Theorem polfvalN 39883

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 3457	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		polfval.p	. . 3 ⊢ 𝑃 = (⊥𝑃‘𝐾)
3		fveq2 6822	. . . . . . 7 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = (Atoms‘𝐾))
4		polfval.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2782	. . . . . 6 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = 𝐴)
6	5	pweqd 4568	. . . . 5 ⊢ (ℎ = 𝐾 → 𝒫 (Atoms‘ℎ) = 𝒫 𝐴)
7		fveq2 6822	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = (pmap‘𝐾))
8		polfval.m	. . . . . . . . . 10 ⊢ 𝑀 = (pmap‘𝐾)
9	7, 8	eqtr4di 2782	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = 𝑀)
10		fveq2 6822	. . . . . . . . . . 11 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = (oc‘𝐾))
11		polfval.o	. . . . . . . . . . 11 ⊢ ⊥ = (oc‘𝐾)
12	10, 11	eqtr4di 2782	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = ⊥ )
13	12	fveq1d 6824	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → ((oc‘ℎ)‘𝑝) = ( ⊥ ‘𝑝))
14	9, 13	fveq12d 6829	. . . . . . . 8 ⊢ (ℎ = 𝐾 → ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = (𝑀‘( ⊥ ‘𝑝)))
15	14	adantr 480	. . . . . . 7 ⊢ ((ℎ = 𝐾 ∧ 𝑝 ∈ 𝑚) → ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = (𝑀‘( ⊥ ‘𝑝)))
16	15	iineq2dv 4967	. . . . . 6 ⊢ (ℎ = 𝐾 → ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))
17	5, 16	ineq12d 4172	. . . . 5 ⊢ (ℎ = 𝐾 → ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))
18	6, 17	mpteq12dv 5179	. . . 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 39882	. . . 4 ⊢ ⊥𝑃 = (ℎ ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘ℎ) ↦ ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)))))
20	4	fvexi 6836	. . . . . 6 ⊢ 𝐴 ∈ V
21	20	pwex 5319	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
22	21	mptex 7159	. . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) ∈ V
23	18, 19, 22	fvmpt 6930	. . 3 ⊢ (𝐾 ∈ V → (⊥𝑃‘𝐾) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))))
24	2, 23	eqtrid 2776	. 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 1540   ∈ wcel 2109  Vcvv 3436   ∩ cin 3902  𝒫 cpw 4551  ∩ ciin 4942   ↦ cmpt 5173  ‘cfv 6482  occoc 17169  Atomscatm 39242  pmapcpmap 39476  ⊥_𝑃cpolN 39881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-polarityN 39882

This theorem is referenced by:  polvalN  39884