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Metamath Proof Explorer

Theorem norm-iii

Description: Theorem 3.3(iii) of Beran p. 97. (Contributed by NM, 25-Oct-1999) (New usage is discouraged.)

Ref Expression
Assertion norm-iii ⊢ A ∈ ℂ ∧ B ∈ ℋ → norm ℎ ⁡ A ⋅ ℎ B = A ⁢ norm ℎ ⁡ B

Proof

Step Hyp Ref Expression
1 fvoveq1 ⊢ A = if A ∈ ℂ A 0 → norm ℎ ⁡ A ⋅ ℎ B = norm ℎ ⁡ if A ∈ ℂ A 0 ⋅ ℎ B
2 fveq2 ⊢ A = if A ∈ ℂ A 0 → A = if A ∈ ℂ A 0
3 2 oveq1d ⊢ A = if A ∈ ℂ A 0 → A ⁢ norm ℎ ⁡ B = if A ∈ ℂ A 0 ⁢ norm ℎ ⁡ B
4 1 3 eqeq12d ⊢ A = if A ∈ ℂ A 0 → norm ℎ ⁡ A ⋅ ℎ B = A ⁢ norm ℎ ⁡ B ↔ norm ℎ ⁡ if A ∈ ℂ A 0 ⋅ ℎ B = if A ∈ ℂ A 0 ⁢ norm ℎ ⁡ B
5 oveq2 ⊢ B = if B ∈ ℋ B 0 ℎ → if A ∈ ℂ A 0 ⋅ ℎ B = if A ∈ ℂ A 0 ⋅ ℎ if B ∈ ℋ B 0 ℎ
6 5 fveq2d ⊢ B = if B ∈ ℋ B 0 ℎ → norm ℎ ⁡ if A ∈ ℂ A 0 ⋅ ℎ B = norm ℎ ⁡ if A ∈ ℂ A 0 ⋅ ℎ if B ∈ ℋ B 0 ℎ
7 fveq2 ⊢ B = if B ∈ ℋ B 0 ℎ → norm ℎ ⁡ B = norm ℎ ⁡ if B ∈ ℋ B 0 ℎ
8 7 oveq2d ⊢ B = if B ∈ ℋ B 0 ℎ → if A ∈ ℂ A 0 ⁢ norm ℎ ⁡ B = if A ∈ ℂ A 0 ⁢ norm ℎ ⁡ if B ∈ ℋ B 0 ℎ
9 6 8 eqeq12d ⊢ B = if B ∈ ℋ B 0 ℎ → norm ℎ ⁡ if A ∈ ℂ A 0 ⋅ ℎ B = if A ∈ ℂ A 0 ⁢ norm ℎ ⁡ B ↔ norm ℎ ⁡ if A ∈ ℂ A 0 ⋅ ℎ if B ∈ ℋ B 0 ℎ = if A ∈ ℂ A 0 ⁢ norm ℎ ⁡ if B ∈ ℋ B 0 ℎ
10 0cn ⊢ 0 ∈ ℂ
11 10 elimel ⊢ if A ∈ ℂ A 0 ∈ ℂ
12 ifhvhv0 ⊢ if B ∈ ℋ B 0 ℎ ∈ ℋ
13 11 12 norm-iii-i ⊢ norm ℎ ⁡ if A ∈ ℂ A 0 ⋅ ℎ if B ∈ ℋ B 0 ℎ = if A ∈ ℂ A 0 ⁢ norm ℎ ⁡ if B ∈ ℋ B 0 ℎ
14 4 9 13 dedth2h ⊢ A ∈ ℂ ∧ B ∈ ℋ → norm ℎ ⁡ A ⋅ ℎ B = A ⁢ norm ℎ ⁡ B