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Theorem unocv 21100
Description: The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypothesis
Ref Expression
inocv.o βŠ₯ = (ocvβ€˜π‘Š)
Assertion
Ref Expression
unocv ( βŠ₯ β€˜(𝐴 βˆͺ 𝐡)) = (( βŠ₯ β€˜π΄) ∩ ( βŠ₯ β€˜π΅))

Proof of Theorem unocv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unss 4145 . . . . . . 7 ((𝐴 βŠ† (Baseβ€˜π‘Š) ∧ 𝐡 βŠ† (Baseβ€˜π‘Š)) ↔ (𝐴 βˆͺ 𝐡) βŠ† (Baseβ€˜π‘Š))
21bicomi 223 . . . . . 6 ((𝐴 βˆͺ 𝐡) βŠ† (Baseβ€˜π‘Š) ↔ (𝐴 βŠ† (Baseβ€˜π‘Š) ∧ 𝐡 βŠ† (Baseβ€˜π‘Š)))
3 ralunb 4152 . . . . . 6 (βˆ€π‘¦ ∈ (𝐴 βˆͺ 𝐡)(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ (βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
42, 3anbi12i 628 . . . . 5 (((𝐴 βˆͺ 𝐡) βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ (𝐴 βˆͺ 𝐡)(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ ((𝐴 βŠ† (Baseβ€˜π‘Š) ∧ 𝐡 βŠ† (Baseβ€˜π‘Š)) ∧ (βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
5 an4 655 . . . . 5 (((𝐴 βŠ† (Baseβ€˜π‘Š) ∧ 𝐡 βŠ† (Baseβ€˜π‘Š)) ∧ (βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))) ↔ ((𝐴 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ∧ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
64, 5bitri 275 . . . 4 (((𝐴 βˆͺ 𝐡) βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ (𝐴 βˆͺ 𝐡)(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ ((𝐴 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ∧ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
76anbi2i 624 . . 3 ((𝑧 ∈ (Baseβ€˜π‘Š) ∧ ((𝐴 βˆͺ 𝐡) βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ (𝐴 βˆͺ 𝐡)(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))) ↔ (𝑧 ∈ (Baseβ€˜π‘Š) ∧ ((𝐴 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ∧ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))))
8 eqid 2733 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
9 eqid 2733 . . . . 5 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
10 eqid 2733 . . . . 5 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
11 eqid 2733 . . . . 5 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
12 inocv.o . . . . 5 βŠ₯ = (ocvβ€˜π‘Š)
138, 9, 10, 11, 12elocv 21088 . . . 4 (𝑧 ∈ ( βŠ₯ β€˜(𝐴 βˆͺ 𝐡)) ↔ ((𝐴 βˆͺ 𝐡) βŠ† (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ (𝐴 βˆͺ 𝐡)(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
14 3anan12 1097 . . . 4 (((𝐴 βˆͺ 𝐡) βŠ† (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ (𝐴 βˆͺ 𝐡)(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ (𝑧 ∈ (Baseβ€˜π‘Š) ∧ ((𝐴 βˆͺ 𝐡) βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ (𝐴 βˆͺ 𝐡)(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
1513, 14bitri 275 . . 3 (𝑧 ∈ ( βŠ₯ β€˜(𝐴 βˆͺ 𝐡)) ↔ (𝑧 ∈ (Baseβ€˜π‘Š) ∧ ((𝐴 βˆͺ 𝐡) βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ (𝐴 βˆͺ 𝐡)(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
168, 9, 10, 11, 12elocv 21088 . . . . . 6 (𝑧 ∈ ( βŠ₯ β€˜π΄) ↔ (𝐴 βŠ† (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
17 3anan12 1097 . . . . . 6 ((𝐴 βŠ† (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ (𝑧 ∈ (Baseβ€˜π‘Š) ∧ (𝐴 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
1816, 17bitri 275 . . . . 5 (𝑧 ∈ ( βŠ₯ β€˜π΄) ↔ (𝑧 ∈ (Baseβ€˜π‘Š) ∧ (𝐴 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
198, 9, 10, 11, 12elocv 21088 . . . . . 6 (𝑧 ∈ ( βŠ₯ β€˜π΅) ↔ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
20 3anan12 1097 . . . . . 6 ((𝐡 βŠ† (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ (𝑧 ∈ (Baseβ€˜π‘Š) ∧ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
2119, 20bitri 275 . . . . 5 (𝑧 ∈ ( βŠ₯ β€˜π΅) ↔ (𝑧 ∈ (Baseβ€˜π‘Š) ∧ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
2218, 21anbi12i 628 . . . 4 ((𝑧 ∈ ( βŠ₯ β€˜π΄) ∧ 𝑧 ∈ ( βŠ₯ β€˜π΅)) ↔ ((𝑧 ∈ (Baseβ€˜π‘Š) ∧ (𝐴 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))) ∧ (𝑧 ∈ (Baseβ€˜π‘Š) ∧ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))))
23 elin 3927 . . . 4 (𝑧 ∈ (( βŠ₯ β€˜π΄) ∩ ( βŠ₯ β€˜π΅)) ↔ (𝑧 ∈ ( βŠ₯ β€˜π΄) ∧ 𝑧 ∈ ( βŠ₯ β€˜π΅)))
24 anandi 675 . . . 4 ((𝑧 ∈ (Baseβ€˜π‘Š) ∧ ((𝐴 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ∧ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))) ↔ ((𝑧 ∈ (Baseβ€˜π‘Š) ∧ (𝐴 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))) ∧ (𝑧 ∈ (Baseβ€˜π‘Š) ∧ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))))
2522, 23, 243bitr4i 303 . . 3 (𝑧 ∈ (( βŠ₯ β€˜π΄) ∩ ( βŠ₯ β€˜π΅)) ↔ (𝑧 ∈ (Baseβ€˜π‘Š) ∧ ((𝐴 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐴 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ∧ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))))
267, 15, 253bitr4i 303 . 2 (𝑧 ∈ ( βŠ₯ β€˜(𝐴 βˆͺ 𝐡)) ↔ 𝑧 ∈ (( βŠ₯ β€˜π΄) ∩ ( βŠ₯ β€˜π΅)))
2726eqriv 2730 1 ( βŠ₯ β€˜(𝐴 βˆͺ 𝐡)) = (( βŠ₯ β€˜π΄) ∩ ( βŠ₯ β€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   βˆͺ cun 3909   ∩ cin 3910   βŠ† wss 3911  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141  Β·π‘–cip 17143  0gc0g 17326  ocvcocv 21080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-ocv 21083
This theorem is referenced by:  cssincl  21108
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