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Metamath Proof Explorer

Theorem iunocv

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

Ref Expression
Hypotheses inocv.o ⊢ ⊥ = ( ocv ‘ 𝑊 )
iunocv.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
Assertion iunocv ( ⊥ ‘ ∪ 𝑥 ∈ 𝐴 𝐵 ) = ( 𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 inocv.o ⊢ ⊥ = ( ocv ‘ 𝑊 )
2 iunocv.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
3 iunss ⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 )
4 eliun ⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 )
5 4 imbi1i ⊢ ( ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
6 r19.23v ⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
7 5 6 bitr4i ⊢ ( ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
8 7 albii ⊢ ( ∀ 𝑦 ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ↔ ∀ 𝑦 ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
9 df-ral ⊢ ( ∀ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ↔ ∀ 𝑦 ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
10 df-ral ⊢ ( ∀ 𝑦 ∈ 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
11 10 ralbii ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ∈ 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
12 ralcom4 ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ( 𝑦 ∈ 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ↔ ∀ 𝑦 ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
13 11 12 bitri ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ↔ ∀ 𝑦 ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
14 8 9 13 3bitr4i ⊢ ( ∀ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) )
15 3 14 anbi12i ⊢ ( ( ∪ 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 ) ) ) )
16 r19.26 ⊢ ( ∀ 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 ) ) ) )
17 15 16 bitr4i ⊢ ( ( ∪ 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 ) ) ) )
18 eliin ⊢ ( 𝑧 ∈ 𝑉 → ( 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐴 𝑧 ∈ ( ⊥ ‘ 𝐵 ) ) )
19 eqid ⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 )
20 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
21 eqid ⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
22 2 19 20 21 1 elocv ⊢ ( 𝑧 ∈ ( ⊥ ‘ 𝐵 ) ↔ ( 𝐵 ⊆ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ ∀ 𝑦 ∈ 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
23 3anan12 ⊢ ( ( 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 ⊢ ( 𝑧 ∈ ( ⊥ ‘ 𝐵 ) ↔ ( 𝑧 ∈ 𝑉 ∧ ( 𝐵 ⊆ 𝑉 ∧ ∀ 𝑦 ∈ 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) )
25 24 baib ⊢ ( 𝑧 ∈ 𝑉 → ( 𝑧 ∈ ( ⊥ ‘ 𝐵 ) ↔ ( 𝐵 ⊆ 𝑉 ∧ ∀ 𝑦 ∈ 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) )
26 25 ralbidv ⊢ ( 𝑧 ∈ 𝑉 → ( ∀ 𝑥 ∈ 𝐴 𝑧 ∈ ( ⊥ ‘ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐵 ⊆ 𝑉 ∧ ∀ 𝑦 ∈ 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) )
27 18 26 bitr2d ⊢ ( 𝑧 ∈ 𝑉 → ( ∀ 𝑥 ∈ 𝐴 ( 𝐵 ⊆ 𝑉 ∧ ∀ 𝑦 ∈ 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ↔ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘ 𝐵 ) ) )
28 17 27 bitrid ⊢ ( 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 ⊢ ( ( 𝑧 ∈ 𝑉 ∧ ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ ∀ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) ↔ ( 𝑧 ∈ 𝑉 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘ 𝐵 ) ) )
30 2 19 20 21 1 elocv ⊢ ( 𝑧 ∈ ( ⊥ ‘ ∪ 𝑥 ∈ 𝐴 𝐵 ) ↔ ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ ∀ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
31 3anan12 ⊢ ( ( ∪ 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 ⊢ ( 𝑧 ∈ ( ⊥ ‘ ∪ 𝑥 ∈ 𝐴 𝐵 ) ↔ ( 𝑧 ∈ 𝑉 ∧ ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝑉 ∧ ∀ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ( 𝑧 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ) )
33 elin ⊢ ( 𝑧 ∈ ( 𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘ 𝐵 ) ) ↔ ( 𝑧 ∈ 𝑉 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘ 𝐵 ) ) )
34 29 32 33 3bitr4i ⊢ ( 𝑧 ∈ ( ⊥ ‘ ∪ 𝑥 ∈ 𝐴 𝐵 ) ↔ 𝑧 ∈ ( 𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘ 𝐵 ) ) )
35 34 eqriv ⊢ ( ⊥ ‘ ∪ 𝑥 ∈ 𝐴 𝐵 ) = ( 𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘ 𝐵 ) )