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Theorem djavalN 39601
Description: Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
djaval.h 𝐻 = (LHypβ€˜πΎ)
djaval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
djaval.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
djaval.n βŠ₯ = ((ocAβ€˜πΎ)β€˜π‘Š)
djaval.j 𝐽 = ((vAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
djavalN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑇 ∧ π‘Œ βŠ† 𝑇)) β†’ (π‘‹π½π‘Œ) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))))

Proof of Theorem djavalN
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 djaval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
2 djaval.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
3 djaval.i . . . . 5 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
4 djaval.n . . . . 5 βŠ₯ = ((ocAβ€˜πΎ)β€˜π‘Š)
5 djaval.j . . . . 5 𝐽 = ((vAβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5djafvalN 39600 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐽 = (π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))))
76adantr 482 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑇 ∧ π‘Œ βŠ† 𝑇)) β†’ 𝐽 = (π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))))
87oveqd 7375 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑇 ∧ π‘Œ βŠ† 𝑇)) β†’ (π‘‹π½π‘Œ) = (𝑋(π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦))))π‘Œ))
92fvexi 6857 . . . . . 6 𝑇 ∈ V
109elpw2 5303 . . . . 5 (𝑋 ∈ 𝒫 𝑇 ↔ 𝑋 βŠ† 𝑇)
1110biimpri 227 . . . 4 (𝑋 βŠ† 𝑇 β†’ 𝑋 ∈ 𝒫 𝑇)
1211ad2antrl 727 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑇 ∧ π‘Œ βŠ† 𝑇)) β†’ 𝑋 ∈ 𝒫 𝑇)
139elpw2 5303 . . . . 5 (π‘Œ ∈ 𝒫 𝑇 ↔ π‘Œ βŠ† 𝑇)
1413biimpri 227 . . . 4 (π‘Œ βŠ† 𝑇 β†’ π‘Œ ∈ 𝒫 𝑇)
1514ad2antll 728 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑇 ∧ π‘Œ βŠ† 𝑇)) β†’ π‘Œ ∈ 𝒫 𝑇)
16 fvexd 6858 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑇 ∧ π‘Œ βŠ† 𝑇)) β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))) ∈ V)
17 fveq2 6843 . . . . . 6 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜π‘₯) = ( βŠ₯ β€˜π‘‹))
1817ineq1d 4172 . . . . 5 (π‘₯ = 𝑋 β†’ (( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)) = (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘¦)))
1918fveq2d 6847 . . . 4 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦))) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘¦))))
20 fveq2 6843 . . . . . 6 (𝑦 = π‘Œ β†’ ( βŠ₯ β€˜π‘¦) = ( βŠ₯ β€˜π‘Œ))
2120ineq2d 4173 . . . . 5 (𝑦 = π‘Œ β†’ (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘¦)) = (( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ)))
2221fveq2d 6847 . . . 4 (𝑦 = π‘Œ β†’ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘¦))) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))))
23 eqid 2737 . . . 4 (π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦)))) = (π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦))))
2419, 22, 23ovmpog 7515 . . 3 ((𝑋 ∈ 𝒫 𝑇 ∧ π‘Œ ∈ 𝒫 𝑇 ∧ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))) ∈ V) β†’ (𝑋(π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( 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† 𝑇 ∧ π‘Œ βŠ† 𝑇)) β†’ (𝑋(π‘₯ ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( βŠ₯ β€˜(( βŠ₯ β€˜π‘₯) ∩ ( βŠ₯ β€˜π‘¦))))π‘Œ) = ( βŠ₯ β€˜(( βŠ₯ β€˜π‘‹) ∩ ( βŠ₯ β€˜π‘Œ))))
268, 25eqtrd 2777 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 βŠ† 𝑇 ∧ π‘Œ βŠ† 𝑇)) β†’ (π‘‹π½π‘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 3446   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  HLchlt 37815  LHypclh 38450  LTrncltrn 38567  DIsoAcdia 39494  ocAcocaN 39585  vAcdjaN 39597
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-djaN 39598
This theorem is referenced by:  djaclN  39602  djajN  39603
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