Theorem ocv0 21806

Description: The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)

Hypotheses

Ref	Expression
ocvz.v	⊢ 𝑉 = (Base‘𝑊)
ocvz.o	⊢ ⊥ = (ocv‘𝑊)

Assertion

Ref	Expression
ocv0	⊢ ( ⊥ ‘∅) = 𝑉

Proof of Theorem ocv0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ss 4356	. . 3 ⊢ ∅ ⊆ 𝑉
2		ocvz.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
3		eqid 2761	. . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
4		eqid 2761	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2761	. . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
6		ocvz.o	. . . 4 ⊢ ⊥ = (ocv‘𝑊)
7	2, 3, 4, 5, 6	ocvval 21796	. . 3 ⊢ (∅ ⊆ 𝑉 → ( ⊥ ‘∅) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))})
8	1, 7	ax-mp 5	. 2 ⊢ ( ⊥ ‘∅) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}
9		ral0 4458	. . . 4 ⊢ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))
10	9	rgenw 3081	. . 3 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))
11		rabid2 3447	. . 3 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))} ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
12	10, 11	mpbir 234	. 2 ⊢ 𝑉 = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ ∅ (𝑥(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))}
13	8, 12	eqtr4i 2787	1 ⊢ ( ⊥ ‘∅) = 𝑉

Syntax hints:   = wceq 1568  ∀wral 3077  {crab 3414   ⊆ wss 3904  ∅c0 4285  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  Scalarcsca 17312  ·_𝑖cip 17314  0_gc0g 17491  ocvcocv 21789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-ocv 21792

This theorem is referenced by:  ocvz  21807  css1  21819