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Theorem unocv 20824
 Description: The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypothesis
Ref Expression
inocv.o = (ocv‘𝑊)
Assertion
Ref Expression
unocv ( ‘(𝐴𝐵)) = (( 𝐴) ∩ ( 𝐵))

Proof of Theorem unocv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unss 4146 . . . . . . 7 ((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ↔ (𝐴𝐵) ⊆ (Base‘𝑊))
21bicomi 227 . . . . . 6 ((𝐴𝐵) ⊆ (Base‘𝑊) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)))
3 ralunb 4153 . . . . . 6 (∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ↔ (∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
42, 3anbi12i 629 . . . . 5 (((𝐴𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ ((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ∧ (∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
5 an4 655 . . . . 5 (((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ∧ (∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) ↔ ((𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
64, 5bitri 278 . . . 4 (((𝐴𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ ((𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
76anbi2i 625 . . 3 ((𝑧 ∈ (Base‘𝑊) ∧ ((𝐴𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) ↔ (𝑧 ∈ (Base‘𝑊) ∧ ((𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))))
8 eqid 2824 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
9 eqid 2824 . . . . 5 (·𝑖𝑊) = (·𝑖𝑊)
10 eqid 2824 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
11 eqid 2824 . . . . 5 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
12 inocv.o . . . . 5 = (ocv‘𝑊)
138, 9, 10, 11, 12elocv 20812 . . . 4 (𝑧 ∈ ( ‘(𝐴𝐵)) ↔ ((𝐴𝐵) ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
14 3anan12 1093 . . . 4 (((𝐴𝐵) ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (𝑧 ∈ (Base‘𝑊) ∧ ((𝐴𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
1513, 14bitri 278 . . 3 (𝑧 ∈ ( ‘(𝐴𝐵)) ↔ (𝑧 ∈ (Base‘𝑊) ∧ ((𝐴𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
168, 9, 10, 11, 12elocv 20812 . . . . . 6 (𝑧 ∈ ( 𝐴) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
17 3anan12 1093 . . . . . 6 ((𝐴 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
1816, 17bitri 278 . . . . 5 (𝑧 ∈ ( 𝐴) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
198, 9, 10, 11, 12elocv 20812 . . . . . 6 (𝑧 ∈ ( 𝐵) ↔ (𝐵 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
20 3anan12 1093 . . . . . 6 ((𝐵 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
2119, 20bitri 278 . . . . 5 (𝑧 ∈ ( 𝐵) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
2218, 21anbi12i 629 . . . 4 ((𝑧 ∈ ( 𝐴) ∧ 𝑧 ∈ ( 𝐵)) ↔ ((𝑧 ∈ (Base‘𝑊) ∧ (𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))))
23 elin 3935 . . . 4 (𝑧 ∈ (( 𝐴) ∩ ( 𝐵)) ↔ (𝑧 ∈ ( 𝐴) ∧ 𝑧 ∈ ( 𝐵)))
24 anandi 675 . . . 4 ((𝑧 ∈ (Base‘𝑊) ∧ ((𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))) ↔ ((𝑧 ∈ (Base‘𝑊) ∧ (𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) ∧ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))))
2522, 23, 243bitr4i 306 . . 3 (𝑧 ∈ (( 𝐴) ∩ ( 𝐵)) ↔ (𝑧 ∈ (Base‘𝑊) ∧ ((𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))))
267, 15, 253bitr4i 306 . 2 (𝑧 ∈ ( ‘(𝐴𝐵)) ↔ 𝑧 ∈ (( 𝐴) ∩ ( 𝐵)))
2726eqriv 2821 1 ( ‘(𝐴𝐵)) = (( 𝐴) ∩ ( 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3133   ∪ cun 3917   ∩ cin 3918   ⊆ wss 3919  ‘cfv 6343  (class class class)co 7149  Basecbs 16483  Scalarcsca 16568  ·𝑖cip 16570  0gc0g 16713  ocvcocv 20804 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-ocv 20807 This theorem is referenced by:  cssincl  20832
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