Theorem ocvi 21601

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	⊢ , = (·𝑖‘𝑊)
ocvfval.f	⊢ 𝐹 = (Scalar‘𝑊)
ocvfval.z	⊢ 0 = (0g‘𝐹)
ocvfval.o	⊢ ⊥ = (ocv‘𝑊)

Assertion

Ref	Expression
ocvi	⊢ ((𝐴 ∈ ( ⊥ ‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝐴 , 𝐵) = 0 )

Proof of Theorem ocvi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ocvfval.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2		ocvfval.i	. . . 4 ⊢ , = (·𝑖‘𝑊)
3		ocvfval.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
4		ocvfval.z	. . . 4 ⊢ 0 = (0g‘𝐹)
5		ocvfval.o	. . . 4 ⊢ ⊥ = (ocv‘𝑊)
6	1, 2, 3, 4, 5	elocv 21600	. . 3 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ))
7	6	simp3bi 1147	. 2 ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) → ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )
8		oveq2 7349	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵))
9	8	eqeq1d 2733	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 , 𝑥) = 0 ↔ (𝐴 , 𝐵) = 0 ))
10	9	rspccva 3571	. 2 ⊢ ((∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 ∧ 𝐵 ∈ 𝑆) → (𝐴 , 𝐵) = 0 )
11	7, 10	sylan 580	1 ⊢ ((𝐴 ∈ ( ⊥ ‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝐴 , 𝐵) = 0 )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  Scalarcsca 17159  ·_𝑖cip 17161  0_gc0g 17338  ocvcocv 21592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-ocv 21595

This theorem is referenced by:  ocvocv  21603  ocvlss  21604  ocvin  21606  lsmcss  21624  clsocv  25172