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Theorem dochfval 40221
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
dochfval ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ 𝑁 = (π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)})))))
Distinct variable groups:   𝑦,𝐡   π‘₯,𝑦,𝐾   π‘₯,𝑉   π‘₯,π‘Š,𝑦
Allowed substitution hints:   𝐡(π‘₯)   π‘ˆ(π‘₯,𝑦)   𝐺(π‘₯,𝑦)   𝐻(π‘₯,𝑦)   𝐼(π‘₯,𝑦)   𝑁(π‘₯,𝑦)   βŠ₯ (π‘₯,𝑦)   𝑉(𝑦)   𝑋(π‘₯,𝑦)

Proof of Theorem dochfval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 dochval.n . . 3 𝑁 = ((ocHβ€˜πΎ)β€˜π‘Š)
2 dochval.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
3 dochval.g . . . . 5 𝐺 = (glbβ€˜πΎ)
4 dochval.o . . . . 5 βŠ₯ = (ocβ€˜πΎ)
5 dochval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
62, 3, 4, 5dochffval 40220 . . . 4 (𝐾 ∈ 𝑋 β†’ (ocHβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)}))))))
76fveq1d 6894 . . 3 (𝐾 ∈ 𝑋 β†’ ((ocHβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)})))))β€˜π‘Š))
81, 7eqtrid 2785 . 2 (𝐾 ∈ 𝑋 β†’ 𝑁 = ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)})))))β€˜π‘Š))
9 fveq2 6892 . . . . . . . 8 (𝑀 = π‘Š β†’ ((DVecHβ€˜πΎ)β€˜π‘€) = ((DVecHβ€˜πΎ)β€˜π‘Š))
10 dochval.u . . . . . . . 8 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
119, 10eqtr4di 2791 . . . . . . 7 (𝑀 = π‘Š β†’ ((DVecHβ€˜πΎ)β€˜π‘€) = π‘ˆ)
1211fveq2d 6896 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (Baseβ€˜π‘ˆ))
13 dochval.v . . . . . 6 𝑉 = (Baseβ€˜π‘ˆ)
1412, 13eqtr4di 2791 . . . . 5 (𝑀 = π‘Š β†’ (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝑉)
1514pweqd 4620 . . . 4 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝒫 𝑉)
16 fveq2 6892 . . . . . 6 (𝑀 = π‘Š β†’ ((DIsoHβ€˜πΎ)β€˜π‘€) = ((DIsoHβ€˜πΎ)β€˜π‘Š))
17 dochval.i . . . . . 6 𝐼 = ((DIsoHβ€˜πΎ)β€˜π‘Š)
1816, 17eqtr4di 2791 . . . . 5 (𝑀 = π‘Š β†’ ((DIsoHβ€˜πΎ)β€˜π‘€) = 𝐼)
1918fveq1d 6894 . . . . . . . . 9 (𝑀 = π‘Š β†’ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦) = (πΌβ€˜π‘¦))
2019sseq2d 4015 . . . . . . . 8 (𝑀 = π‘Š β†’ (π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦) ↔ π‘₯ βŠ† (πΌβ€˜π‘¦)))
2120rabbidv 3441 . . . . . . 7 (𝑀 = π‘Š β†’ {𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)} = {𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)})
2221fveq2d 6896 . . . . . 6 (𝑀 = π‘Š β†’ (πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)}) = (πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)}))
2322fveq2d 6896 . . . . 5 (𝑀 = π‘Š β†’ ( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)})) = ( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)})))
2418, 23fveq12d 6899 . . . 4 (𝑀 = π‘Š β†’ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)}))) = (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)}))))
2515, 24mpteq12dv 5240 . . 3 (𝑀 = π‘Š β†’ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)})))) = (π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)})))))
26 eqid 2733 . . 3 (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)}))))) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)})))))
2713fvexi 6906 . . . . 5 𝑉 ∈ V
2827pwex 5379 . . . 4 𝒫 𝑉 ∈ V
2928mptex 7225 . . 3 (π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)})))) ∈ V
3025, 26, 29fvmpt 6999 . 2 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (((DIsoHβ€˜πΎ)β€˜π‘€)β€˜π‘¦)})))))β€˜π‘Š) = (π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)})))))
318, 30sylan9eq 2793 1 ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ 𝑁 = (π‘₯ ∈ 𝒫 𝑉 ↦ (πΌβ€˜( βŠ₯ β€˜(πΊβ€˜{𝑦 ∈ 𝐡 ∣ π‘₯ βŠ† (πΌβ€˜π‘¦)})))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3433   βŠ† wss 3949  π’« cpw 4603   ↦ cmpt 5232  β€˜cfv 6544  Basecbs 17144  occoc 17205  glbcglb 18263  LHypclh 38855  DVecHcdvh 39949  DIsoHcdih 40099  ocHcoch 40218
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-doch 40219
This theorem is referenced by:  dochval  40222  dochfN  40227
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