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Theorem dochval 40856
Description: Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)
Hypotheses
Ref Expression
dochval.b 𝐡 = (Baseβ€˜πΎ)
dochval.g 𝐺 = (glbβ€˜πΎ)
dochval.o βŠ₯ = (ocβ€˜πΎ)
dochval.h 𝐻 = (LHypβ€˜πΎ)
dochval.i 𝐼 = ((DIsoHβ€˜πΎ)β€˜π‘Š)
dochval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
dochval.v 𝑉 = (Baseβ€˜π‘ˆ)
dochval.n 𝑁 = ((ocHβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dochval (((𝐾 ∈ π‘Œ ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (π‘β€˜π‘‹) = (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)}))))
Distinct variable groups:   𝑦,𝐡   𝑦,𝐾   𝑦,π‘Š   𝑦,𝑋
Allowed substitution hints:   π‘ˆ(𝑦)   𝐺(𝑦)   𝐻(𝑦)   𝐼(𝑦)   𝑁(𝑦)   βŠ₯ (𝑦)   𝑉(𝑦)   π‘Œ(𝑦)

Proof of Theorem dochval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 dochval.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
2 dochval.g . . . . 5 𝐺 = (glbβ€˜πΎ)
3 dochval.o . . . . 5 βŠ₯ = (ocβ€˜πΎ)
4 dochval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
5 dochval.i . . . . 5 𝐼 = ((DIsoHβ€˜πΎ)β€˜π‘Š)
6 dochval.u . . . . 5 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
7 dochval.v . . . . 5 𝑉 = (Baseβ€˜π‘ˆ)
8 dochval.n . . . . 5 𝑁 = ((ocHβ€˜πΎ)β€˜π‘Š)
91, 2, 3, 4, 5, 6, 7, 8dochfval 40855 . . . 4 ((𝐾 ∈ π‘Œ ∧ π‘Š ∈ 𝐻) β†’ 𝑁 = (π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)})))))
109adantr 479 . . 3 (((𝐾 ∈ π‘Œ ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑁 = (π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)})))))
1110fveq1d 6904 . 2 (((𝐾 ∈ π‘Œ ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (π‘β€˜π‘‹) = ((π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)}))))β€˜π‘‹))
127fvexi 6916 . . . . . 6 𝑉 ∈ V
1312elpw2 5351 . . . . 5 (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 βŠ† 𝑉)
1413biimpri 227 . . . 4 (𝑋 βŠ† 𝑉 β†’ 𝑋 ∈ 𝒫 𝑉)
1514adantl 480 . . 3 (((𝐾 ∈ π‘Œ ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑋 ∈ 𝒫 𝑉)
16 fvex 6915 . . 3 (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)}))) ∈ V
17 sseq1 4007 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† (πΌβ€˜π‘¦) ↔ 𝑋 βŠ† (πΌβ€˜π‘¦)))
1817rabbidv 3438 . . . . . . 7 (π‘₯ = 𝑋 β†’ {𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)} = {𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)})
1918fveq2d 6906 . . . . . 6 (π‘₯ = 𝑋 β†’ (πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)}) = (πΊβ€˜{𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)}))
2019fveq2d 6906 . . . . 5 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)})) = ( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)})))
2120fveq2d 6906 . . . 4 (π‘₯ = 𝑋 β†’ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)}))) = (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)}))))
22 eqid 2728 . . . 4 (π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)})))) = (π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)}))))
2321, 22fvmptg 7008 . . 3 ((𝑋 ∈ 𝒫 𝑉 ∧ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)}))) ∈ V) β†’ ((π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)}))))β€˜π‘‹) = (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)}))))
2415, 16, 23sylancl 584 . 2 (((𝐾 ∈ π‘Œ ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ((π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)}))))β€˜π‘‹) = (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)}))))
2511, 24eqtrd 2768 1 (((𝐾 ∈ π‘Œ ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (π‘β€˜π‘‹) = (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)}))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {crab 3430  Vcvv 3473   βŠ† wss 3949  π’« cpw 4606   ↦ cmpt 5235  β€˜cfv 6553  Basecbs 17187  occoc 17248  glbcglb 18309  LHypclh 39489  DVecHcdvh 40583  DIsoHcdih 40733  ocHcoch 40852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-doch 40853
This theorem is referenced by:  dochval2  40857  dochcl  40858  dochvalr  40862  dochss  40870
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