Theorem docafvalN 41321

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)
docaval.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
docaval.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
docaval.n	⊢ 𝑁 = ((ocA‘𝐾)‘𝑊)

Assertion

Ref	Expression
docafvalN	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))))

Distinct variable groups:   𝑥,𝑧,𝐾   𝑥,𝐼,𝑧   𝑥,𝑇   𝑥,𝑊,𝑧

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

Proof of Theorem docafvalN

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		docaval.n	. . 3 ⊢ 𝑁 = ((ocA‘𝐾)‘𝑊)
2		docaval.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
3		docaval.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
4		docaval.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
5		docaval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
6	2, 3, 4, 5	docaffvalN 41320	. . . 4 ⊢ (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)))))
7	6	fveq1d 6834	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((ocA‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤))))‘𝑊))
8	1, 7	eqtrid 2781	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝑁 = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤))))‘𝑊))
9		fveq2 6832	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
10		docaval.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11	9, 10	eqtr4di 2787	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
12	11	pweqd 4569	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 ((LTrn‘𝐾)‘𝑤) = 𝒫 𝑇)
13		fveq2 6832	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = ((DIsoA‘𝐾)‘𝑊))
14		docaval.i	. . . . . 6 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
15	13, 14	eqtr4di 2787	. . . . 5 ⊢ (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = 𝐼)
16	15	cnveqd 5822	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ◡((DIsoA‘𝐾)‘𝑤) = ◡𝐼)
17	15	rneqd 5885	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ran ((DIsoA‘𝐾)‘𝑤) = ran 𝐼)
18	17	rabeqdv 3412	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧} = {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})
19	18	inteqd 4905	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧} = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})
20	16, 19	fveq12d 6839	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧}) = (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧}))
21	20	fveq2d 6836	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) = ( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})))
22		fveq2 6832	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ⊥ ‘𝑤) = ( ⊥ ‘𝑊))
23	21, 22	oveq12d 7374	. . . . . 6 ⊢ (𝑤 = 𝑊 → (( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) = (( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)))
24		id 22	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
25	23, 24	oveq12d 7374	. . . . 5 ⊢ (𝑤 = 𝑊 → ((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤) = ((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))
26	15, 25	fveq12d 6839	. . . 4 ⊢ (𝑤 = 𝑊 → (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤)) = (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))
27	12, 26	mpteq12dv 5183	. . 3 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤))) = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))))
28		eqid 2734	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤)))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤))))
29	10	fvexi 6846	. . . . 5 ⊢ 𝑇 ∈ V
30	29	pwex 5323	. . . 4 ⊢ 𝒫 𝑇 ∈ V
31	30	mptex 7167	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))) ∈ V
32	27, 28, 31	fvmpt 6939	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤))))‘𝑊) = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))))
33	8, 32	sylan9eq 2789	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3397   ⊆ wss 3899  𝒫 cpw 4552  ∩ cint 4900   ↦ cmpt 5177  ^◡ccnv 5621  ran crn 5623  ‘cfv 6490  (class class class)co 7356  occoc 17183  joincjn 18232  meetcmee 18233  LHypclh 40183  LTrncltrn 40300  DIsoAcdia 41227  ocAcocaN 41318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  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-ov 7359  df-docaN 41319

This theorem is referenced by:  docavalN  41322