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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 ( ‘(𝐴𝐵)) = (( 𝐴) ∩ ( 𝐵))

Proof of Theorem unocv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unss 4051 . . . . . . 7 ((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ↔ (𝐴𝐵) ⊆ (Base‘𝑊))
21bicomi 216 . . . . . 6 ((𝐴𝐵) ⊆ (Base‘𝑊) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)))
3 ralunb 4058 . . . . . 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 618 . . . . 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 644 . . . . 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 267 . . . 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 614 . . 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 2780 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
9 eqid 2780 . . . . 5 (·𝑖𝑊) = (·𝑖𝑊)
10 eqid 2780 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
11 eqid 2780 . . . . 5 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
12 inocv.o . . . . 5 = (ocv‘𝑊)
138, 9, 10, 11, 12elocv 20529 . . . 4 (𝑧 ∈ ( ‘(𝐴𝐵)) ↔ ((𝐴𝐵) ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
14 3anan12 1078 . . . 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 267 . . 3 (𝑧 ∈ ( ‘(𝐴𝐵)) ↔ (𝑧 ∈ (Base‘𝑊) ∧ ((𝐴𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
168, 9, 10, 11, 12elocv 20529 . . . . . 6 (𝑧 ∈ ( 𝐴) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
17 3anan12 1078 . . . . . 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 267 . . . . 5 (𝑧 ∈ ( 𝐴) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
198, 9, 10, 11, 12elocv 20529 . . . . . 6 (𝑧 ∈ ( 𝐵) ↔ (𝐵 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
20 3anan12 1078 . . . . . 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 267 . . . . 5 (𝑧 ∈ ( 𝐵) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
2218, 21anbi12i 618 . . . 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 4060 . . . 4 (𝑧 ∈ (( 𝐴) ∩ ( 𝐵)) ↔ (𝑧 ∈ ( 𝐴) ∧ 𝑧 ∈ ( 𝐵)))
24 anandi 664 . . . 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 295 . . 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 295 . 2 (𝑧 ∈ ( ‘(𝐴𝐵)) ↔ 𝑧 ∈ (( 𝐴) ∩ ( 𝐵)))
2726eqriv 2777 1 ( ‘(𝐴𝐵)) = (( 𝐴) ∩ ( 𝐵))
Colors of variables: wff setvar class
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  0gc0g 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
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