Theorem polfvalN 38775

Description: 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
polfvalN	⊢ (𝐾 ∈ 𝐵 → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))))

Distinct variable groups:   𝐴,𝑚   𝑚,𝑝,𝐾

Allowed substitution hints:   𝐴(𝑝)   𝐵(𝑚,𝑝)   𝑃(𝑚,𝑝)   𝑀(𝑚,𝑝)   ⊥ (𝑚,𝑝)

Proof of Theorem polfvalN

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3493	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		polfval.p	. . 3 ⊢ 𝑃 = (⊥𝑃‘𝐾)
3		fveq2 6892	. . . . . . 7 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = (Atoms‘𝐾))
4		polfval.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2791	. . . . . 6 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = 𝐴)
6	5	pweqd 4620	. . . . 5 ⊢ (ℎ = 𝐾 → 𝒫 (Atoms‘ℎ) = 𝒫 𝐴)
7		fveq2 6892	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = (pmap‘𝐾))
8		polfval.m	. . . . . . . . . 10 ⊢ 𝑀 = (pmap‘𝐾)
9	7, 8	eqtr4di 2791	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = 𝑀)
10		fveq2 6892	. . . . . . . . . . 11 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = (oc‘𝐾))
11		polfval.o	. . . . . . . . . . 11 ⊢ ⊥ = (oc‘𝐾)
12	10, 11	eqtr4di 2791	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = ⊥ )
13	12	fveq1d 6894	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → ((oc‘ℎ)‘𝑝) = ( ⊥ ‘𝑝))
14	9, 13	fveq12d 6899	. . . . . . . 8 ⊢ (ℎ = 𝐾 → ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = (𝑀‘( ⊥ ‘𝑝)))
15	14	adantr 482	. . . . . . 7 ⊢ ((ℎ = 𝐾 ∧ 𝑝 ∈ 𝑚) → ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = (𝑀‘( ⊥ ‘𝑝)))
16	15	iineq2dv 5023	. . . . . 6 ⊢ (ℎ = 𝐾 → ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))
17	5, 16	ineq12d 4214	. . . . 5 ⊢ (ℎ = 𝐾 → ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))
18	6, 17	mpteq12dv 5240	. . . 4 ⊢ (ℎ = 𝐾 → (𝑚 ∈ 𝒫 (Atoms‘ℎ) ↦ ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))))
19		df-polarityN 38774	. . . 4 ⊢ ⊥𝑃 = (ℎ ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘ℎ) ↦ ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)))))
20	4	fvexi 6906	. . . . . 6 ⊢ 𝐴 ∈ V
21	20	pwex 5379	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
22	21	mptex 7225	. . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) ∈ V
23	18, 19, 22	fvmpt 6999	. . 3 ⊢ (𝐾 ∈ V → (⊥𝑃‘𝐾) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))))
24	2, 23	eqtrid 2785	. 2 ⊢ (𝐾 ∈ V → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))))
25	1, 24	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948  𝒫 cpw 4603  ∩ ciin 4999   ↦ cmpt 5232  ‘cfv 6544  occoc 17205  Atomscatm 38133  pmapcpmap 38368  ⊥_𝑃cpolN 38773

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-polarityN 38774

This theorem is referenced by:  polvalN  38776