Theorem unocv 20797

Description: The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)

Hypothesis

Ref	Expression
inocv.o	⊢ ⊥ = (ocv‘𝑊)

Assertion

Ref	Expression
unocv	⊢ ( ⊥ ‘(𝐴 ∪ 𝐵)) = (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵))

Proof of Theorem unocv

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

Step	Hyp	Ref	Expression
1		unss 4114	. . . . . . 7 ⊢ ((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ↔ (𝐴 ∪ 𝐵) ⊆ (Base‘𝑊))
2	1	bicomi 223	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)))
3		ralunb 4121	. . . . . 6 ⊢ (∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ↔ (∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
4	2, 3	anbi12i 626	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ ((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ∧ (∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
5		an4 652	. . . . 5 ⊢ (((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ∧ (∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) ↔ ((𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
6	4, 5	bitri 274	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ ((𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
7	6	anbi2i 622	. . 3 ⊢ ((𝑧 ∈ (Base‘𝑊) ∧ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) ↔ (𝑧 ∈ (Base‘𝑊) ∧ ((𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))))
8		eqid 2738	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
9		eqid 2738	. . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
10		eqid 2738	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
11		eqid 2738	. . . . 5 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
12		inocv.o	. . . . 5 ⊢ ⊥ = (ocv‘𝑊)
13	8, 9, 10, 11, 12	elocv 20785	. . . 4 ⊢ (𝑧 ∈ ( ⊥ ‘(𝐴 ∪ 𝐵)) ↔ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
14		3anan12 1094	. . . 4 ⊢ (((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (𝑧 ∈ (Base‘𝑊) ∧ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
15	13, 14	bitri 274	. . 3 ⊢ (𝑧 ∈ ( ⊥ ‘(𝐴 ∪ 𝐵)) ↔ (𝑧 ∈ (Base‘𝑊) ∧ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
16	8, 9, 10, 11, 12	elocv 20785	. . . . . 6 ⊢ (𝑧 ∈ ( ⊥ ‘𝐴) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
17		3anan12 1094	. . . . . 6 ⊢ ((𝐴 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
18	16, 17	bitri 274	. . . . 5 ⊢ (𝑧 ∈ ( ⊥ ‘𝐴) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
19	8, 9, 10, 11, 12	elocv 20785	. . . . . 6 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝐵 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
20		3anan12 1094	. . . . . 6 ⊢ ((𝐵 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
21	19, 20	bitri 274	. . . . 5 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
22	18, 21	anbi12i 626	. . . 4 ⊢ ((𝑧 ∈ ( ⊥ ‘𝐴) ∧ 𝑧 ∈ ( ⊥ ‘𝐵)) ↔ ((𝑧 ∈ (Base‘𝑊) ∧ (𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))))
23		elin 3899	. . . 4 ⊢ (𝑧 ∈ (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵)) ↔ (𝑧 ∈ ( ⊥ ‘𝐴) ∧ 𝑧 ∈ ( ⊥ ‘𝐵)))
24		anandi 672	. . . 4 ⊢ ((𝑧 ∈ (Base‘𝑊) ∧ ((𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))) ↔ ((𝑧 ∈ (Base‘𝑊) ∧ (𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))))
25	22, 23, 24	3bitr4i 302	. . 3 ⊢ (𝑧 ∈ (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵)) ↔ (𝑧 ∈ (Base‘𝑊) ∧ ((𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))))
26	7, 15, 25	3bitr4i 302	. 2 ⊢ (𝑧 ∈ ( ⊥ ‘(𝐴 ∪ 𝐵)) ↔ 𝑧 ∈ (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵)))
27	26	eqriv 2735	1 ⊢ ( ⊥ ‘(𝐴 ∪ 𝐵)) = (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵))

Syntax hints:   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Scalarcsca 16891  ·_𝑖cip 16893  0_gc0g 17067  ocvcocv 20777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-ocv 20780

This theorem is referenced by:  cssincl  20805