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Theorem unocv 21543
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 4177 . . . . . . 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 4184 . . . . . 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 626 . . . . 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 653 . . . . 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 622 . . 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 2724 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
9 eqid 2724 . . . . 5 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
10 eqid 2724 . . . . 5 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
11 eqid 2724 . . . . 5 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
12 inocv.o . . . . 5 βŠ₯ = (ocvβ€˜π‘Š)
138, 9, 10, 11, 12elocv 21531 . . . 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 1093 . . . 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 21531 . . . . . 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 1093 . . . . . 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 21531 . . . . . 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 1093 . . . . . 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 626 . . . 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 3957 . . . 4 (𝑧 ∈ (( βŠ₯ β€˜π΄) ∩ ( βŠ₯ β€˜π΅)) ↔ (𝑧 ∈ ( βŠ₯ β€˜π΄) ∧ 𝑧 ∈ ( βŠ₯ β€˜π΅)))
24 anandi 673 . . . 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 2721 1 ( βŠ₯ β€˜(𝐴 βˆͺ 𝐡)) = (( βŠ₯ β€˜π΄) ∩ ( βŠ₯ β€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   βˆͺ cun 3939   ∩ cin 3940   βŠ† wss 3941  β€˜cfv 6534  (class class class)co 7402  Basecbs 17145  Scalarcsca 17201  Β·π‘–cip 17203  0gc0g 17386  ocvcocv 21523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-ocv 21526
This theorem is referenced by:  cssincl  21551
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