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Theorem ocvi 21221
Description: Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
ocvfval.v 𝑉 = (Baseβ€˜π‘Š)
ocvfval.i , = (Β·π‘–β€˜π‘Š)
ocvfval.f 𝐹 = (Scalarβ€˜π‘Š)
ocvfval.z 0 = (0gβ€˜πΉ)
ocvfval.o βŠ₯ = (ocvβ€˜π‘Š)
Assertion
Ref Expression
ocvi ((𝐴 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝐡 ∈ 𝑆) β†’ (𝐴 , 𝐡) = 0 )

Proof of Theorem ocvi
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ocvfval.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
2 ocvfval.i . . . 4 , = (Β·π‘–β€˜π‘Š)
3 ocvfval.f . . . 4 𝐹 = (Scalarβ€˜π‘Š)
4 ocvfval.z . . . 4 0 = (0gβ€˜πΉ)
5 ocvfval.o . . . 4 βŠ₯ = (ocvβ€˜π‘Š)
61, 2, 3, 4, 5elocv 21220 . . 3 (𝐴 ∈ ( βŠ₯ β€˜π‘†) ↔ (𝑆 βŠ† 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 ))
76simp3bi 1147 . 2 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 )
8 oveq2 7416 . . . 4 (π‘₯ = 𝐡 β†’ (𝐴 , π‘₯) = (𝐴 , 𝐡))
98eqeq1d 2734 . . 3 (π‘₯ = 𝐡 β†’ ((𝐴 , π‘₯) = 0 ↔ (𝐴 , 𝐡) = 0 ))
109rspccva 3611 . 2 ((βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 ∧ 𝐡 ∈ 𝑆) β†’ (𝐴 , 𝐡) = 0 )
117, 10sylan 580 1 ((𝐴 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝐡 ∈ 𝑆) β†’ (𝐴 , 𝐡) = 0 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  Scalarcsca 17199  Β·π‘–cip 17201  0gc0g 17384  ocvcocv 21212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-ocv 21215
This theorem is referenced by:  ocvocv  21223  ocvlss  21224  ocvin  21226  lsmcss  21244  clsocv  24766
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