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Theorem iunocv 21101
Description: The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
inocv.o βŠ₯ = (ocvβ€˜π‘Š)
iunocv.v 𝑉 = (Baseβ€˜π‘Š)
Assertion
Ref Expression
iunocv ( βŠ₯ β€˜βˆͺ π‘₯ ∈ 𝐴 𝐡) = (𝑉 ∩ ∩ π‘₯ ∈ 𝐴 ( βŠ₯ β€˜π΅))
Distinct variable groups:   π‘₯,𝑉   π‘₯,π‘Š
Allowed substitution hints:   𝐴(π‘₯)   𝐡(π‘₯)   βŠ₯ (π‘₯)

Proof of Theorem iunocv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iunss 5006 . . . . . . 7 (βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉 ↔ βˆ€π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉)
2 eliun 4959 . . . . . . . . . . 11 (𝑦 ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡 ↔ βˆƒπ‘₯ ∈ 𝐴 𝑦 ∈ 𝐡)
32imbi1i 350 . . . . . . . . . 10 ((𝑦 ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ (βˆƒπ‘₯ ∈ 𝐴 𝑦 ∈ 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
4 r19.23v 3176 . . . . . . . . . 10 (βˆ€π‘₯ ∈ 𝐴 (𝑦 ∈ 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ (βˆƒπ‘₯ ∈ 𝐴 𝑦 ∈ 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
53, 4bitr4i 278 . . . . . . . . 9 ((𝑦 ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ βˆ€π‘₯ ∈ 𝐴 (𝑦 ∈ 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
65albii 1822 . . . . . . . 8 (βˆ€π‘¦(𝑦 ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ βˆ€π‘¦βˆ€π‘₯ ∈ 𝐴 (𝑦 ∈ 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
7 df-ral 3062 . . . . . . . 8 (βˆ€π‘¦ ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ βˆ€π‘¦(𝑦 ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
8 df-ral 3062 . . . . . . . . . 10 (βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ βˆ€π‘¦(𝑦 ∈ 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
98ralbii 3093 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦(𝑦 ∈ 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
10 ralcom4 3268 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦(𝑦 ∈ 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ βˆ€π‘¦βˆ€π‘₯ ∈ 𝐴 (𝑦 ∈ 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
119, 10bitri 275 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ βˆ€π‘¦βˆ€π‘₯ ∈ 𝐴 (𝑦 ∈ 𝐡 β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
126, 7, 113bitr4i 303 . . . . . . 7 (βˆ€π‘¦ ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
131, 12anbi12i 628 . . . . . 6 ((βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ (βˆ€π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
14 r19.26 3111 . . . . . 6 (βˆ€π‘₯ ∈ 𝐴 (𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ (βˆ€π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
1513, 14bitr4i 278 . . . . 5 ((βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ βˆ€π‘₯ ∈ 𝐴 (𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
16 eliin 4960 . . . . . 6 (𝑧 ∈ 𝑉 β†’ (𝑧 ∈ ∩ π‘₯ ∈ 𝐴 ( βŠ₯ β€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐴 𝑧 ∈ ( βŠ₯ β€˜π΅)))
17 iunocv.v . . . . . . . . . 10 𝑉 = (Baseβ€˜π‘Š)
18 eqid 2733 . . . . . . . . . 10 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
19 eqid 2733 . . . . . . . . . 10 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
20 eqid 2733 . . . . . . . . . 10 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
21 inocv.o . . . . . . . . . 10 βŠ₯ = (ocvβ€˜π‘Š)
2217, 18, 19, 20, 21elocv 21088 . . . . . . . . 9 (𝑧 ∈ ( βŠ₯ β€˜π΅) ↔ (𝐡 βŠ† 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
23 3anan12 1097 . . . . . . . . 9 ((𝐡 βŠ† 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ (𝑧 ∈ 𝑉 ∧ (𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
2422, 23bitri 275 . . . . . . . 8 (𝑧 ∈ ( βŠ₯ β€˜π΅) ↔ (𝑧 ∈ 𝑉 ∧ (𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
2524baib 537 . . . . . . 7 (𝑧 ∈ 𝑉 β†’ (𝑧 ∈ ( βŠ₯ β€˜π΅) ↔ (𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
2625ralbidv 3171 . . . . . 6 (𝑧 ∈ 𝑉 β†’ (βˆ€π‘₯ ∈ 𝐴 𝑧 ∈ ( βŠ₯ β€˜π΅) ↔ βˆ€π‘₯ ∈ 𝐴 (𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
2716, 26bitr2d 280 . . . . 5 (𝑧 ∈ 𝑉 β†’ (βˆ€π‘₯ ∈ 𝐴 (𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ 𝐡 (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ 𝑧 ∈ ∩ π‘₯ ∈ 𝐴 ( βŠ₯ β€˜π΅)))
2815, 27bitrid 283 . . . 4 (𝑧 ∈ 𝑉 β†’ ((βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ 𝑧 ∈ ∩ π‘₯ ∈ 𝐴 ( βŠ₯ β€˜π΅)))
2928pm5.32i 576 . . 3 ((𝑧 ∈ 𝑉 ∧ (βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))) ↔ (𝑧 ∈ 𝑉 ∧ 𝑧 ∈ ∩ π‘₯ ∈ 𝐴 ( βŠ₯ β€˜π΅)))
3017, 18, 19, 20, 21elocv 21088 . . . 4 (𝑧 ∈ ( βŠ₯ β€˜βˆͺ π‘₯ ∈ 𝐴 𝐡) ↔ (βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ βˆ€π‘¦ ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
31 3anan12 1097 . . . 4 ((βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ βˆ€π‘¦ ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ (𝑧 ∈ 𝑉 ∧ (βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
3230, 31bitri 275 . . 3 (𝑧 ∈ ( βŠ₯ β€˜βˆͺ π‘₯ ∈ 𝐴 𝐡) ↔ (𝑧 ∈ 𝑉 ∧ (βˆͺ π‘₯ ∈ 𝐴 𝐡 βŠ† 𝑉 ∧ βˆ€π‘¦ ∈ βˆͺ π‘₯ ∈ 𝐴 𝐡(𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))))
33 elin 3927 . . 3 (𝑧 ∈ (𝑉 ∩ ∩ π‘₯ ∈ 𝐴 ( βŠ₯ β€˜π΅)) ↔ (𝑧 ∈ 𝑉 ∧ 𝑧 ∈ ∩ π‘₯ ∈ 𝐴 ( βŠ₯ β€˜π΅)))
3429, 32, 333bitr4i 303 . 2 (𝑧 ∈ ( βŠ₯ β€˜βˆͺ π‘₯ ∈ 𝐴 𝐡) ↔ 𝑧 ∈ (𝑉 ∩ ∩ π‘₯ ∈ 𝐴 ( βŠ₯ β€˜π΅)))
3534eqriv 2730 1 ( βŠ₯ β€˜βˆͺ π‘₯ ∈ 𝐴 𝐡) = (𝑉 ∩ ∩ π‘₯ ∈ 𝐴 ( βŠ₯ β€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3910   βŠ† wss 3911  βˆͺ ciun 4955  βˆ© ciin 4956  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-iun 4957  df-iin 4958  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: (None)
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