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Metamath Proof Explorer

Theorem ofmulrt

Description: The set of roots of a product is the union of the roots of the terms. (Contributed by Mario Carneiro, 28-Jul-2014)

Ref Expression
Assertion ofmulrt ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → ( ◡ ( 𝐹 ∘f · 𝐺 ) “ { 0 } ) = ( ( ◡ 𝐹 “ { 0 } ) ∪ ( ◡ 𝐺 “ { 0 } ) ) )

Proof

Step Hyp Ref Expression
1 simp2 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → 𝐹 : 𝐴 ⟶ ℂ )
2 1 ffnd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → 𝐹 Fn 𝐴 )
3 simp3 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → 𝐺 : 𝐴 ⟶ ℂ )
4 3 ffnd ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → 𝐺 Fn 𝐴 )
5 simp1 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → 𝐴 ∈ 𝑉 )
6 inidm ⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴
7 eqidd ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
8 eqidd ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) )
9 2 4 5 5 6 7 8 ofval ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ∘f · 𝐺 ) ‘ 𝑥 ) = ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) )
10 9 eqeq1d ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝐹 ∘f · 𝐺 ) ‘ 𝑥 ) = 0 ↔ ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) = 0 ) )
11 1 ffvelcdmda ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℂ )
12 3 ffvelcdmda ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) ∈ ℂ )
13 11 12 mul0ord ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝐹 ‘ 𝑥 ) · ( 𝐺 ‘ 𝑥 ) ) = 0 ↔ ( ( 𝐹 ‘ 𝑥 ) = 0 ∨ ( 𝐺 ‘ 𝑥 ) = 0 ) ) )
14 10 13 bitrd ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝐹 ∘f · 𝐺 ) ‘ 𝑥 ) = 0 ↔ ( ( 𝐹 ‘ 𝑥 ) = 0 ∨ ( 𝐺 ‘ 𝑥 ) = 0 ) ) )
15 14 pm5.32da ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → ( ( 𝑥 ∈ 𝐴 ∧ ( ( 𝐹 ∘f · 𝐺 ) ‘ 𝑥 ) = 0 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( ( 𝐹 ‘ 𝑥 ) = 0 ∨ ( 𝐺 ‘ 𝑥 ) = 0 ) ) ) )
16 2 4 5 5 6 offn ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → ( 𝐹 ∘f · 𝐺 ) Fn 𝐴 )
17 fniniseg ⊢ ( ( 𝐹 ∘f · 𝐺 ) Fn 𝐴 → ( 𝑥 ∈ ( ◡ ( 𝐹 ∘f · 𝐺 ) “ { 0 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( ( 𝐹 ∘f · 𝐺 ) ‘ 𝑥 ) = 0 ) ) )
18 16 17 syl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → ( 𝑥 ∈ ( ◡ ( 𝐹 ∘f · 𝐺 ) “ { 0 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( ( 𝐹 ∘f · 𝐺 ) ‘ 𝑥 ) = 0 ) ) )
19 fniniseg ⊢ ( 𝐹 Fn 𝐴 → ( 𝑥 ∈ ( ◡ 𝐹 “ { 0 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑥 ) = 0 ) ) )
20 2 19 syl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → ( 𝑥 ∈ ( ◡ 𝐹 “ { 0 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑥 ) = 0 ) ) )
21 fniniseg ⊢ ( 𝐺 Fn 𝐴 → ( 𝑥 ∈ ( ◡ 𝐺 “ { 0 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑥 ) = 0 ) ) )
22 4 21 syl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → ( 𝑥 ∈ ( ◡ 𝐺 “ { 0 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑥 ) = 0 ) ) )
23 20 22 orbi12d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → ( ( 𝑥 ∈ ( ◡ 𝐹 “ { 0 } ) ∨ 𝑥 ∈ ( ◡ 𝐺 “ { 0 } ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑥 ) = 0 ) ∨ ( 𝑥 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑥 ) = 0 ) ) ) )
24 elun ⊢ ( 𝑥 ∈ ( ( ◡ 𝐹 “ { 0 } ) ∪ ( ◡ 𝐺 “ { 0 } ) ) ↔ ( 𝑥 ∈ ( ◡ 𝐹 “ { 0 } ) ∨ 𝑥 ∈ ( ◡ 𝐺 “ { 0 } ) ) )
25 andi ⊢ ( ( 𝑥 ∈ 𝐴 ∧ ( ( 𝐹 ‘ 𝑥 ) = 0 ∨ ( 𝐺 ‘ 𝑥 ) = 0 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑥 ) = 0 ) ∨ ( 𝑥 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑥 ) = 0 ) ) )
26 23 24 25 3bitr4g ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → ( 𝑥 ∈ ( ( ◡ 𝐹 “ { 0 } ) ∪ ( ◡ 𝐺 “ { 0 } ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( ( 𝐹 ‘ 𝑥 ) = 0 ∨ ( 𝐺 ‘ 𝑥 ) = 0 ) ) ) )
27 15 18 26 3bitr4d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → ( 𝑥 ∈ ( ◡ ( 𝐹 ∘f · 𝐺 ) “ { 0 } ) ↔ 𝑥 ∈ ( ( ◡ 𝐹 “ { 0 } ) ∪ ( ◡ 𝐺 “ { 0 } ) ) ) )
28 27 eqrdv ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ ℂ ∧ 𝐺 : 𝐴 ⟶ ℂ ) → ( ◡ ( 𝐹 ∘f · 𝐺 ) “ { 0 } ) = ( ( ◡ 𝐹 “ { 0 } ) ∪ ( ◡ 𝐺 “ { 0 } ) ) )