Theorem unocv 21224

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 4183	. . . . . . 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 4190	. . . . . 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 627	. . . . 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 654	. . . . 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 623	. . 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 2732	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
9		eqid 2732	. . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
10		eqid 2732	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
11		eqid 2732	. . . . 5 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
12		inocv.o	. . . . 5 ⊢ ⊥ = (ocv‘𝑊)
13	8, 9, 10, 11, 12	elocv 21212	. . . 4 ⊢ (𝑧 ∈ ( ⊥ ‘(𝐴 ∪ 𝐵)) ↔ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
14		3anan12 1096	. . . 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 21212	. . . . . 6 ⊢ (𝑧 ∈ ( ⊥ ‘𝐴) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
17		3anan12 1096	. . . . . 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 21212	. . . . . 6 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝐵 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
20		3anan12 1096	. . . . . 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 627	. . . 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 3963	. . . 4 ⊢ (𝑧 ∈ (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵)) ↔ (𝑧 ∈ ( ⊥ ‘𝐴) ∧ 𝑧 ∈ ( ⊥ ‘𝐵)))
24		anandi 674	. . . 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 2729	1 ⊢ ( ⊥ ‘(𝐴 ∪ 𝐵)) = (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵))

Syntax hints:   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  Scalarcsca 17196  ·_𝑖cip 17198  0_gc0g 17381  ocvcocv 21204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-ocv 21207

This theorem is referenced by:  cssincl  21232