Theorem iunocv 21218

Description: The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.)

Hypotheses

Ref	Expression
inocv.o	⊢ ⊥ = (ocv‘𝑊)
iunocv.v	⊢ 𝑉 = (Base‘𝑊)

Assertion

Ref	Expression
iunocv	⊢ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) = (𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵))

Distinct variable groups:   𝑥,𝑉   𝑥,𝑊

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   ⊥ (𝑥)

Proof of Theorem iunocv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunss 5047	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉)
2		eliun 5000	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	imbi1i 350	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
4		r19.23v 3183	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
5	3, 4	bitr4i 278	. . . . . . . . 9 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
6	5	albii 1822	. . . . . . . 8 ⊢ (∀𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
7		df-ral 3063	. . . . . . . 8 ⊢ (∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ↔ ∀𝑦(𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
8		df-ral 3063	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ↔ ∀𝑦(𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
9	8	ralbii 3094	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
10		ralcom4 3284	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
11	9, 10	bitri 275	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
12	6, 7, 11	3bitr4i 303	. . . . . . 7 ⊢ (∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))
13	1, 12	anbi12i 628	. . . . . 6 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
14		r19.26 3112	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
15	13, 14	bitr4i 278	. . . . 5 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
16		eliin 5001	. . . . . 6 ⊢ (𝑧 ∈ 𝑉 → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ ( ⊥ ‘𝐵)))
17		iunocv.v	. . . . . . . . . 10 ⊢ 𝑉 = (Base‘𝑊)
18		eqid 2733	. . . . . . . . . 10 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
19		eqid 2733	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
20		eqid 2733	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
21		inocv.o	. . . . . . . . . 10 ⊢ ⊥ = (ocv‘𝑊)
22	17, 18, 19, 20, 21	elocv 21205	. . . . . . . . 9 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝐵 ⊆ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
23		3anan12 1097	. . . . . . . . 9 ⊢ ((𝐵 ⊆ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (𝑧 ∈ 𝑉 ∧ (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
24	22, 23	bitri 275	. . . . . . . 8 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝑧 ∈ 𝑉 ∧ (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
25	24	baib 537	. . . . . . 7 ⊢ (𝑧 ∈ 𝑉 → (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
26	25	ralbidv 3178	. . . . . 6 ⊢ (𝑧 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 𝑧 ∈ ( ⊥ ‘𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
27	16, 26	bitr2d 280	. . . . 5 ⊢ (𝑧 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵)))
28	15, 27	bitrid 283	. . . 4 ⊢ (𝑧 ∈ 𝑉 → ((∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵)))
29	28	pm5.32i 576	. . 3 ⊢ ((𝑧 ∈ 𝑉 ∧ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))) ↔ (𝑧 ∈ 𝑉 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵)))
30	17, 18, 19, 20, 21	elocv 21205	. . . 4 ⊢ (𝑧 ∈ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) ↔ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
31		3anan12 1097	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ (𝑧 ∈ 𝑉 ∧ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
32	30, 31	bitri 275	. . 3 ⊢ (𝑧 ∈ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑧 ∈ 𝑉 ∧ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
33		elin 3963	. . 3 ⊢ (𝑧 ∈ (𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵)) ↔ (𝑧 ∈ 𝑉 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵)))
34	29, 32, 33	3bitr4i 303	. 2 ⊢ (𝑧 ∈ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) ↔ 𝑧 ∈ (𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵)))
35	34	eqriv 2730	1 ⊢ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) = (𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088  ∀wal 1540   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ∩ cin 3946   ⊆ wss 3947  ∪ ciun 4996  ∩ ciin 4997  ‘cfv 6540  (class class class)co 7404  Basecbs 17140  Scalarcsca 17196  ·_𝑖cip 17198  0_gc0g 17381  ocvcocv 21197

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 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720

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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7407  df-ocv 21200

This theorem is referenced by: (None)