Theorem iunocv 21735

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 5004	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉)
2		eliun 4955	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	imbi1i 351	. . . . . . . . . 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 3191	. . . . . . . . . 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 280	. . . . . . . . 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 1841	. . . . . . . 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 3079	. . . . . . . 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 3079	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ↔ ∀𝑦(𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
9	8	ralbii 3110	. . . . . . . . 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 3290	. . . . . . . . 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 277	. . . . . . . 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 305	. . . . . . 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 637	. . . . . 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 3124	. . . . . 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 280	. . . . 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 4956	. . . . . 6 ⊢ (𝑧 ∈ 𝑉 → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ ( ⊥ ‘𝐵)))
17		iunocv.v	. . . . . . . . . 10 ⊢ 𝑉 = (Base‘𝑊)
18		eqid 2764	. . . . . . . . . 10 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
19		eqid 2764	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
20		eqid 2764	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
21		inocv.o	. . . . . . . . . 10 ⊢ ⊥ = (ocv‘𝑊)
22	17, 18, 19, 20, 21	elocv 21722	. . . . . . . . 9 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝐵 ⊆ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
23		3anan12 1108	. . . . . . . . 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 277	. . . . . . . 8 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝑧 ∈ 𝑉 ∧ (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
25	24	baib 543	. . . . . . 7 ⊢ (𝑧 ∈ 𝑉 → (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
26	25	ralbidv 3187	. . . . . 6 ⊢ (𝑧 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 𝑧 ∈ ( ⊥ ‘𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
27	16, 26	bitr2d 282	. . . . 5 ⊢ (𝑧 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))) ↔ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵)))
28	15, 27	bitrid 285	. . . 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 582	. . 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 21722	. . . 4 ⊢ (𝑧 ∈ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) ↔ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
31		3anan12 1108	. . . 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 277	. . 3 ⊢ (𝑧 ∈ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑧 ∈ 𝑉 ∧ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
33		elin 3922	. . 3 ⊢ (𝑧 ∈ (𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵)) ↔ (𝑧 ∈ 𝑉 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵)))
34	29, 32, 33	3bitr4i 305	. 2 ⊢ (𝑧 ∈ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) ↔ 𝑧 ∈ (𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵)))
35	34	eqriv 2761	1 ⊢ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) = (𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099  ∀wal 1560   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088   ∩ cin 3905   ⊆ wss 3906  ∪ ciun 4951  ∩ ciin 4952  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  Scalarcsca 17291  ·_𝑖cip 17293  0_gc0g 17470  ocvcocv 21714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-ov 7401  df-ocv 21717

This theorem is referenced by: (None)