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Theorem ocvi 20741
Description: Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
ocvfval.v 𝑉 = (Base‘𝑊)
ocvfval.i , = (·𝑖𝑊)
ocvfval.f 𝐹 = (Scalar‘𝑊)
ocvfval.z 0 = (0g𝐹)
ocvfval.o = (ocv‘𝑊)
Assertion
Ref Expression
ocvi ((𝐴 ∈ ( 𝑆) ∧ 𝐵𝑆) → (𝐴 , 𝐵) = 0 )

Proof of Theorem ocvi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ocvfval.v . . . 4 𝑉 = (Base‘𝑊)
2 ocvfval.i . . . 4 , = (·𝑖𝑊)
3 ocvfval.f . . . 4 𝐹 = (Scalar‘𝑊)
4 ocvfval.z . . . 4 0 = (0g𝐹)
5 ocvfval.o . . . 4 = (ocv‘𝑊)
61, 2, 3, 4, 5elocv 20740 . . 3 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
76simp3bi 1139 . 2 (𝐴 ∈ ( 𝑆) → ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )
8 oveq2 7153 . . . 4 (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵))
98eqeq1d 2820 . . 3 (𝑥 = 𝐵 → ((𝐴 , 𝑥) = 0 ↔ (𝐴 , 𝐵) = 0 ))
109rspccva 3619 . 2 ((∀𝑥𝑆 (𝐴 , 𝑥) = 0𝐵𝑆) → (𝐴 , 𝐵) = 0 )
117, 10sylan 580 1 ((𝐴 ∈ ( 𝑆) ∧ 𝐵𝑆) → (𝐴 , 𝐵) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933  cfv 6348  (class class class)co 7145  Basecbs 16471  Scalarcsca 16556  ·𝑖cip 16558  0gc0g 16701  ocvcocv 20732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-ocv 20735
This theorem is referenced by:  ocvocv  20743  ocvlss  20744  ocvin  20746  lsmcss  20764  clsocv  23780
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