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Theorem unocv 20246
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 3930 . . . . . . 7 ((𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)) ↔ (𝐴𝐵) ⊆ (Base‘𝑊))
21bicomi 214 . . . . . 6 ((𝐴𝐵) ⊆ (Base‘𝑊) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝐵 ⊆ (Base‘𝑊)))
3 ralunb 3937 . . . . . 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 735 . . . . 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 900 . . . . 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 264 . . . 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 732 . . 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 2760 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
9 eqid 2760 . . . . 5 (·𝑖𝑊) = (·𝑖𝑊)
10 eqid 2760 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
11 eqid 2760 . . . . 5 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
12 inocv.o . . . . 5 = (ocv‘𝑊)
138, 9, 10, 11, 12elocv 20234 . . . 4 (𝑧 ∈ ( ‘(𝐴𝐵)) ↔ ((𝐴𝐵) ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
14 3anan12 1082 . . . 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 264 . . 3 (𝑧 ∈ ( ‘(𝐴𝐵)) ↔ (𝑧 ∈ (Base‘𝑊) ∧ ((𝐴𝐵) ⊆ (Base‘𝑊) ∧ ∀𝑦 ∈ (𝐴𝐵)(𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
168, 9, 10, 11, 12elocv 20234 . . . . . 6 (𝑧 ∈ ( 𝐴) ↔ (𝐴 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
17 3anan12 1082 . . . . . 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 264 . . . . 5 (𝑧 ∈ ( 𝐴) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐴 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐴 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
198, 9, 10, 11, 12elocv 20234 . . . . . 6 (𝑧 ∈ ( 𝐵) ↔ (𝐵 ⊆ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
20 3anan12 1082 . . . . . 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 264 . . . . 5 (𝑧 ∈ ( 𝐵) ↔ (𝑧 ∈ (Base‘𝑊) ∧ (𝐵 ⊆ (Base‘𝑊) ∧ ∀𝑦𝐵 (𝑧(·𝑖𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
2218, 21anbi12i 735 . . . 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 3939 . . . 4 (𝑧 ∈ (( 𝐴) ∩ ( 𝐵)) ↔ (𝑧 ∈ ( 𝐴) ∧ 𝑧 ∈ ( 𝐵)))
24 anandi 906 . . . 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 292 . . 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 292 . 2 (𝑧 ∈ ( ‘(𝐴𝐵)) ↔ 𝑧 ∈ (( 𝐴) ∩ ( 𝐵)))
2726eqriv 2757 1 ( ‘(𝐴𝐵)) = (( 𝐴) ∩ ( 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  cun 3713  cin 3714  wss 3715  cfv 6049  (class class class)co 6814  Basecbs 16079  Scalarcsca 16166  ·𝑖cip 16168  0gc0g 16322  ocvcocv 20226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-ocv 20229
This theorem is referenced by:  cssincl  20254
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