Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dochval Structured version   Visualization version   GIF version

Theorem dochval 39843
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 39842 . . . 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 482 . . 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 6849 . 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 6861 . . . . . 6 𝑉 ∈ V
1312elpw2 5307 . . . . 5 (𝑋 ∈ 𝒫 𝑉 ↔ 𝑋 βŠ† 𝑉)
1413biimpri 227 . . . 4 (𝑋 βŠ† 𝑉 β†’ 𝑋 ∈ 𝒫 𝑉)
1514adantl 483 . . 3 (((𝐾 ∈ π‘Œ ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑋 ∈ 𝒫 𝑉)
16 fvex 6860 . . 3 (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)}))) ∈ V
17 sseq1 3974 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† (πΌβ€˜π‘¦) ↔ 𝑋 βŠ† (πΌβ€˜π‘¦)))
1817rabbidv 3418 . . . . . . 7 (π‘₯ = 𝑋 β†’ {𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)} = {𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)})
1918fveq2d 6851 . . . . . 6 (π‘₯ = 𝑋 β†’ (πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)}) = (πΊβ€˜{𝑦 ∈ 𝐡 ∣ 𝑋 βŠ† (πΌβ€˜π‘¦)}))
2019fveq2d 6851 . . . . 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 6851 . . . 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 2737 . . . 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 6951 . . 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 587 . 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 2777 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 397   = wceq 1542   ∈ wcel 2107  {crab 3410  Vcvv 3448   βŠ† wss 3915  π’« cpw 4565   ↦ cmpt 5193  β€˜cfv 6501  Basecbs 17090  occoc 17148  glbcglb 18206  LHypclh 38476  DVecHcdvh 39570  DIsoHcdih 39720  ocHcoch 39839
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 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389
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 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-doch 39840
This theorem is referenced by:  dochval2  39844  dochcl  39845  dochvalr  39849  dochss  39857
  Copyright terms: Public domain W3C validator