Theorem docavalN 39586

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 39585	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))))
9	8	adantr 481	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))))
10	9	fveq1d 6844	. 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 6856	. . . . . 6 ⊢ 𝑇 ∈ V
12	11	elpw2 5302	. . . . 5 ⊢ (𝑋 ∈ 𝒫 𝑇 ↔ 𝑋 ⊆ 𝑇)
13	12	biimpri 227	. . . 4 ⊢ (𝑋 ⊆ 𝑇 → 𝑋 ∈ 𝒫 𝑇)
14	13	adantl 482	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → 𝑋 ∈ 𝒫 𝑇)
15		fvex 6855	. . 3 ⊢ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)) ∈ V
16		sseq1 3969	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑥 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑧))
17	16	rabbidv 3415	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧} = {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})
18	17	inteqd 4912	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧} = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})
19	18	fveq2d 6846	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧}) = (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))
20	19	fveq2d 6846	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) = ( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))
21	20	oveq1d 7372	. . . . 5 ⊢ (𝑥 = 𝑋 → (( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) = (( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)))
22	21	fvoveq1d 7379	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)) = (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))
23		eqid 2736	. . . 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 6946	. . 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 586	. 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 2776	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝑁‘𝑋) = (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3407  Vcvv 3445   ⊆ wss 3910  𝒫 cpw 4560  ∩ cint 4907   ↦ cmpt 5188  ^◡ccnv 5632  ran crn 5634  ‘cfv 6496  (class class class)co 7357  occoc 17141  joincjn 18200  meetcmee 18201  HLchlt 37812  LHypclh 38447  LTrncltrn 38564  DIsoAcdia 39491  ocAcocaN 39582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-docaN 39583

This theorem is referenced by:  docaclN  39587  diaocN  39588