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Theorem docafvalN 39993
Description: Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
docaval.j ∨ = (joinβ€˜πΎ)
docaval.m ∧ = (meetβ€˜πΎ)
docaval.o βŠ₯ = (ocβ€˜πΎ)
docaval.h 𝐻 = (LHypβ€˜πΎ)
docaval.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
docaval.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
docaval.n 𝑁 = ((ocAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
docafvalN ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝑁 = (π‘₯ ∈ 𝒫 𝑇 ↦ (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š))))
Distinct variable groups:   π‘₯,𝑧,𝐾   π‘₯,𝐼,𝑧   π‘₯,𝑇   π‘₯,π‘Š,𝑧
Allowed substitution hints:   𝑇(𝑧)   𝐻(π‘₯,𝑧)   ∨ (π‘₯,𝑧)   ∧ (π‘₯,𝑧)   𝑁(π‘₯,𝑧)   βŠ₯ (π‘₯,𝑧)   𝑉(π‘₯,𝑧)

Proof of Theorem docafvalN
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 docaval.n . . 3 𝑁 = ((ocAβ€˜πΎ)β€˜π‘Š)
2 docaval.j . . . . 5 ∨ = (joinβ€˜πΎ)
3 docaval.m . . . . 5 ∧ = (meetβ€˜πΎ)
4 docaval.o . . . . 5 βŠ₯ = (ocβ€˜πΎ)
5 docaval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
62, 3, 4, 5docaffvalN 39992 . . . 4 (𝐾 ∈ 𝑉 β†’ (ocAβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀)))))
76fveq1d 6894 . . 3 (𝐾 ∈ 𝑉 β†’ ((ocAβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀))))β€˜π‘Š))
81, 7eqtrid 2785 . 2 (𝐾 ∈ 𝑉 β†’ 𝑁 = ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀))))β€˜π‘Š))
9 fveq2 6892 . . . . . 6 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
10 docaval.t . . . . . 6 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
119, 10eqtr4di 2791 . . . . 5 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
1211pweqd 4620 . . . 4 (𝑀 = π‘Š β†’ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) = 𝒫 𝑇)
13 fveq2 6892 . . . . . 6 (𝑀 = π‘Š β†’ ((DIsoAβ€˜πΎ)β€˜π‘€) = ((DIsoAβ€˜πΎ)β€˜π‘Š))
14 docaval.i . . . . . 6 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
1513, 14eqtr4di 2791 . . . . 5 (𝑀 = π‘Š β†’ ((DIsoAβ€˜πΎ)β€˜π‘€) = 𝐼)
1615cnveqd 5876 . . . . . . . . 9 (𝑀 = π‘Š β†’ β—‘((DIsoAβ€˜πΎ)β€˜π‘€) = ◑𝐼)
1715rneqd 5938 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ ran ((DIsoAβ€˜πΎ)β€˜π‘€) = ran 𝐼)
1817rabeqdv 3448 . . . . . . . . . 10 (𝑀 = π‘Š β†’ {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧} = {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})
1918inteqd 4956 . . . . . . . . 9 (𝑀 = π‘Š β†’ ∩ {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧} = ∩ {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})
2016, 19fveq12d 6899 . . . . . . . 8 (𝑀 = π‘Š β†’ (β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧}) = (β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧}))
2120fveq2d 6896 . . . . . . 7 (𝑀 = π‘Š β†’ ( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) = ( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})))
22 fveq2 6892 . . . . . . 7 (𝑀 = π‘Š β†’ ( βŠ₯ β€˜π‘€) = ( βŠ₯ β€˜π‘Š))
2321, 22oveq12d 7427 . . . . . 6 (𝑀 = π‘Š β†’ (( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) = (( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)))
24 id 22 . . . . . 6 (𝑀 = π‘Š β†’ 𝑀 = π‘Š)
2523, 24oveq12d 7427 . . . . 5 (𝑀 = π‘Š β†’ ((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀) = ((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š))
2615, 25fveq12d 6899 . . . 4 (𝑀 = π‘Š β†’ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀)) = (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š)))
2712, 26mpteq12dv 5240 . . 3 (𝑀 = π‘Š β†’ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀))) = (π‘₯ ∈ 𝒫 𝑇 ↦ (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š))))
28 eqid 2733 . . 3 (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀)))) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀))))
2910fvexi 6906 . . . . 5 𝑇 ∈ V
3029pwex 5379 . . . 4 𝒫 𝑇 ∈ V
3130mptex 7225 . . 3 (π‘₯ ∈ 𝒫 𝑇 ↦ (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š))) ∈ V
3227, 28, 31fvmpt 6999 . 2 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ 𝒫 ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜((( βŠ₯ β€˜(β—‘((DIsoAβ€˜πΎ)β€˜π‘€)β€˜βˆ© {𝑧 ∈ ran ((DIsoAβ€˜πΎ)β€˜π‘€) ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘€)) ∧ 𝑀))))β€˜π‘Š) = (π‘₯ ∈ 𝒫 𝑇 ↦ (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š))))
338, 32sylan9eq 2793 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝑁 = (π‘₯ ∈ 𝒫 𝑇 ↦ (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š))))
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  βˆ© cint 4951   ↦ cmpt 5232  β—‘ccnv 5676  ran crn 5678  β€˜cfv 6544  (class class class)co 7409  occoc 17205  joincjn 18264  meetcmee 18265  LHypclh 38855  LTrncltrn 38972  DIsoAcdia 39899  ocAcocaN 39990
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-int 4952  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-ov 7412  df-docaN 39991
This theorem is referenced by:  docavalN  39994
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