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Theorem djavalN 41137
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 41136 . . . 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 7448 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋𝐽𝑌) = (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌))
92fvexi 6920 . . . . . 6 𝑇 ∈ V
109elpw2 5334 . . . . 5 (𝑋 ∈ 𝒫 𝑇𝑋𝑇)
1110biimpri 228 . . . 4 (𝑋𝑇𝑋 ∈ 𝒫 𝑇)
1211ad2antrl 728 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → 𝑋 ∈ 𝒫 𝑇)
139elpw2 5334 . . . . 5 (𝑌 ∈ 𝒫 𝑇𝑌𝑇)
1413biimpri 228 . . . 4 (𝑌𝑇𝑌 ∈ 𝒫 𝑇)
1514ad2antll 729 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → 𝑌 ∈ 𝒫 𝑇)
16 fvexd 6921 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V)
17 fveq2 6906 . . . . . 6 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1817ineq1d 4219 . . . . 5 (𝑥 = 𝑋 → (( 𝑥) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑦)))
1918fveq2d 6910 . . . 4 (𝑥 = 𝑋 → ( ‘(( 𝑥) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑦))))
20 fveq2 6906 . . . . . 6 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2120ineq2d 4220 . . . . 5 (𝑦 = 𝑌 → (( 𝑋) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑌)))
2221fveq2d 6910 . . . 4 (𝑦 = 𝑌 → ( ‘(( 𝑋) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑌))))
23 eqid 2737 . . . 4 (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))
2419, 22, 23ovmpog 7592 . . 3 ((𝑋 ∈ 𝒫 𝑇𝑌 ∈ 𝒫 𝑇 ∧ ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V) → (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
2512, 15, 16, 24syl3anc 1373 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
268, 25eqtrd 2777 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋𝐽𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600  cfv 6561  (class class class)co 7431  cmpo 7433  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  DIsoAcdia 41030  ocAcocaN 41121  vAcdjaN 41133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-djaN 41134
This theorem is referenced by:  djaclN  41138  djajN  41139
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