Theorem ocvi 21585

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 21584	. . 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 7398	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵))
9	8	eqeq1d 2732	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 , 𝑥) = 0 ↔ (𝐴 , 𝐵) = 0 ))
10	9	rspccva 3590	. 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 1540   ∈ wcel 2109  ∀wral 3045   ⊆ wss 3917  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  Scalarcsca 17230  ·_𝑖cip 17232  0_gc0g 17409  ocvcocv 21576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-ocv 21579

This theorem is referenced by:  ocvocv  21587  ocvlss  21588  ocvin  21590  lsmcss  21608  clsocv  25157