Theorem docafvalN 37190

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 37189	. . . 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 6435	. . 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	syl5eq 2873	. 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 6433	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
10		docaval.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11	9, 10	syl6eqr 2879	. . . . 5 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
12	11	pweqd 4383	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 ((LTrn‘𝐾)‘𝑤) = 𝒫 𝑇)
13		fveq2 6433	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = ((DIsoA‘𝐾)‘𝑊))
14		docaval.i	. . . . . 6 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
15	13, 14	syl6eqr 2879	. . . . 5 ⊢ (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = 𝐼)
16	15	cnveqd 5530	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ◡((DIsoA‘𝐾)‘𝑤) = ◡𝐼)
17	15	rneqd 5585	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ran ((DIsoA‘𝐾)‘𝑤) = ran 𝐼)
18		rabeq 3405	. . . . . . . . . . 11 ⊢ (ran ((DIsoA‘𝐾)‘𝑤) = ran 𝐼 → {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧} = {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧} = {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})
20	19	inteqd 4702	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧} = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})
21	16, 20	fveq12d 6440	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧}) = (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧}))
22	21	fveq2d 6437	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) = ( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})))
23		fveq2 6433	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ⊥ ‘𝑤) = ( ⊥ ‘𝑊))
24	22, 23	oveq12d 6923	. . . . . 6 ⊢ (𝑤 = 𝑊 → (( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) = (( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)))
25		id 22	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
26	24, 25	oveq12d 6923	. . . . 5 ⊢ (𝑤 = 𝑊 → ((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤) = ((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))
27	15, 26	fveq12d 6440	. . . 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)))
28	12, 27	mpteq12dv 4956	. . 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))))
29		eqid 2825	. . 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))))
30	10	fvexi 6447	. . . . 5 ⊢ 𝑇 ∈ V
31	30	pwex 5080	. . . 4 ⊢ 𝒫 𝑇 ∈ V
32	31	mptex 6742	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))) ∈ V
33	28, 29, 32	fvmpt 6529	. 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))))
34	8, 33	sylan9eq 2881	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164  {crab 3121   ⊆ wss 3798  𝒫 cpw 4378  ∩ cint 4697   ↦ cmpt 4952  ^◡ccnv 5341  ran crn 5343  ‘cfv 6123  (class class class)co 6905  occoc 16313  joincjn 17297  meetcmee 17298  LHypclh 36052  LTrncltrn 36169  DIsoAcdia 37096  ocAcocaN 37187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-docaN 37188

This theorem is referenced by:  docavalN  37191