Theorem unocv 20024

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 3787	. . . . . . 7 ⊢ ((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ↔ (𝐴 ∪ 𝐵) ⊆ (Base‘𝑊))
2	1	bicomi 214	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)))
3		ralunb 3794	. . . . . 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 733	. . . . 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 865	. . . . 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 264	. . . 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 730	. . 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 2622	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
9		eqid 2622	. . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
10		eqid 2622	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
11		eqid 2622	. . . . 5 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
12		inocv.o	. . . . 5 ⊢ ⊥ = (ocv‘𝑊)
13	8, 9, 10, 11, 12	elocv 20012	. . . 4 ⊢ (𝑧 ∈ ( ⊥ ‘(𝐴 ∪ 𝐵)) ↔ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
14		3anan12 1051	. . . 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 264	. . 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 20012	. . . . . 6 ⊢ (𝑧 ∈ ( ⊥ ‘𝐴) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
17		3anan12 1051	. . . . . 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 264	. . . . 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 20012	. . . . . 6 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝐵 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
20		3anan12 1051	. . . . . 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 264	. . . . 5 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
22	18, 21	anbi12i 733	. . . 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 3796	. . . 4 ⊢ (𝑧 ∈ (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵)) ↔ (𝑧 ∈ ( ⊥ ‘𝐴) ∧ 𝑧 ∈ ( ⊥ ‘𝐵)))
24		anandi 871	. . . 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 292	. . 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 292	. 2 ⊢ (𝑧 ∈ ( ⊥ ‘(𝐴 ∪ 𝐵)) ↔ 𝑧 ∈ (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵)))
27	26	eqriv 2619	1 ⊢ ( ⊥ ‘(𝐴 ∪ 𝐵)) = (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵))

Syntax hints:   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Scalarcsca 15944  ·_𝑖cip 15946  0_gc0g 16100  ocvcocv 20004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-ocv 20007

This theorem is referenced by:  cssincl  20032