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Theorem djhval 39149
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 39148 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
76adantr 484 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
87oveqd 7230 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋 𝑌) = (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌))
93fvexi 6731 . . . . . 6 𝑉 ∈ V
109elpw2 5238 . . . . 5 (𝑋 ∈ 𝒫 𝑉𝑋𝑉)
1110biimpri 231 . . . 4 (𝑋𝑉𝑋 ∈ 𝒫 𝑉)
1211ad2antrl 728 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → 𝑋 ∈ 𝒫 𝑉)
139elpw2 5238 . . . . 5 (𝑌 ∈ 𝒫 𝑉𝑌𝑉)
1413biimpri 231 . . . 4 (𝑌𝑉𝑌 ∈ 𝒫 𝑉)
1514ad2antll 729 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → 𝑌 ∈ 𝒫 𝑉)
16 fvexd 6732 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V)
17 fveq2 6717 . . . . . 6 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1817ineq1d 4126 . . . . 5 (𝑥 = 𝑋 → (( 𝑥) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑦)))
1918fveq2d 6721 . . . 4 (𝑥 = 𝑋 → ( ‘(( 𝑥) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑦))))
20 fveq2 6717 . . . . . 6 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2120ineq2d 4127 . . . . 5 (𝑦 = 𝑌 → (( 𝑋) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑌)))
2221fveq2d 6721 . . . 4 (𝑦 = 𝑌 → ( ‘(( 𝑋) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑌))))
23 eqid 2737 . . . 4 (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))
2419, 22, 23ovmpog 7368 . . 3 ((𝑋 ∈ 𝒫 𝑉𝑌 ∈ 𝒫 𝑉 ∧ ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V) → (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
2512, 15, 16, 24syl3anc 1373 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
268, 25eqtrd 2777 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋 𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  Vcvv 3408  cin 3865  wss 3866  𝒫 cpw 4513  cfv 6380  (class class class)co 7213  cmpo 7215  Basecbs 16760  HLchlt 37101  LHypclh 37735  DVecHcdvh 38829  ocHcoch 39098  joinHcdjh 39145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-djh 39146
This theorem is referenced by:  djhval2  39150  djhcl  39151  djhlj  39152  djhexmid  39162
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