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Theorem unocv 21605
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 4180 . . . . . . 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 4187 . . . . . 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 627 . . . . 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 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 2728 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
9 eqid 2728 . . . . 5 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
10 eqid 2728 . . . . 5 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
11 eqid 2728 . . . . 5 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
12 inocv.o . . . . 5 βŠ₯ = (ocvβ€˜π‘Š)
138, 9, 10, 11, 12elocv 21593 . . . 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 1094 . . . 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 21593 . . . . . 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 1094 . . . . . 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 21593 . . . . . 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 1094 . . . . . 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 627 . . . 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 3961 . . . 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 2725 1 ( βŠ₯ β€˜(𝐴 βˆͺ 𝐡)) = (( βŠ₯ β€˜π΄) ∩ ( βŠ₯ β€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3057   βˆͺ cun 3943   ∩ cin 3944   βŠ† wss 3945  β€˜cfv 6542  (class class class)co 7414  Basecbs 17173  Scalarcsca 17229  Β·π‘–cip 17231  0gc0g 17414  ocvcocv 21585
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 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-ocv 21588
This theorem is referenced by:  cssincl  21613
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