Theorem iunocv 21653

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 5002	. . . . . . 7 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉)
2		eliun 4952	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	imbi1i 349	. . . . . . . . . 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 3165	. . . . . . . . . 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 1821	. . . . . . . 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 3053	. . . . . . . 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 3053	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)) ↔ ∀𝑦(𝑦 ∈ 𝐵 → (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
9	8	ralbii 3084	. . . . . . . . 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 3264	. . . . . . . . 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 629	. . . . . 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 3098	. . . . . 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 4953	. . . . . 6 ⊢ (𝑧 ∈ 𝑉 → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ ( ⊥ ‘𝐵)))
17		iunocv.v	. . . . . . . . . 10 ⊢ 𝑉 = (Base‘𝑊)
18		eqid 2737	. . . . . . . . . 10 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
19		eqid 2737	. . . . . . . . . 10 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
20		eqid 2737	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
21		inocv.o	. . . . . . . . . 10 ⊢ ⊥ = (ocv‘𝑊)
22	17, 18, 19, 20, 21	elocv 21640	. . . . . . . . 9 ⊢ (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝐵 ⊆ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
23		3anan12 1096	. . . . . . . . 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 535	. . . . . . 7 ⊢ (𝑧 ∈ 𝑉 → (𝑧 ∈ ( ⊥ ‘𝐵) ↔ (𝐵 ⊆ 𝑉 ∧ ∀𝑦 ∈ 𝐵 (𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊)))))
26	25	ralbidv 3161	. . . . . 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 574	. . 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 21640	. . . 4 ⊢ (𝑧 ∈ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) ↔ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵(𝑧(·𝑖‘𝑊)𝑦) = (0g‘(Scalar‘𝑊))))
31		3anan12 1096	. . . 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 3919	. . 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 2734	1 ⊢ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) = (𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3902   ⊆ wss 3903  ∪ ciun 4948  ∩ ciin 4949  ‘cfv 6502  (class class class)co 7370  Basecbs 17150  Scalarcsca 17194  ·_𝑖cip 17196  0_gc0g 17373  ocvcocv 21632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  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 6458  df-fun 6504  df-fn 6505  df-f 6506  df-fv 6510  df-ov 7373  df-ocv 21635

This theorem is referenced by: (None)