Theorem unocv 20541

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 4051	. . . . . . 7 ⊢ ((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ↔ (𝐴 ∪ 𝐵) ⊆ (Base‘𝑊))
2	1	bicomi 216	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)))
3		ralunb 4058	. . . . . 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 618	. . . . 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 644	. . . . 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 267	. . . 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 614	. . 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 2780	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
9		eqid 2780	. . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
10		eqid 2780	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
11		eqid 2780	. . . . 5 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
12		inocv.o	. . . . 5 ⊢ ⊥ = (ocv‘𝑊)
13	8, 9, 10, 11, 12	elocv 20529	. . . 4 ⊢ (𝑧 ∈ ( ⊥ ‘(𝐴 ∪ 𝐵)) ↔ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
14		3anan12 1078	. . . 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 267	. . 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 20529	. . . . . 6 ⊢ (𝑧 ∈ ( ⊥ ‘𝐴) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
17		3anan12 1078	. . . . . 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 267	. . . . 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 20529	. . . . . 6 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝐵 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
20		3anan12 1078	. . . . . 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 267	. . . . 5 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
22	18, 21	anbi12i 618	. . . 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 4060	. . . 4 ⊢ (𝑧 ∈ (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵)) ↔ (𝑧 ∈ ( ⊥ ‘𝐴) ∧ 𝑧 ∈ ( ⊥ ‘𝐵)))
24		anandi 664	. . . 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 295	. . 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 295	. 2 ⊢ (𝑧 ∈ ( ⊥ ‘(𝐴 ∪ 𝐵)) ↔ 𝑧 ∈ (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵)))
27	26	eqriv 2777	1 ⊢ ( ⊥ ‘(𝐴 ∪ 𝐵)) = (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵))

Syntax hints:   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051  ∀wral 3090   ∪ cun 3829   ∩ cin 3830   ⊆ wss 3831  ‘cfv 6193  (class class class)co 6982  Basecbs 16345  Scalarcsca 16430  ·_𝑖cip 16432  0_gc0g 16575  ocvcocv 20521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3419  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-op 4451  df-uni 4718  df-br 4935  df-opab 4997  df-mpt 5014  df-id 5316  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-fv 6201  df-ov 6985  df-ocv 20524

This theorem is referenced by:  cssincl  20549