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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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