Theorem docafvalN 41996

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 41995	. . . 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 6881	. . 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 2807	. 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 6879	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
10		docaval.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11	9, 10	eqtr4di 2813	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
12	11	pweqd 4574	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 ((LTrn‘𝐾)‘𝑤) = 𝒫 𝑇)
13		fveq2 6879	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = ((DIsoA‘𝐾)‘𝑊))
14		docaval.i	. . . . . 6 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
15	13, 14	eqtr4di 2813	. . . . 5 ⊢ (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = 𝐼)
16	15	cnveqd 5855	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ◡((DIsoA‘𝐾)‘𝑤) = ◡𝐼)
17	15	rneqd 5922	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ran ((DIsoA‘𝐾)‘𝑤) = ran 𝐼)
18	17	rabeqdv 3427	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧} = {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})
19	18	inteqd 4912	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧} = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})
20	16, 19	fveq12d 6886	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧}) = (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧}))
21	20	fveq2d 6883	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) = ( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})))
22		fveq2 6879	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ⊥ ‘𝑤) = ( ⊥ ‘𝑊))
23	21, 22	oveq12d 7432	. . . . . 6 ⊢ (𝑤 = 𝑊 → (( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) = (( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)))
24		id 23	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
25	23, 24	oveq12d 7432	. . . . 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 6886	. . . 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 5192	. . 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 2760	. . 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 6893	. . . . 5 ⊢ 𝑇 ∈ V
30	29	pwex 5345	. . . 4 ⊢ 𝒫 𝑇 ∈ V
31	30	mptex 7223	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))) ∈ V
32	27, 28, 31	fvmpt 6987	. 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 2815	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412   ⊆ wss 3899  𝒫 cpw 4557  ∩ cint 4907   ↦ cmpt 5186  ^◡ccnv 5654  ran crn 5656  ‘cfv 6533  (class class class)co 7414  occoc 17351  joincjn 18400  meetcmee 18401  LHypclh 40858  LTrncltrn 40975  DIsoAcdia 41902  ocAcocaN 41993

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-docaN 41994

This theorem is used by:  docavalN  41997