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Theorem djhval 40258
Description: Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)
Hypotheses
Ref Expression
djhval.h 𝐻 = (LHypβ€˜πΎ)
djhval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
djhval.v 𝑉 = (Baseβ€˜π‘ˆ)
djhval.o βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)
djhval.j ∨ = ((joinHβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
djhval (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉)) β†’ (𝑋 ∨ π‘Œ) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))))

Proof of Theorem djhval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 djhval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
2 djhval.u . . . . 5 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
3 djhval.v . . . . 5 𝑉 = (Baseβ€˜π‘ˆ)
4 djhval.o . . . . 5 βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)
5 djhval.j . . . . 5 ∨ = ((joinHβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5djhfval 40257 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ∨ = (π‘₯ ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))))
76adantr 482 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉)) β†’ ∨ = (π‘₯ ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))))
87oveqd 7423 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉)) β†’ (𝑋 ∨ π‘Œ) = (𝑋(π‘₯ ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦))))π‘Œ))
93fvexi 6903 . . . . . 6 𝑉 ∈ V
109elpw2 5345 . . . . 5 (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 βŠ† 𝑉)
1110biimpri 227 . . . 4 (𝑋 βŠ† 𝑉 β†’ 𝑋 ∈ 𝒫 𝑉)
1211ad2antrl 727 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉)) β†’ 𝑋 ∈ 𝒫 𝑉)
139elpw2 5345 . . . . 5 (π‘Œ ∈ 𝒫 𝑉 ↔ π‘Œ βŠ† 𝑉)
1413biimpri 227 . . . 4 (π‘Œ βŠ† 𝑉 β†’ π‘Œ ∈ 𝒫 𝑉)
1514ad2antll 728 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉)) β†’ π‘Œ ∈ 𝒫 𝑉)
16 fvexd 6904 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉)) β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))) ∈ V)
17 fveq2 6889 . . . . . 6 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜π‘₯) = ( βŠ₯ β€˜π‘‹))
1817ineq1d 4211 . . . . 5 (π‘₯ = 𝑋 β†’ (( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)) = (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘¦)))
1918fveq2d 6893 . . . 4 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦))) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘¦))))
20 fveq2 6889 . . . . . 6 (𝑦 = π‘Œ β†’ ( βŠ₯ β€˜π‘¦) = ( βŠ₯ β€˜π‘Œ))
2120ineq2d 4212 . . . . 5 (𝑦 = π‘Œ β†’ (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘¦)) = (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ)))
2221fveq2d 6893 . . . 4 (𝑦 = π‘Œ β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘¦))) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))))
23 eqid 2733 . . . 4 (π‘₯ ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))) = (π‘₯ ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦))))
2419, 22, 23ovmpog 7564 . . 3 ((𝑋 ∈ 𝒫 𝑉 ∧ π‘Œ ∈ 𝒫 𝑉 ∧ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))) ∈ V) β†’ (𝑋(π‘₯ ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦))))π‘Œ) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))))
2512, 15, 16, 24syl3anc 1372 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉)) β†’ (𝑋(π‘₯ ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦))))π‘Œ) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))))
268, 25eqtrd 2773 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑉 ∧ π‘Œ βŠ† 𝑉)) β†’ (𝑋 ∨ π‘Œ) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  β€˜cfv 6541  (class class class)co 7406   ∈ cmpo 7408  Basecbs 17141  HLchlt 38209  LHypclh 38844  DVecHcdvh 39938  ocHcoch 40207  joinHcdjh 40254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-djh 40255
This theorem is referenced by:  djhval2  40259  djhcl  40260  djhlj  40261  djhexmid  40271
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