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Theorem unocv 21678
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 4185 . . . . . . 7 ((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ↔ (𝐴𝐵) ⊆ (Base‘𝑊))
21bicomi 223 . . . . . 6 ((𝐴𝐵) ⊆ (Base‘𝑊) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)))
3 ralunb 4192 . . . . . 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 626 . . . . 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 654 . . . . 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 274 . . . 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 621 . . 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 2726 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
9 eqid 2726 . . . . 5 (·𝑖𝑊) = (·𝑖𝑊)
10 eqid 2726 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
11 eqid 2726 . . . . 5 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
12 inocv.o . . . . 5 = (ocv‘𝑊)
138, 9, 10, 11, 12elocv 21666 . . . 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 274 . . 3 (𝑧 ∈ ( ‘(𝐴𝐵)) ↔ (𝑧 ∈ (Base‘𝑊) ∧ ((𝐴𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
168, 9, 10, 11, 12elocv 21666 . . . . . 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 274 . . . . 5 (𝑧 ∈ ( 𝐴) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
198, 9, 10, 11, 12elocv 21666 . . . . . 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 274 . . . . 5 (𝑧 ∈ ( 𝐵) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
2218, 21anbi12i 626 . . . 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 3963 . . . 4 (𝑧 ∈ (( 𝐴) ∩ ( 𝐵)) ↔ (𝑧 ∈ ( 𝐴) ∧ 𝑧 ∈ ( 𝐵)))
24 anandi 674 . . . 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 302 . . 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 302 . 2 (𝑧 ∈ ( ‘(𝐴𝐵)) ↔ 𝑧 ∈ (( 𝐴) ∩ ( 𝐵)))
2726eqriv 2723 1 ( ‘(𝐴𝐵)) = (( 𝐴) ∩ ( 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  cun 3945  cin 3946  wss 3947  cfv 6556  (class class class)co 7426  Basecbs 17215  Scalarcsca 17271  ·𝑖cip 17273  0gc0g 17456  ocvcocv 21658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-fv 6564  df-ov 7429  df-ocv 21661
This theorem is referenced by:  cssincl  21686
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