Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  djhval Structured version   Visualization version   GIF version

Theorem djhval 37004
Description: Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)
Hypotheses
Ref Expression
djhval.h 𝐻 = (LHyp‘𝐾)
djhval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
djhval.v 𝑉 = (Base‘𝑈)
djhval.o = ((ocH‘𝐾)‘𝑊)
djhval.j = ((joinH‘𝐾)‘𝑊)
Assertion
Ref Expression
djhval (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋 𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))

Proof of Theorem djhval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 djhval.h . . . . 5 𝐻 = (LHyp‘𝐾)
2 djhval.u . . . . 5 𝑈 = ((DVecH‘𝐾)‘𝑊)
3 djhval.v . . . . 5 𝑉 = (Base‘𝑈)
4 djhval.o . . . . 5 = ((ocH‘𝐾)‘𝑊)
5 djhval.j . . . . 5 = ((joinH‘𝐾)‘𝑊)
61, 2, 3, 4, 5djhfval 37003 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
76adantr 480 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
87oveqd 6707 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋 𝑌) = (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌))
9 fvex 6239 . . . . . . 7 (Base‘𝑈) ∈ V
103, 9eqeltri 2726 . . . . . 6 𝑉 ∈ V
1110elpw2 4858 . . . . 5 (𝑋 ∈ 𝒫 𝑉𝑋𝑉)
1211biimpri 218 . . . 4 (𝑋𝑉𝑋 ∈ 𝒫 𝑉)
1312ad2antrl 764 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → 𝑋 ∈ 𝒫 𝑉)
1410elpw2 4858 . . . . 5 (𝑌 ∈ 𝒫 𝑉𝑌𝑉)
1514biimpri 218 . . . 4 (𝑌𝑉𝑌 ∈ 𝒫 𝑉)
1615ad2antll 765 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → 𝑌 ∈ 𝒫 𝑉)
17 fvexd 6241 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V)
18 fveq2 6229 . . . . . 6 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1918ineq1d 3846 . . . . 5 (𝑥 = 𝑋 → (( 𝑥) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑦)))
2019fveq2d 6233 . . . 4 (𝑥 = 𝑋 → ( ‘(( 𝑥) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑦))))
21 fveq2 6229 . . . . . 6 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2221ineq2d 3847 . . . . 5 (𝑦 = 𝑌 → (( 𝑋) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑌)))
2322fveq2d 6233 . . . 4 (𝑦 = 𝑌 → ( ‘(( 𝑋) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑌))))
24 eqid 2651 . . . 4 (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))
2520, 23, 24ovmpt2g 6837 . . 3 ((𝑋 ∈ 𝒫 𝑉𝑌 ∈ 𝒫 𝑉 ∧ ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V) → (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
2613, 16, 17, 25syl3anc 1366 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
278, 26eqtrd 2685 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋 𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191  cfv 5926  (class class class)co 6690  cmpt2 6692  Basecbs 15904  HLchlt 34955  LHypclh 35588  DVecHcdvh 36684  ocHcoch 36953  joinHcdjh 37000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-djh 37001
This theorem is referenced by:  djhval2  37005  djhcl  37006  djhlj  37007  djhexmid  37017
  Copyright terms: Public domain W3C validator