Theorem ocvi 21629

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 21628	. . 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 7413	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 , 𝑥) = (𝐴 , 𝐵))
9	8	eqeq1d 2737	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 , 𝑥) = 0 ↔ (𝐴 , 𝐵) = 0 ))
10	9	rspccva 3600	. 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 2108  ∀wral 3051   ⊆ wss 3926  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  Scalarcsca 17274  ·_𝑖cip 17276  0_gc0g 17453  ocvcocv 21620

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7408  df-ocv 21623

This theorem is referenced by:  ocvocv  21631  ocvlss  21632  ocvin  21634  lsmcss  21652  clsocv  25202