑚 = 𝑋 → ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)) = ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))
11	10	ineq2d 3957	. . . 4 ⊢ (𝑚 = 𝑋 → (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
12		eqid 2760	. . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))
13	3	inex1 4951	. . . 4 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))) ∈ V
14	11, 12, 13	fvmpt 6445	. . 3 ⊢ (𝑋 ∈ 𝒫 𝐴 → ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
15	9, 14	sylan9eq 2814	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
16	4, 15	sylan2br 494	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))

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