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Theorem ocvi 21096
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 21095 . . 3 (𝐴 ∈ ( βŠ₯ β€˜π‘†) ↔ (𝑆 βŠ† 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 ))
76simp3bi 1148 . 2 (𝐴 ∈ ( βŠ₯ β€˜π‘†) β†’ βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 )
8 oveq2 7369 . . . 4 (π‘₯ = 𝐡 β†’ (𝐴 , π‘₯) = (𝐴 , 𝐡))
98eqeq1d 2735 . . 3 (π‘₯ = 𝐡 β†’ ((𝐴 , π‘₯) = 0 ↔ (𝐴 , 𝐡) = 0 ))
109rspccva 3582 . 2 ((βˆ€π‘₯ ∈ 𝑆 (𝐴 , π‘₯) = 0 ∧ 𝐡 ∈ 𝑆) β†’ (𝐴 , 𝐡) = 0 )
117, 10sylan 581 1 ((𝐴 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝐡 ∈ 𝑆) β†’ (𝐴 , 𝐡) = 0 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   βŠ† wss 3914  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  Scalarcsca 17144  Β·π‘–cip 17146  0gc0g 17329  ocvcocv 21087
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-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-ocv 21090
This theorem is referenced by:  ocvocv  21098  ocvlss  21099  ocvin  21101  lsmcss  21119  clsocv  24637
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