Theorem docaffvalN 41895

Description: Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
docaval.j	⊢ ∨ = (join‘𝐾)
docaval.m	⊢ ∧ = (meet‘𝐾)
docaval.o	⊢ ⊥ = (oc‘𝐾)
docaval.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
docaffvalN	⊢ (𝐾 ∈ 𝑉 → (ocA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤)))))

Distinct variable groups:   𝑤,𝐻   𝑥,𝑤,𝑧,𝐾

Allowed substitution hints:   𝐻(𝑥,𝑧)   ∨ (𝑥,𝑧,𝑤)   ∧ (𝑥,𝑧,𝑤)   ⊥ (𝑥,𝑧,𝑤)   𝑉(𝑥,𝑧,𝑤)

Proof of Theorem docaffvalN

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3476	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6881	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		docaval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2816	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
6	5	fveq1d 6883	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7	6	pweqd 4579	. . . . 5 ⊢ (𝑘 = 𝐾 → 𝒫 ((LTrn‘𝑘)‘𝑤) = 𝒫 ((LTrn‘𝐾)‘𝑤))
8		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (DIsoA‘𝑘) = (DIsoA‘𝐾))
9	8	fveq1d 6883	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((DIsoA‘𝑘)‘𝑤) = ((DIsoA‘𝐾)‘𝑤))
10		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
11		docaval.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
12	10, 11	eqtr4di 2816	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
13		fveq2 6881	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14		docaval.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
15	13, 14	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
16		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17		docaval.o	. . . . . . . . . 10 ⊢ ⊥ = (oc‘𝐾)
18	16, 17	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = ⊥ )
19	9	cnveqd 5861	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ◡((DIsoA‘𝑘)‘𝑤) = ◡((DIsoA‘𝐾)‘𝑤))
20	9	rneqd 5928	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → ran ((DIsoA‘𝑘)‘𝑤) = ran ((DIsoA‘𝐾)‘𝑤))
21	20	rabeqdv 3431	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧} = {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})
22	21	inteqd 4917	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧} = ∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})
23	19, 22	fveq12d 6888	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}) = (◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))
24	18, 23	fveq12d 6888	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) = ( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})))
25	18	fveq1d 6883	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑤) = ( ⊥ ‘𝑤))
26	15, 24, 25	oveq123d 7431	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤)) = (( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)))
27		eqidd 2764	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑤 = 𝑤)
28	12, 26, 27	oveq123d 7431	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤) = ((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤))
29	9, 28	fveq12d 6888	. . . . 5 ⊢ (𝑘 = 𝐾 → (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)) = (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤)))
30	7, 29	mpteq12dv 5198	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤))) = (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤))))
31	4, 30	mpteq12dv 5198	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤)))))
32		df-docaN 41894	. . 3 ⊢ ocA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))
33	31, 32, 3	mptfvmpt 7226	. 2 ⊢ (𝐾 ∈ V → (ocA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤)))))
34	1, 33	syl 18	1 ⊢ (𝐾 ∈ 𝑉 → (ocA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤)))))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562  ∩ cint 4912   ↦ cmpt 5192  ^◡ccnv 5660  ran crn 5662  ‘cfv 6536  (class class class)co 7410  occoc 17313  joincjn 18362  meetcmee 18363  LHypclh 40758  LTrncltrn 40875  DIsoAcdia 41802  ocAcocaN 41893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-docaN 41894

This theorem is referenced by:  docafvalN  41896