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Theorem djavalN 37156
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 37155 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
76adantr 473 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))
87oveqd 6895 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋𝐽𝑌) = (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌))
92fvexi 6425 . . . . . 6 𝑇 ∈ V
109elpw2 5020 . . . . 5 (𝑋 ∈ 𝒫 𝑇𝑋𝑇)
1110biimpri 220 . . . 4 (𝑋𝑇𝑋 ∈ 𝒫 𝑇)
1211ad2antrl 720 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → 𝑋 ∈ 𝒫 𝑇)
139elpw2 5020 . . . . 5 (𝑌 ∈ 𝒫 𝑇𝑌𝑇)
1413biimpri 220 . . . 4 (𝑌𝑇𝑌 ∈ 𝒫 𝑇)
1514ad2antll 721 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → 𝑌 ∈ 𝒫 𝑇)
16 fvexd 6426 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → ( ‘(( 𝑋) ∩ ( 𝑌))) ∈ V)
17 fveq2 6411 . . . . . 6 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1817ineq1d 4011 . . . . 5 (𝑥 = 𝑋 → (( 𝑥) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑦)))
1918fveq2d 6415 . . . 4 (𝑥 = 𝑋 → ( ‘(( 𝑥) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑦))))
20 fveq2 6411 . . . . . 6 (𝑦 = 𝑌 → ( 𝑦) = ( 𝑌))
2120ineq2d 4012 . . . . 5 (𝑦 = 𝑌 → (( 𝑋) ∩ ( 𝑦)) = (( 𝑋) ∩ ( 𝑌)))
2221fveq2d 6415 . . . 4 (𝑦 = 𝑌 → ( ‘(( 𝑋) ∩ ( 𝑦))) = ( ‘(( 𝑋) ∩ ( 𝑌))))
23 eqid 2799 . . . 4 (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))) = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))
2419, 22, 23ovmpt2g 7029 . . 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 1491 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋(𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦))))𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
268, 25eqtrd 2833 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋𝐽𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  cin 3768  wss 3769  𝒫 cpw 4349  cfv 6101  (class class class)co 6878  cmpt2 6880  HLchlt 35371  LHypclh 36005  LTrncltrn 36122  DIsoAcdia 37049  ocAcocaN 37140  vAcdjaN 37152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-djaN 37153
This theorem is referenced by:  djaclN  37157  djajN  37158
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