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Theorem djhval 37201
 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 37200 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
76adantr 466 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
87oveqd 6809 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋 𝑌) = (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌))
9 fvex 6342 . . . . . . 7 (Base‘𝑈) ∈ V
103, 9eqeltri 2845 . . . . . 6 𝑉 ∈ V
1110elpw2 4956 . . . . 5 (𝑋 ∈ 𝒫 𝑉𝑋𝑉)
1211biimpri 218 . . . 4 (𝑋𝑉𝑋 ∈ 𝒫 𝑉)
1312ad2antrl 699 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → 𝑋 ∈ 𝒫 𝑉)
1410elpw2 4956 . . . . 5 (𝑌 ∈ 𝒫 𝑉𝑌𝑉)
1514biimpri 218 . . . 4 (𝑌𝑉𝑌 ∈ 𝒫 𝑉)
1615ad2antll 700 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → 𝑌 ∈ 𝒫 𝑉)
17 fvexd 6344 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V)
18 fveq2 6332 . . . . . 6 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1918ineq1d 3962 . . . . 5 (𝑥 = 𝑋 → (( 𝑥) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑦)))
2019fveq2d 6336 . . . 4 (𝑥 = 𝑋 → ( ‘(( 𝑥) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑦))))
21 fveq2 6332 . . . . . 6 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2221ineq2d 3963 . . . . 5 (𝑦 = 𝑌 → (( 𝑋) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑌)))
2322fveq2d 6336 . . . 4 (𝑦 = 𝑌 → ( ‘(( 𝑋) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑌))))
24 eqid 2770 . . . 4 (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))
2520, 23, 24ovmpt2g 6941 . . 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 1475 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋(𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
278, 26eqtrd 2804 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋 𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144  Vcvv 3349   ∩ cin 3720   ⊆ wss 3721  𝒫 cpw 4295  ‘cfv 6031  (class class class)co 6792   ↦ cmpt2 6794  Basecbs 16063  HLchlt 35152  LHypclh 35785  DVecHcdvh 36881  ocHcoch 37150  joinHcdjh 37197 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-djh 37198 This theorem is referenced by:  djhval2  37202  djhcl  37203  djhlj  37204  djhexmid  37214
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