Theorem unocv 20827

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 4163	. . . . . . 7 ⊢ ((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ↔ (𝐴 ∪ 𝐵) ⊆ (Base‘𝑊))
2	1	bicomi 226	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)))
3		ralunb 4170	. . . . . 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 628	. . . . 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 277	. . . 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 624	. . 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 2824	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
9		eqid 2824	. . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
10		eqid 2824	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
11		eqid 2824	. . . . 5 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
12		inocv.o	. . . . 5 ⊢ ⊥ = (ocv‘𝑊)
13	8, 9, 10, 11, 12	elocv 20815	. . . 4 ⊢ (𝑧 ∈ ( ⊥ ‘(𝐴 ∪ 𝐵)) ↔ ((𝐴 ∪ 𝐵) ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
14		3anan12 1092	. . . 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 277	. . 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 20815	. . . . . 6 ⊢ (𝑧 ∈ ( ⊥ ‘𝐴) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐴 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
17		3anan12 1092	. . . . . 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 277	. . . . 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 20815	. . . . . 6 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝐵 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
20		3anan12 1092	. . . . . 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 277	. . . . 5 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
22	18, 21	anbi12i 628	. . . 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 4172	. . . 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 305	. . 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 305	. 2 ⊢ (𝑧 ∈ ( ⊥ ‘(𝐴 ∪ 𝐵)) ↔ 𝑧 ∈ (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵)))
27	26	eqriv 2821	1 ⊢ ( ⊥ ‘(𝐴 ∪ 𝐵)) = (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵))

Syntax hints:   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∀wral 3141   ∪ cun 3937   ∩ cin 3938   ⊆ wss 3939  ‘cfv 6358  (class class class)co 7159  Basecbs 16486  Scalarcsca 16571  ·_𝑖cip 16573  0_gc0g 16716  ocvcocv 20807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7162  df-ocv 20810

This theorem is referenced by:  cssincl  20835