Theorem docavalN 40297

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
docavalN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝑁‘𝑋) = (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))

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

Allowed substitution hints:   𝐻(𝑧)   ∨ (𝑧)   ∧ (𝑧)   𝑁(𝑧)   ⊥ (𝑧)

Proof of Theorem docavalN

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		docaval.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
2		docaval.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
3		docaval.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
4		docaval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		docaval.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6		docaval.i	. . . . 5 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
7		docaval.n	. . . . 5 ⊢ 𝑁 = ((ocA‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	docafvalN 40296	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))))
9	8	adantr 479	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))))
10	9	fveq1d 6892	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝑁‘𝑋) = ((𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))‘𝑋))
11	5	fvexi 6904	. . . . . 6 ⊢ 𝑇 ∈ V
12	11	elpw2 5344	. . . . 5 ⊢ (𝑋 ∈ 𝒫 𝑇 ↔ 𝑋 ⊆ 𝑇)
13	12	biimpri 227	. . . 4 ⊢ (𝑋 ⊆ 𝑇 → 𝑋 ∈ 𝒫 𝑇)
14	13	adantl 480	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝑋 ∈ 𝒫 𝑇)
15		fvex 6903	. . 3 ⊢ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)) ∈ V
16		sseq1 4006	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑧))
17	16	rabbidv 3438	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧} = {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})
18	17	inteqd 4954	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧} = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})
19	18	fveq2d 6894	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧}) = (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))
20	19	fveq2d 6894	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) = ( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))
21	20	oveq1d 7426	. . . . 5 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) = (( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)))
22	21	fvoveq1d 7433	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)) = (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))
23		eqid 2730	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))) = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))
24	22, 23	fvmptg 6995	. . 3 ⊢ ((𝑋 ∈ 𝒫 𝑇 ∧ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)) ∈ V) → ((𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))‘𝑋) = (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))
25	14, 15, 24	sylancl 584	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ((𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))‘𝑋) = (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))
26	10, 25	eqtrd 2770	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝑁‘𝑋) = (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {crab 3430  Vcvv 3472   ⊆ wss 3947  𝒫 cpw 4601  ∩ cint 4949   ↦ cmpt 5230  ^◡ccnv 5674  ran crn 5676  ‘cfv 6542  (class class class)co 7411  occoc 17209  joincjn 18268  meetcmee 18269  HLchlt 38523  LHypclh 39158  LTrncltrn 39275  DIsoAcdia 40202  ocAcocaN 40293

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-docaN 40294

This theorem is referenced by:  docaclN  40298  diaocN  40299