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Theorem ocvi 20361
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 20360 . . 3 (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))
76simp3bi 1144 . 2 (𝐴 ∈ ( 𝑆) → ∀𝑥𝑆 (𝐴 , 𝑥) = 0 )
8 oveq2 7147 . . . 4 (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵))
98eqeq1d 2803 . . 3 (𝑥 = 𝐵 → ((𝐴 , 𝑥) = 0 ↔ (𝐴 , 𝐵) = 0 ))
109rspccva 3573 . 2 ((∀𝑥𝑆 (𝐴 , 𝑥) = 0𝐵𝑆) → (𝐴 , 𝐵) = 0 )
117, 10sylan 583 1 ((𝐴 ∈ ( 𝑆) ∧ 𝐵𝑆) → (𝐴 , 𝐵) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  wss 3884  cfv 6328  (class class class)co 7139  Basecbs 16478  Scalarcsca 16563  ·𝑖cip 16565  0gc0g 16708  ocvcocv 20352
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-ocv 20355
This theorem is referenced by:  ocvocv  20363  ocvlss  20364  ocvin  20366  lsmcss  20384  clsocv  23857
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