Theorem polvalN 40342

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 6846	. . 3 ⊢ 𝐴 ∈ V
3	2	elpw2 5269	. 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 40341	. . . 4 ⊢ (𝐾 ∈ 𝐵 → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))))
8	7	fveq1d 6834	. . 3 ⊢ (𝐾 ∈ 𝐵 → (𝑃‘𝑋) = ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))‘𝑋))
9		iineq1 4952	. . . . 5 ⊢ (𝑚 = 𝑋 → ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)) = ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))
10	9	ineq2d 4161	. . . 4 ⊢ (𝑚 = 𝑋 → (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
11		eqid 2737	. . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))
12	2	inex1 5252	. . . 4 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))) ∈ V
13	10, 11, 12	fvmpt 6939	. . 3 ⊢ (𝑋 ∈ 𝒫 𝐴 → ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
14	8, 13	sylan9eq 2792	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))
15	3, 14	sylan2br 596	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ∩ ciin 4935   ↦ cmpt 5167  ‘cfv 6490  occoc 17186  Atomscatm 39700  pmapcpmap 39934  ⊥_𝑃cpolN 40339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-polarityN 40340

This theorem is referenced by:  polval2N  40343  pol0N  40346  polcon3N  40354  polatN  40368