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Theorem docavalN 39589
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
docavalN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (π‘β€˜π‘‹) = (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š)))
Distinct variable groups:   𝑧,𝐾   𝑧,𝐼   𝑧,π‘Š   𝑧,𝑇   𝑧,𝑋
Allowed substitution hints:   𝐻(𝑧)   ∨ (𝑧)   ∧ (𝑧)   𝑁(𝑧)   βŠ₯ (𝑧)

Proof of Theorem docavalN
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 docaval.j . . . . 5 ∨ = (joinβ€˜πΎ)
2 docaval.m . . . . 5 ∧ = (meetβ€˜πΎ)
3 docaval.o . . . . 5 βŠ₯ = (ocβ€˜πΎ)
4 docaval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
5 docaval.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
6 docaval.i . . . . 5 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
7 docaval.n . . . . 5 𝑁 = ((ocAβ€˜πΎ)β€˜π‘Š)
81, 2, 3, 4, 5, 6, 7docafvalN 39588 . . . 4 ((𝐾 ∈ HL ∧ 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))))
98adantr 482 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) 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))))
109fveq1d 6845 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (π‘β€˜π‘‹) = ((π‘₯ ∈ 𝒫 𝑇 ↦ (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š)))β€˜π‘‹))
115fvexi 6857 . . . . . 6 𝑇 ∈ V
1211elpw2 5303 . . . . 5 (𝑋 ∈ 𝒫 𝑇 ↔ 𝑋 βŠ† 𝑇)
1312biimpri 227 . . . 4 (𝑋 βŠ† 𝑇 β†’ 𝑋 ∈ 𝒫 𝑇)
1413adantl 483 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ 𝑋 ∈ 𝒫 𝑇)
15 fvex 6856 . . 3 (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š)) ∈ V
16 sseq1 3970 . . . . . . . . . 10 (π‘₯ = 𝑋 β†’ (π‘₯ βŠ† 𝑧 ↔ 𝑋 βŠ† 𝑧))
1716rabbidv 3416 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧} = {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})
1817inteqd 4913 . . . . . . . 8 (π‘₯ = 𝑋 β†’ ∩ {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧} = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})
1918fveq2d 6847 . . . . . . 7 (π‘₯ = 𝑋 β†’ (β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧}) = (β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧}))
2019fveq2d 6847 . . . . . 6 (π‘₯ = 𝑋 β†’ ( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) = ( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})))
2120oveq1d 7373 . . . . 5 (π‘₯ = 𝑋 β†’ (( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) = (( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)))
2221fvoveq1d 7380 . . . 4 (π‘₯ = 𝑋 β†’ (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š)) = (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š)))
23 eqid 2737 . . . 4 (π‘₯ ∈ 𝒫 𝑇 ↦ (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ 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)))
2422, 23fvmptg 6947 . . 3 ((𝑋 ∈ 𝒫 𝑇 ∧ (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š)) ∈ V) β†’ ((π‘₯ ∈ 𝒫 𝑇 ↦ (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š)))β€˜π‘‹) = (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š)))
2514, 15, 24sylancl 587 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ ((π‘₯ ∈ 𝒫 𝑇 ↦ (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ π‘₯ βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š)))β€˜π‘‹) = (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 βŠ† 𝑧})) ∨ ( βŠ₯ β€˜π‘Š)) ∧ π‘Š)))
2610, 25eqtrd 2777 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑇) β†’ (π‘β€˜π‘‹) = (πΌβ€˜((( βŠ₯ β€˜(β—‘πΌβ€˜βˆ© {𝑧 ∈ ran 𝐼 ∣ 𝑋 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 3408  Vcvv 3446   βŠ† wss 3911  π’« cpw 4561  βˆ© cint 4908   ↦ cmpt 5189  β—‘ccnv 5633  ran crn 5635  β€˜cfv 6497  (class class class)co 7358  occoc 17142  joincjn 18201  meetcmee 18202  HLchlt 37815  LHypclh 38450  LTrncltrn 38567  DIsoAcdia 39494  ocAcocaN 39585
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
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-int 4909  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-docaN 39586
This theorem is referenced by:  docaclN  39590  diaocN  39591
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