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Theorem djhval 41097
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 41096 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
76adantr 479 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
87oveqd 7441 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋 𝑌) = (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌))
93fvexi 6915 . . . . . 6 𝑉 ∈ V
109elpw2 5352 . . . . 5 (𝑋 ∈ 𝒫 𝑉𝑋𝑉)
1110biimpri 227 . . . 4 (𝑋𝑉𝑋 ∈ 𝒫 𝑉)
1211ad2antrl 726 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → 𝑋 ∈ 𝒫 𝑉)
139elpw2 5352 . . . . 5 (𝑌 ∈ 𝒫 𝑉𝑌𝑉)
1413biimpri 227 . . . 4 (𝑌𝑉𝑌 ∈ 𝒫 𝑉)
1514ad2antll 727 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → 𝑌 ∈ 𝒫 𝑉)
16 fvexd 6916 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V)
17 fveq2 6901 . . . . . 6 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1817ineq1d 4212 . . . . 5 (𝑥 = 𝑋 → (( 𝑥) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑦)))
1918fveq2d 6905 . . . 4 (𝑥 = 𝑋 → ( ‘(( 𝑥) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑦))))
20 fveq2 6901 . . . . . 6 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2120ineq2d 4213 . . . . 5 (𝑦 = 𝑌 → (( 𝑋) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑌)))
2221fveq2d 6905 . . . 4 (𝑦 = 𝑌 → ( ‘(( 𝑋) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑌))))
23 eqid 2726 . . . 4 (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))
2419, 22, 23ovmpog 7585 . . 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 1368 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
268, 25eqtrd 2766 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋 𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  cin 3946  wss 3947  𝒫 cpw 4607  cfv 6554  (class class class)co 7424  cmpo 7426  Basecbs 17213  HLchlt 39048  LHypclh 39683  DVecHcdvh 40777  ocHcoch 41046  joinHcdjh 41093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-djh 41094
This theorem is referenced by:  djhval2  41098  djhcl  41099  djhlj  41100  djhexmid  41110
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