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Theorem djavalN 35442
 Description: Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
djaval.h 𝐻 = (LHyp‘𝐾)
djaval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
djaval.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
djaval.n = ((ocA‘𝐾)‘𝑊)
djaval.j 𝐽 = ((vA‘𝐾)‘𝑊)
Assertion
Ref Expression
djavalN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋𝐽𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))

Proof of Theorem djavalN
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 djaval.h . . . . 5 𝐻 = (LHyp‘𝐾)
2 djaval.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
3 djaval.i . . . . 5 𝐼 = ((DIsoA‘𝐾)‘𝑊)
4 djaval.n . . . . 5 = ((ocA‘𝐾)‘𝑊)
5 djaval.j . . . . 5 𝐽 = ((vA‘𝐾)‘𝑊)
61, 2, 3, 4, 5djafvalN 35441 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
76adantr 480 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
87oveqd 6566 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋𝐽𝑌) = (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌))
9 fvex 6113 . . . . . . 7 ((LTrn‘𝐾)‘𝑊) ∈ V
102, 9eqeltri 2684 . . . . . 6 𝑇 ∈ V
1110elpw2 4755 . . . . 5 (𝑋 ∈ 𝒫 𝑇𝑋𝑇)
1211biimpri 217 . . . 4 (𝑋𝑇𝑋 ∈ 𝒫 𝑇)
1312ad2antrl 760 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → 𝑋 ∈ 𝒫 𝑇)
1410elpw2 4755 . . . . 5 (𝑌 ∈ 𝒫 𝑇𝑌𝑇)
1514biimpri 217 . . . 4 (𝑌𝑇𝑌 ∈ 𝒫 𝑇)
1615ad2antll 761 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → 𝑌 ∈ 𝒫 𝑇)
17 fvex 6113 . . . 4 ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V
1817a1i 11 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V)
19 fveq2 6103 . . . . . 6 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
2019ineq1d 3775 . . . . 5 (𝑥 = 𝑋 → (( 𝑥) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑦)))
2120fveq2d 6107 . . . 4 (𝑥 = 𝑋 → ( ‘(( 𝑥) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑦))))
22 fveq2 6103 . . . . . 6 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2322ineq2d 3776 . . . . 5 (𝑦 = 𝑌 → (( 𝑋) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑌)))
2423fveq2d 6107 . . . 4 (𝑦 = 𝑌 → ( ‘(( 𝑋) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑌))))
25 eqid 2610 . . . 4 (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))
2621, 24, 25ovmpt2g 6693 . . 3 ((𝑋 ∈ 𝒫 𝑇𝑌 ∈ 𝒫 𝑇 ∧ ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V) → (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
2713, 16, 18, 26syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
288, 27eqtrd 2644 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋𝐽𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  DIsoAcdia 35335  ocAcocaN 35426  vAcdjaN 35438 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-djaN 35439 This theorem is referenced by:  djaclN  35443  djajN  35444
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